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5 Tips About the Art and Craft of Vocal Production,
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Woohoo! Science! Snarky Mars crafts and hot dudes with Mohawks!
The Curiosity Rover drew me in with its Twitter feed.
Then SarcasticRover turned up. (I kind of prefer the real one’s humor).
But nothing beats NASA Mohawk dude (aka Bobak Ferdowski). There’s a tumblr page devoted to images and memes.
Not only do I watch Community, I teach at Lane Community College; love the Starburns reference.
Bobak Mountain! Yes! According to his Twitter feed, he liked it too.
| 78,175
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Q. What Are Your Hours?
A. The shop itself is open Monday through Friday from 8:00 to 5:30, we do however reply to emails and voicemails left after hours and on weekends.
Q. Can All Wheel Drive Auto Perform Factory Warranty Work?
A. No, All Wheel Drive Auto.
Q. Does All Wheel Drive Auto Work With Aftermarket Warranty Companies Or Insurance Claims?
A. Yes, we work with most major third Party Aftermarket Warranty Companies and most major Insurance companies as well.
Q. Will Subaru Void My Factory Warranty If I Have Work Performed Outside Of the Subaru Dealers?
A.. See The Magnuson Moss Act for more information. Magnuson Moss Act
Q. Does All Wheel Drive Auto Guaranty Their Work?
A. Absolutely yes, all service and repair work is guaranteed by All Wheel Drive Auto, most service and repairs on your Subaru come with a 1 year 15 month warranty, some repairs such as engine replacements or transmission replacements come with longer stated warranties, some services such as a repair made with a used part to save you some money may come with a shorter warranty term.
Q. Do You Provide Loaner Cars?
A. Yes, All Wheel Drive Auto has Subaru loaner cars for you to use, loaner cars are typically reserved for larger maintenance services & Subaru Repairs. We ask that you have proof of full coverage insurance, valid drivers license and replace the fuel you used so there’s some for the next customer.
Q. Do You Provide a Shuttle Service?
A. Yes we provide a local Shuttle service, we typically shuttle within a ten mile radius of the shop but exceptions can be made if needed.
Q. What To Do If My Subaru Has Broken Down After Hours?
A. If you have a tow company you have used in the past its best to contact them and have it Towed to our shop. If you have AAA you can call the number on your membership card. If you need a tow company we use Totem Lake Towing (425 821 9004). We have night drop envelopes and pens just to the right of our first service bay door for you or a tow truck driver to use. Please email us service@allwheeldriveauto.com and let us know the car is coming, or if you prefer call and leave us a voicemail. We will do our best to get it looked at the next day after it arrives and get you back on the road.
Q. Can I Bring My Subaru To You After Hours?
A. Yes, Please park and lock your car in a parking stall. We have night drop envelopes and pens just to the right of our first service bay door for you to fill out to the best of your ability, place the keys in the envelope and the envelope in the slot. We will contact you with any questions after checking the vehicle in, if needed.
Q. Does All Wheel Drive Auto Perform Auto Body Or Collision repair.
A. Actually no we suggest L-M Auto Body in Kirkland (425-821-8074) we have been using them for years our selves, they are good people and will take great care of you and your Subaru.
| 264,746
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The Archbishop of Dublin Diarmuid Martin is denying asking for the resignation of two axillary bishops this week.
Responding to Irish television host's suggestion that the Bishop's authority had been undermined by the pope who allegedly this week rejected the offer of resignation from auxiliary bishops Eamonn Walsh and Ray Field, Martin said “I asked for accountability, and I believe that is something . . . I haven’t always been successful in doing that,."
Martin said accountability was when people said “this is the level of my responsibility . . . I stand over that and I am not just going to say sit back and say nothing’’. Asked if the pope had second-guessed him, Dr Martin replied: “No. I know what the pope thinks.’’
In November 2009, there were calls for Field and Walsh to resign from their post in the wake of the publication of the Murphy Report.
On Christmas Eve 2009 at Midnight Mass, it was announced that Field and fellow bishop Eamonn Walsh had resigned.
In their joint resignation speech they said that: "evening informed Archbishop Diarmuid Martin that we are offering our resignation to His Holiness, Pope Benedict XVI, as Auxiliary Bishops to the Archbishop of Dublin.
"As we celebrate the Feast of Christmas, the Birth of our Saviour, the Prince of Peace,."
On 11 August, 2010 it was announced that Pope Benedict XVI did not accept Bishop Field's resignation and would return to ministry within the archdiocese along with Bishop Eamonn Walsh.
Dublin Archbishop Diarmuid Martin
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TITLE: What is the electric field due to a coherent radiation? Is it polarized?
QUESTION [1 upvotes]: What is the time-structure (i.e., how does the magnitude and direction change with time) of the expectation value of the electric field $\langle\textbf{E}\rangle(t)$ of a radiation field described by a quantum coherent state $|\alpha\rangle$?
Actually, I heard that laser can be described by coherent states and from the time-structure, I want to see whether the expected electric field is polarized or not. I know little to no quantum optics and not sure how to extract the information about the nature of the observable electric field due to a coherent state such as a laser.
REPLY [3 votes]: Your question is ill-posed.
Basically, when you say that you want to consider
a radiation field described by a quantum coherent state $|\alpha\rangle$
you are already implicitly considering the polarization of the electric field. This is because to speak of a quantum coherent state $|\alpha\rangle$ over a single mode you are already implicitly basing your treatment on a separation of the electromagnetic field observables over a basis of classical modes, i.e. on an expansion of the form
$$
{\mathbf E}(\mathbf r,t) = \sum_n\bigg[a_n(t)\mathbf f_n(\mathbf r)+a_n(t)^*\mathbf f_n^*(\mathbf r)\bigg],
$$
where $\mathbf f_n(\mathbf r)$ are vector-valued functions of position and the (complex) mode amplitudes $a_n(t)$ carry all of the dynamical information about the classical state of the field, and which you then translate via canonical quantization to an expansion of the form
$$
\hat{\mathbf E}(\mathbf r) = \sum_n\bigg[\hat{a}_n\mathbf f_n(\mathbf r)+\hat {a}_n^\dagger\mathbf f_n^*(\mathbf r)\bigg],
$$
where the classical mode amplitudes $a_n(t)$ get replaced by bosonic annihilation operators $\hat a_n$. (A previous appearance of this description on answer to a question of yours is here.)
Generically speaking, moreover, saying that the radiation is described by a coherent state $|\alpha\rangle$ implicitly implies that it is one of those modes that carries the coherent state in question, and that all the other modes are in the QED vacuum state.
Thus,
is the electric field polarized? yes, by implicit assumption.
what polarization does it have? it depends ─ whatever polarization is defined by the mode function $\mathbf f_n(\mathbf r)$.
Moreover, that polarization can be anything: it can be linearly polarized, it can be circularly polarized, and it can also be a space-dependent polarization like a radial or azimuthal polarization or more complicated combinations like, say, ellipse-point polarization fields.
On the other hand all of this can, of course, be quantified: in the Schrödinger picture we normally have the time dependence $t\mapsto |\alpha e^{-i\omega t}\rangle$, where the frequency $\omega$ comes from demanding that $\mathbf f_n(\mathbf r)$ be a divergenceless Helmholtz eigenfunction, and this will then give the time dependence
$$
\langle \hat{\mathbf E}(\mathbf r) \rangle
=
\alpha e^{-i\omega t}\mathbf f_n(\mathbf r)+\alpha^* e^{+i\omega t}\mathbf f_n^*(\mathbf r).
$$
If your mode was linearly polarized to begin with then you can set e.g. $\mathbf f_n(\mathbf r) = \mathbf e_x e^{ikx}$, or if you wanted a circular polarization you could swap the $\mathbf e_x$ for $\mathbf e_x+ i \mathbf e_y$, or you could have a space-dependent polarization. In the 'worst'-case scenario, $\mathbf f_n(\mathbf r)$ could be a space-dependent polarization field that varies on length scales much shorter than what your detector can resolve, in which case the field will look de-polarized to your detector.
However:
it is important to note that the assignment of the coherent state $|\alpha\rangle$ to a laser field is a good model in most conditions but not in all conditions. The clearest division is that this description is only valid for CW lasers, but there is also a huge cross-section of lasers that are pulsed instead of CW, and those require a multi-mode description to even begin to make sense.
This is part of a wider point, in that lasers are a varied bunch, and there are few properties that hold universally. In the particular case of polarization, most lasers do indeed produce a polarized output, but there are multiple cases that produce unpolarized or partially-polarized light, through a variety of physical mechanisms (like the space-dependent polarization mentioned above, or a time-varying output due to temperature or other fluctuations).
| 50,876
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\begin{document}
\title{On the radicals of exponential Lie groups}
\author{S.G. Dani}
\maketitle
\begin{abstract}
Let $G$ be a connected exponential Lie group and $R$ be the solvable radical
of $G$. We describe a condition on $G/R$ under which one can then conclude
that $R$ is an exponential Lie group. The condition holds in particular when
$G$ is a complex Lie group and this yilds a stronger version of a result of
Moskowitz and Sacksteder \cite{MS} on the center of a complex exponential Lie
group being connected. Along the way we prove a criterion for elements
from certain subsets of a solvable Lie group to be exponential, which would
be of independent interest.
\end{abstract}
\medskip
\noindent {\it Keywords}: Exponential Lie groups, solvable radical, connectedness of center.
\section{Introduction}
Let $G$ be a connected Lie group, $\frak G$ be the Lie algebra of $G$,
and let $\exp: \frak G \to G$ be the exponential map.
An element $g\in G$ is said to be
{\it exponential} if there exists $X\in \frak G$ such that $g=\exp X$, namely
if $g$ is contained in the image of the exponential map. The Lie group $G$ is said to be exponential if every element is exponential, or equivaently if
the exponential map is surjective. There has been considerable interest
in literature to understand which Lie groups are exponential. On the whole
satisfactory results are known when $G$ is either solvable or semisimple
(see \cite{DH} for the status until the mid-1990's and \cite{W}, \cite{W2} and \cite{W3}
for later work). A general Lie group is an almost
semidirect product of a semisimple and a solvable subgroup (namely
a semisimple Levi subgroup and the solvable radical respectively),
but not much clarity is attained so far as to which semidirect products
are exponential Lie groups. Some results in this respect may be found in
\cite{DM}, \cite{Dj}, \cite{MS} and \cite{C}.
In this context we consider the following question.
Given an exponential Lie group $G$ when can we
conclude that its radical $R$ is exponential?
We describe a condition on $G/R$, the semisimple quotient Lie group,
which enables such a conclusion (see Theorem~\ref{thm});
the condition involved holds for a large class
of semisimple Lie groups (see Theorem~\ref{ss}).
The question as above arose in the context of the following.
The main theorem (Theorem~1) in \cite{MS} states that a
connected complex Lie group $G$ is exponential if and only if its center is
connected and the adjoint group $\Ad (G)$ is exponential; as has been noted
in the review of the paper in Mathematical Reviews the proof of the theorem
given in the paper is valid only under an additional condition that $G$
is ``semi-algebraic'' (we shall not go into the details of the condition
as it does not concern us in the sequel). The ``if'' part,
and also part of the converse, is relatively easy to see, so the main content
of such a statement (when valid) is that the center of a connected complex
exponential Lie group is connected; moreover,
a complex semisimple exponential Lie group has trivial center, and hence conclusion is
equivalent to that the center of the radical is connected.
This would follow directly, if we conclude that the radical is exponential,
since the center of an exponential solvable Lie group is always connected (see
in particular, Corollary~\ref{cor}, infra).
Our results apply to this case and in particular yield
the above mentioned statement from \cite{MS} as a corollary in a special
case. Our technique
however is independent of~\cite{MS}.
We now formulate the condition involved and state the main results.
To begin, we recall that a one-parameter subgroup of a Lie group $L$ is
said to be
{\it unipotent} if it is of the form $\{\exp tX\}_{t\in \R}$, where $X$
is an element of the Lie algebra of $L$ such that the adjoint transformation
$\ad X$ is a nilpotent linear tranformation of the Lie algebra.
\begin{definition}{\rm We say that a connected Lie group $L$ satisfies
{\it condition $\frak U$} if there exists a unipotent one-parameter subgroup
$U$ of $L$ such that the centralizer of $U$ in $L$, viz. $\{x\in L\mid xu=ux
\hbox{ \rm for all } u\in U\}$, does not contain any compact subgroup of
positive dimension. }
\end{definition}
\begin{theorem}\label{thm}
Let $G$ be an exponential Lie group and $R$ be the radical of $G$.
Let $S=G/R$ and suppose that $S$ satisfies
condition $\frak U$.
Then $R$ is an exponential Lie group.
\end{theorem}
In the context of the theorem it would be of interest to know when a
semisimple Lie group satisfies condition $\frak U$.
Let $S$ be a semisimple Lie group and let $S=KAN$ be an Iwasawa
decomposition of $S$, namely $K, A$ and $N$ are closed connected
subgroups of $S$, and if $S^*$ is the adjoint group of $S$ and $\Ad :S\to S^*$
is the adjoint representatiton, then $\Ad (K)$ is a maximal compact subgroup
of $S^*$, $\Ad (A)$ is a maximal
connected subgroup whose action on the Lie algebra is diagonalisable
over $\R$, and
$N$ is a simply connected nilpotent Lie subgroup of $S$ normalised by
$A$ (see for example \cite{Hel}); we note that $K$ contains the center
of $S$ and $K$ itself is not compact when the center is infinite, as in the case of
some nonlinear semisimple Lie groups. Let $M$ be the centraliser of $A$ in
$K$. Then $M$ is
a closed (not necessarily connected) subgroup of $S$. Also $M$ normalises
$N$ and $P=MAN$ is
a minimal parabolic subgroup of $S$.
With regard to condition $\frak U$ we prove the
following.
\begin{theorem}\label{ss}
Let the notation be as above and suppose that $M$ is abelian. Then $S$ satisfies condition
$\frak U$. In particular if $S$
is a $\R$-split semisimple Lie group or a complex semisimple Lie group then
it satisfies condition $\frak U$.
\end{theorem}
\begin{remark}
{\rm It has also been possible to prove (without reference to $M$ being
abelian) that if $N$ contains an
$\R$-regular unipotent element of $S$, in
the sense of \cite{A}, which is not centralised by
any compact subgroup of $P$ of positive dimension then $S$ satisfies
condition $\frak U$. The condition holds when $M$ is abelian.
It has however not been possible to ascertain whether there are semisimple Lie groups
$S$ with $M$ nonabelian for which this holds. We shall therefore not go
into the technical details
of this possible generalisation, at this stage.
}
\end{remark}
\iffalse
Let $\frak A$ and $\frak N$ be the Lie subalgebras corresponding to $A$ and $N$
respectively. Then we have the
decomposition of $\frak N$ with respect to the adjoint action of $\ad \,\frak A$ as
$\frak N=\sum_{\varphi \in \Phi}\frak N_\varphi$, where $\Phi$ is a "system of positive
$\R$-roots" consisting of linear forms $\varphi :\frak A\to \R$, and for each
$\varphi \in \Phi$, $$ \frak N_\varphi= \{X\in \frak N\mid \ad \,\alpha (X)=\varphi (\alpha)X \hbox{ \rm
for all } \alpha \in \frak A\}.$$ Let $\Delta$ be the set of simple roots from $\Phi$. For each
$\alpha \in \Delta $, $\sum_{\varphi \in \Phi, \varphi \neq \alpha} \frak N_\varphi$ is
a Lie ideal, we shall denote by $N'_\alpha$ the corresponding simply connected Lie
subgroup of $N$. We say that an element $n\in N$ is {\it $\R$-regular} if it is not contained in
$N'_\alpha$ for any $\alpha \in \Delta$; see \cite{A}, and also \S 4 for some details.
\begin{theorem}\label{ss}
Let the notation be as above. Suppose that one of the following conditions
is satisfied.
i) there exists a $\R$-regular element in $N$ which is not centralised by
any compact subgroup of $P$ of positive dimension.
ii) $M$ is abelian.
\noindent Then $S$ satisfies condition $\frak U$. In particular if $S$
is a $\R$-split semisimple Lie group or a complex semisimple Lie group then
it satisfies condition $\frak U$.
\end{theorem}
\fi
When $S$ is the group of $\R$-elements of a semisimple algebraic group
$\underline S$ defined
over $\R$ (or the connected component of the identity in such a group) the
condition of $M$ being abelian is equivalent to $\underline S$ being a quasi-split
group, namely one admitting a Borel subgroup defined over $\R$ (see \cite{R}
and \cite{C}). Thus
Theorem~\ref{ss}
applies in this case; we note however that $S$ as in Theorem~\ref{ss} need not
be linear.
Theorems~\ref{thm} and \ref{ss} together imply the following, which yields
in particular the statement from \cite{MS} recalled above.
\begin{corollary}
Let $G$ be an exponential Lie group, $R$ be the radical of $G$ and
$S=G/R$. Suppose $S$ satisfies the condition as in Theorem~\ref{ss}.
Then $R$ is an exponential Lie group. In particular
the center of $R$ is connected.
\end{corollary}
We note that for condition $\frak U$ to hold for a semisimple Lie group
it has to be noncompact. Also, not all noncompact semisimple Lie
groups satisfy condition $\frak U$; this can be readily verified for the group
$SU(n,1)$
for instance, which is in fact one of the exponential Lie groups
(see \cite{DN},\cite{W2}). The question
whether a Lie group $G$ being exponential implies that its radical is
exponential remains open in the cases when $G/R$ is one of these
semisimple Lie groups.
Towards proving Theorem~\ref{thm} we also prove a result on
exponential elements in solvable
Lie groups, which may be of independent interest.
The question of exponentiality of solvable Lie groups was considered earlier
in \cite{W} (and earlier papers cited there), but the focus there has been on
criteria for the group to be
exponential, namely {\it all} elements being exponential. On the
other hand our result
below, Theorem~\ref{solvable}, is about exponentiality of elements from
certain subsets. Moreover,
the argument in \cite{W} involves the theory of Cartan subgroups, which is
technically more intricate; our argument is based on more elementary
considerations.
The paper is organised as follows. The result on exponential elements
in solvable Lie groups,
Theorem~\ref{solvable}, will be taken
up in \S 2, and in \S 3 the results are applied to prove Theorem~\ref{thm}
and some related results. In \S 4 we discuss when condition $\frak U$ is
satisfied, and prove Theorem~\ref{ss}.
\section{Exponential elements in solvable Lie groups}
Let $H$ be a connected solvable Lie group and $N$ be a nilpotent simply
connected
closed normal Lie subgroup of $H$ such that $H/N$ is abelian. We denote
by $\frak N (H)$
the class of closed connected normal subgroups of $H$ contained in $N$.
We shall be considering pairs of the form
$(M,M')$, with $M,M' \in \frak N (H)$, and $M' \subset M$, and
for brevity we shall refer to such a pair as a {\it subquotient} (of $N$, with respect
to the action of $H$).
A subquetient $(M,M')$ is said to be abelian if $M/M'$ is abelian;
as
$N$ is simply connected, in this case $M/M'$ is a vector space over $\R$.
The conjugation action of $H$ on itself induces an action on each $M\in \frak N(H)$, by restriction,
and hence on all subquotients of $H$.
An abelian subquotient $(M,M')$, $M, M'\in \frak N(H)$, will be called an {\it irreducible
subquotient} if the $H$-action on $M/M'$ is irreducible. We
note that for any irreducible subquotient $(M,M')$, $M,M'\in \frak N(H)$, the action of $N$
on $M/M'$ is trivial, and hence the action factors to an irreducible linear action of
$A$ on $M/M'$.
\begin{definition}{\rm Let the notation be as above. Let $a\in A=H/N$ and
$B$ be a one-parameter subgroup of $A$ containing $a$.
We say that the pair $(a,B)$ is of {\it type~$\cal E$} if the following
holds: if for an irreducible
subquotient $(M,M')$, $M,M'\in \frak N(H)$,
the action of $a$ on $M/M'$ is trivial, then the action of $B$ on $M/M'$
is trivial; the element $a$ is said to be type $\cal E$ if there exists
a one-parameter
subgroup $B$ of $A$ such that the pair $(a,B)$ is of type $\cal E$.
}
\end{definition}
We prove the following.
\begin{theorem}\label{solvable}
Let $H$ be a connected solvable Lie group and $N$ be a
nilpotent simply connected
closed normal Lie subgroup of $H$ such that $H/N$ is abelian.
Let $A=H/N$. Let $x\in H$ and $a=xN\in A$. Then
the following conditions are equivalent:
i) $a$ is of type $\cal E$.
ii) for every $n\in N$, the element $xn$ is exponential in $H$.
\end{theorem}
We first consider the following special case, in which we prove some more precise
results. By a {\it vector subgroup} we mean a subgroup which is topologically
isomorphic to $\R^d$ for some $d\geq 1$ (with respect to the induced
topology).
A vector subgroup
will be considered equipped with its canonical structure as a vector space over $\R$.
\begin{proposition}\label{prep}
Let $L$ be a connected Lie group. Let $V$ be a vector subgroup of $L$
and $P$ be a one-parameter subgroup normalizing $V$,
and such that the conjugation action of $P$ on $V$ is
irreducible and nontrivial. Let $H=PV$. Let
$p \in P$ be nontrivial and let $\sigma (p):V\to V$ denote the
conjugation action of $p$ on $V$.
Then the following holds:
i) Every one-parameter subgroup of $L$ contained in $H$ is either
contained in $V$ or is of the form
$ wP w^{-1}$ for some $w\in V$.
ii) If $\sigma (p)$ is nontrivial then for any $v\in V$, $pv$
is contained in a unique one-parameter
subgroup contained in $H$.
iii) If $\sigma (p)$ is trivial, then for $v\in V$,
$v\neq 0$ (the zero element in $V$), $p v$ is not contained
in any one-parameter subgroup of $H$.
\end{proposition}
\proof At the outset we note that the subgroup $H$ as in the hypothesis
need be a closed subgroup of $L$. It is however a Lie subgroup whose
Lie algebra is the sum of the Lie subalgebras of $P$ (which is
one-dimensional) and $V$. Let $\frak H$ be the Lie algebra of $H$;
we shall realize it as $\R \xi \oplus V$, where $\xi$ is a generator
of the Lie algebra of $P$ and the vector space $V$ is identified with
its Lie algebra. We now prove the statements as in the Proposition.
i) For $w\in V$ the Lie subalgebra of $\frak H$
corresponding to the one-parameter subgroup $wP w^{-1}$ is spanned by
$\xi +(\theta (w)-w)$, where $\theta :V\to V$ is given by $\theta (u)=\frac
{dp_{-t}(u)}{dt}$ for all $u\in V$. We see that as the $P$-action on $V $
is nontrivial and irreducible the map $w\mapsto \theta (w)-w$
is surjective, and hence all elements of the form $\xi + v$, $v\in V$, are
among the tangents to $wP w^{-1}$, $w\in V$. Since scalar multiples of
these cover all elements that are not contained in $V$, it follows that
every one-parameter subgroup which is not contained in $V$ is of the form
$wP w^{-1}$ (upto scaling of the parameter). This proves (i).
ii) Let $p\in P$ be such that $\sigma (p)$ is nontrivial. As the $P$-action
on $V$ is irreducible and nontrivial this
implies in particular that $1$ is not an eigenvalue of $\sigma (p)$. Hence
$\sigma (p)^{-1} -I$, where $I$ is the identity transformation,
is invertible. Now let $v\in V$ be given. Then
there exists $w\in V$ such that $v=\sigma (p) ^{-1}(w)-w$.
Hence, in $H$, $v =(p^{-1} wp)w^{-1}$. Thus $p v
=wp w^{-1}$, and it is contained in the one-parameter subgroup $wP
w^{-1}$.
We note that if $pv$ is contained
in $wP w^{-1}$ for some $w\in V$ then in fact $p v=wp w^{-1}$; if
$q\in P$ is such that $pv=wqw^{-1}=q(q^{-1}wqw^{-1})\in qV$, then $p^{-1}q\in V$
and hence $pv=wqw^{-1}=(wpw^{-1})(wp^{-1}qw^{-1})= wpw^{-1}$.
Now if $p v$ is contained in
$w_1P w_1^{-1}$ and $w_2P w_2^{-1}$, $w_1,w_2 \in V$,
we have $w_1p w_1^{-1}=w_2p w_2^{-1}$, which means that $w_2^{-1}w_1$
(or $w_1-w_2$ is addivive notation) is fixed by $\sigma (p)$.
Since $\sigma (p)$ is nontrivial and the $P$-action is irreducible
this implies that $w_1=w_2$. Thus
one-parameter subgroup containing $pv$ is
unique. This proves~(ii).
iii) Now let $p\in P$ be such that $\sigma (p)$ is trivial.
Let $v\in V$, $v\neq 0$ and
suppose that $pv$ is contained in $wP w^{-1}$ for
some $w\in V$. Then, as before, we have $p v=wp w^{-1}$.
Hence $v=p^{-1}wp w^{-1}=
\sigma (p)^{-1}(w)-w=0$, contradicting the assumption that $v\neq 0$. Hence $p v$
is not contained in any
one-parameter subgroup of $H$. This proves~(iii). \qed
In the sequel it will be convenient to use the following terminology.
\begin{definition}
{\rm Given a Lie group $L$, a
closed normal subgroup $M$ of $L$ with $\eta :L\to L/M$ the canonical
quotient map, and
a one-parameter subgroups $\Psi$ of $L/M$, by a {\it lift} of $\Psi$
in $L$ we mean
a one-parameter subgroup $\Phi$ of $L$ such that $\eta (\Phi) = \Psi$.
}
\end{definition}
\begin{proposition}\label{countable}
Let the notations $H,N, A, x$ and $a$ be as in Theorem~\ref{solvable}.
Let $B$ be a one-parameter subgroup of $A$ containing
$a $, such that $(a,B)$ is of type~$\cal E$. Then for any $n\in N$ there
exists a lift of $B$ containing $xn$. Moreover, the collection of
such lifts is countable.
\end{proposition}
\proof
We shall show that if $(M,M')$, where $M,M'\in \frak N(H)$, is an
irreducible subquotient then the following holds:
given a one-parameter subgroup $\Psi$ of $H/M$ which is a lift of $B$
and $y\in H$ such that $yN=a$,
there exist at least one and at most countably many lifts $\Phi$
of $\Psi$ in $H/M'$ containing $yM'$.
Applying this to successive pairs from a sequence $N_0, N_1, \dots , N_k$
in $\frak N(H)$ such that $N_0=N$,
$N_k$ is trivial, and for all $j=0,\dots , k-1$, $N_{j+1} \subset N_j$ and
$(N_j,N_{j+1})$ is an irreducible subquotient, yields
the statement as in the proposition.
Now let $(M,M')$ as above, a one-parameter subgroup $\Psi$ of $H/M$ which is a lift of $B$, and
$y\in H$ such that $yM=a$ be given.
Let $P$ be a lift of $\Psi$ in $H/M'$. We now apply
Proposition~\ref{prep} to $P(M/M')$ (the product of $P$ and $M/M'$ in $H/M'$),
with $v=yM'$. We note that any one-parameter
subgroup of $H/M'$ which is a lift of $\Psi$ is contained in $P(M/M')$,
thus it suffices
to show that there exists a lift $\Phi$ of $\Psi$ in $P(M/M')$
containing $yM'$. If the action of
$a$ on $M/M'$ is nontrivial then this is assured by Proposition~\ref{prep}(ii).
On the other hand if the action of $a$ on $M/M'$ is trivial
then by hypothesis the action of $B$ is also trivial, and since $P$ is a lift
of $B$ this implies that $P(M/M')$ is an abelian Lie group, in
this case the assertion is obvious. This proves the Proposition. \qed
\begin{proposition}\label{one-step}
Let $H,N, A, x$ and $a=xN$ be as in Theorem~\ref{solvable} and let $B$ be a
one-parameter subgroup of $A$ containing $a$. Let $M\in \frak N(H)$ be
an abelian subgroup such that the $A$-action on $M$ is irreducible and
the following holds:
i) for the group $H/M$, with $A$ being viewed canonically as $(H/M)/(N/M)$,
the pair $(a,B)$ is of type $\cal E$.
ii) on $M$ the action of $a$ is trivial but the action of $B$ is not trivial.
\noindent Let $Q$ be a lift of $B$ in $H/M$ and $y\in H$ be such that $yN=a$.
Then there exist a unique $m\in M$
such that $ym$ is contained in a lift of $Q$.
\end{proposition}
\proof Let
$P$ be a lift of $Q$ in $H$. Let $p\in P$ be such that $pM=yM$, and
$m\in M$ such that $p=ym$. Then $pN=yN=a$.
By condition~(ii) in the hypothesis and Proposition~\ref{prep}(iii),
applied to the subgroup $PM$,
we get that $pm'$ is not contained in a lift of $Q$ for any nontrivial
element $m'$ of $M$.
Thus $m$ is the only element of $M$ such that $ym$ is contained
in a lift of $Q$. \qed
\begin{proposition}\label{notE}
Let $H,N, A, x$ and $a=xN$ be as in Theorem~\ref{solvable} and let $B$ be a
one-parameter subgroup of $A$ containing $a$. Suppose that $(a,B)$ is not of
type $\cal E$. Let $E=\{n\in N\mid xn \hbox{ \rm is contained in a lift of }
B \hbox{ \rm in } H\}$. Then
$E$ has $0$ Haar measure in $N$.
\end{proposition}
\proof At the outset we note that the set of exponential elements in a Lie group
is a Borel subset, and hence it follows that $E$ as in the hypothesis is a Borel subset of $N$.
As $(a,B)$ is not of
type $\cal E$ there exists an irreducible subquotient $(M,M')$,
$M,M'\in \frak N (H)$, such that the $a$-action on $M/M'$ is trivial and the
$B$ action is nontrivial, and we may without loss of generality assume $M$ to be
of maximal possible dimension among such pairs. To prove the propositin it
suffices to show that $EM'/M'$ has zero Haar measure in $N/M'$, and hence
passing to $H/M'$, we may without loss of generality assume $M'$ to be trivial. Thus $M$
is a normal vector subgroup of $H$. By the maximality
condition on $M$, $(a,B)$ is of type $\cal E$ for $H/M$. Hence by
Proposition~\ref{countable} for any $n\in N$ there exist only countably
many lifts of $B$ in $H/M$ containing $xnM$. Let $n\in N$ and $Q$ be
any lift of $B$ in $H/M$ containing $xnM$. Let $P$ be a lift of $Q$ in $H$
and $m\in M$ be such that $xnm\in P$.
We now apply
Proposition~\ref{prep}(iii), with $M$ in place of $V$ there. The choice
of $M$ as above shows that the conjugation action of $xnm$ on $M$ is trivial but the
action of $P$ is not trivial. By Proposition~\ref{prep}(iii) therefore $xnm$
is the only element in $xnM$ which is contained in a lift of $Q$. Since
there are only countably many lifts $Q$ of $B$ containing $xnM$, this
shows that there are only contably many $m$ in $M$ such that $xnm$
is contained in a lift of $B$ in $H$. In other words, $E$ as in the
hypothesis intersects each coset $xnM$, where $n\in N$, in only countably many points. It follows
that its Haar measure of $E$ must be $0$. \hfill
\medskip
\noindent{\it Proof of Theorem~\ref{solvable}}:
Statement (i) $\implies$ (ii) follows immediately from (the existence
statement in)
Proposition~\ref{countable}. We now prove the converse. We suppose
that condition~(ii) holds, but $a$ is not of type~$\cal E$, and arrive
at a contradiction.
Let $\cal C$ be the class of $L\in \frak N (H)$, such that $xN/L$
is not of type $\cal E$ for the group $H/L$. Then $\cal C$ is nonempty
since by assumption $a$ is not of type~$\cal E$ for $H$, so
$\cal C$ contains the trivial subgroup. Let $L$ be an
element of $\cal C$ with maximum possible dimension; we note that $L$
is a proper subgroup of $N$, since the action of $A$ on $N/L$
has to be nontrivial.
For notational convenience we shall consider $H/L$ as $H$, and respectively
$N/L$ as $N$, and $xL$ as $x$. Then in the modified notation we have that
$xn$ is exponential in $H$ for all $n\in N$, $xN$ is not of type $\cal E$,
but $xN/M$ is of type $\cal E$ for every $M\in \frak N(H)$ of positive
dimension.
We fix a $M\in \frak N(H)$ of positive dimension such that the action
of $A=H/N$ on $M$ is irreducible. Let $a=xN$. Then the action of $a$ on
$M$ is trivial since otherwise $a$ would be of type~$\cal E$ under the
conditions as above.
By hypothesis, for each $n\in
N$ there exists a one-parameter subgroup containing $xn$.
We note that the set
of one-parameter subgroups of $A$ containing $a=xN$ is countable, say
$\{B_j\}$ with $j$ running over a countable set; it can be a singleton
set as would be the case when $A$ is simply connected.
By Proposition~\ref{notE}, applied to $H/M$ there exists $n\in N$ such
that $xnM$ is not
contained in a lift of $B_j$ for any $j$ such that $(a,B_j)$ is not of
type $\cal E$ for $H/M$; indeed the set of such $n$ is a set of full
Haar measure. We fix such $n$ and consider the elements $xnm$,
$m\in M$. By hypothesis each of them is exponential and hence is contained
in a lift of $B_j$ for some $j$, and by the choice of $n$, the
$j$ must be such that $(a,B_j)$ is of type $\cal E$ for $N/M$. Hence to prove
the theorem it suffices
to show that the action of one of these on $M$ is trivial. Consider
any $B_j$ such that $(a,B_j)$ is of type $\cal E$ for $H/M$. Then
by Proposition~\ref{countable}
for any $j$ there exist only countably many lifts
of $B_j$ to $H/M$ containing $xnM$. Let $Q$ be any lift
of $B_j$ containing $xnM$. If the action of $B_j$
on $M$ is nontrivial then by Proposition~\ref{one-step} there exists
a unique $m\in M$ such that $xnm$ is contained in a lift of $Q$ in $H$.
It follows therefore that if the action of $B_j$ on $M$ is
nontrivial then the set of $m\in M$ for which $xnm$ is contained
in a lift of $B_j$ is countable. But this is a contradiction
since in fact since every $xnm$, $m\in M$, is contained in a
one-parameter subgroup of $H$, which necessarily has to be a lift of
some $B_j$ such that $(a,B_j)$ is of type $\cal E$ on $H/M$. The contradiction shows that $a$ must indeed by
of type~$\cal E$, which proves the theorem. \qed
\begin{remark}
{\rm The criterion in Theorem~\ref{solvable} involves that for
$a\in A$ there exists a {\it common} one-parameter subgroup $B$ such that for
every irreducible subquotient $(M,M')$, $M,M'\in \frak N (H)$,
such that if the action of $a$ on $M/M'$ is trivial the action of $B$
is also trivial. It does not suffice to have such a one-parameter for
each $(M,M')$ individually, and dependent on it. This is illustrated by the following example:
Let $H$ be the semi-direct product of $\T^2=\{(\rho_1,\rho_2)\mid \rho_1,\rho_2 \in \C, |\rho_1|
=|\rho_2|=1\}$ and $\C^2=\{(z_1,z_2)\mid z_1,z_2\in \C\}$, with the
conjugation action of $(\rho_1,\rho_2)$ given by $(z_1,z_2)\mapsto
(\rho_1\rho_2z_1, \rho_1\rho_2^{-1}z_2)$. Then $N$ and $A$ as in the earlier
notation may be seen to correspond to $\C^2$ and $\T^2$ respectively,
(together with the actions involved). The irreducible subquotient
actions of $A$ on $N$ are seen to correspond to actions on the two copies of
$\C$, corresponding to the two coordinates.
Let $a=(-1,-1) \in \T^2$. Then the action of $a$ on $\C^2$ is trivial.
We see that there are one-parameter subgroups $B_1$ and $B_2$
of $\T^2$ containing $a$ acting trivially on the subquotients corresponding to
the first
and second coordinates respectively, but there is no one-parameter subgroup
containing $a$ and acting trivially on both coordinates. Thus the condition
of Theorem~\ref{solvable} is not satisfied. Hence by the Theorem the
group is not exponential, as may also be checked directly.
}
\end{remark}
Theorem~\ref{solvable} can be used to deduce the following property,
known earlier (\cite{W}, Corollary~3.18).
\begin{corollary}\label{cor}
Let $H$ be an exponential solvable Lie group. Then the center of
$H$ is connected.
\end{corollary}
\proof Let $N$ be the nilradical of $H$ and $A=H/N$. Let
$z$ be an element contained in the center of $H$ and $a=zN\in A$. Since
$H$ is exponential $xn$ is exponential for all $n\in N$. Also, $z$ being contained
in the center, the action of $a$
on any irreducible subquotient is trivial. Hence by Theorem~\ref{solvable}
there exists a one-parameter subgroup $B$ of $A$ containing $a$ such that
the action of $B$ on any irreducible subquotient is trivial. Let $L$ be the subgroup of
$H$ containing $N$ and such that $B=L/N$. Then the preceding
observation implies that
$L$ is a connected nilpotent Lie group. Since $N$ is the nilradical we get that $L=N$.
Therefore $z$ is contained in the center of $ N$, say $Z$. Now $Z$ is a connected abelian
Lie group, and the center of $H$ is precisely the set of fixed points of the action of $A$ on $Z$,
induced by the $A$-action on $N$. It can be seen that as $A$ is connected, the set of
fixed points is a connected subgroup of $Z$. Hence the center of $H$ is connected. \qed
We also deduce the following characterization of exponential Lie group,
which is in a sense the main nontechnical part in the characterization of exponentiality
of solvable Lie groups in
Theorem~3.17 of \cite{W}. We follow the terminology as in \cite{W};
however the symbols chosen are different, to avoid clash with the
notation in the rest of this paper.
\begin{corollary}
Let $H$ be a connected solvable Lie group, $\frak H$ be the Lie algebra
of $H$, and $C$ be a Cartan subgroup of $H$.
Then $H$ is an exponential Lie group if and only if for all nilpotent
elements $\nu \in \frak H$ the centralizer
of $\nu$ in $C$, namely $Z_C(\nu):=\{y\in C\mid \Ad (y) (\nu)=\nu\}$, is
connected.
\end{corollary}
\proof Let $N$ be the nilradical of $H$ and $\frak N$ be the Lie subalgebra
corresponding to $N$. Then we have $H=CN$. As $\frak H$ is solvable, all
nilpotent elements in $\frak H$
are contained in $\frak N$. Also, $\frak N$ has a decomposition
as $\oplus_{s\in S}\frak N_s$, with $S$ an indexing set, such that
the following holds: for each $s\in S$,
$\frak N_s$ is
$\Ad C$-invariant, for each $y\in C$ the eigenvalues
of the restriction of $\Ad y$ to
$\frak N_s$ consist of a complex conjugate pair, say $\lambda (y,s)$ and
$\overline {\lambda (y,s)}$ (only one when real), and there exists an order
on $S$,
denoted by $\geq$, such that
for any $t\in S$, $\sum_{s\geq t}\frak N_s$
is a Lie ideal in $\frak H$; more precise relations can be written down for
commutators of the $\frak N_s$'s but we do not need that here. We note
also that $C$ is a connected nilpotent Lie group and hence given $y\in C$
and a one-parameter subgroup $B'$ of $CN/N$ containing $yN$ there exists a
one-parameter $B$ of $C$ containing $y$ such that $B'=BN/N$.
Now suppose that $H$ is exponential and let a nilpotent $\nu$ element be
given. Then we have $\nu =\sum_{s\in S}\nu_s$, with $\nu_s\in \frak N_s$ for
all $s\in S$. Let $y\in C$
and $a=yN/N$. By Theorem~\ref{solvable}, together with one of the
observations
above, there exists a one-parameter
subgroup $B$ of $C$ containing $y$ such that for any irreducible
subquotient $(M,M')$ such that
the action of $y$ on $M/M'$ is trivial, the action of $B$ on $M/M'$ is also
trivial.
Now, $\Ad y (\nu)=\nu$ if and only if $\lambda (y,s)=1$ for all $s$
such that $\nu_s\neq 0$. Consider any $t\in S$ such that $\lambda (y,t)=1$.
Let $M$ be the closed subgroup with Lie algebra $\sum_{s\geq t}\frak N_s$
and $M'$ be a closed connected normal subgroup of $H$, properly contained
in $M$, such that the Lie algebra of $M'$ contains $\frak N_s$, for all $s> t$ (that
is $s\geq t$ and $s\neq t$), and $(M,M')$ is an irreducible quotient.
As $\lambda (y,t)=1$ it follows that
the action of $y$ on $M/M'$ is trivial. Hence the action of $B$ on $M/M'$
is also trivial and in turn $\lambda (b , t)=1$ for all $b\in B$.
Thus we get that $\Ad b (\nu)=\nu$ for all $b\in B$. This shows that
the centralizer of $\nu$ in $C$ is connected.
Now suppose that $Z_C(\nu)$ is connected for all
$\nu \in \frak N$. Let $x\in H$ be given and let $a=xN\in H/N$. Since
$H=CN$ there there exists $y\in C$ such that $a=yN/N$. Let $S'=\{s\in S\mid
\lambda (y,s)=1\}$. There exists an element $\nu=\sum_{s\in S'}\nu_s$, such
that each $\nu_s$, $s\in S'$, is a nonzero element of $\frak N_s$ such
that $\Ad y(\nu_s)=\nu_s$. In particular $\nu$ is fixed by $\Ad y$, and
since $Z_C(\nu)$ is connected and nilpotent we get that there exists a one-parameer
subgroup $B$ of $C$ containing $y$ such that $\Ad b (\nu)=\nu$ for all $b\in B$.
Hence $\Ad b (\nu_s)=\nu_s$, and in turn
$\lambda (b,s)=1$, for all $s\in S'$ and $b\in B$.
Now let $B'=BN/N$. Then $B'$ is a one-parameter subgroup of $H/N$
containing $a$; we shall
show that $(a,B')$ is of type $\cal E$.
Let $(M,M')$ be
an irreducible subquotient such that the action of $a$ on $M/M'$ is trivial.
Then there exists $t\in S'$ such that $M$ is contained in
$\sum_{s\geq t}\frak N_s$ and $M'$ contains $\sum_{s> t}\frak N_s$. Since
$\lambda (b,s)=1$ for all $s\in S'$ and $b\in B$, it follows that
the action of $B'=BN/N$ on $M/M'$ is trivial. Thus $(a,B')$ is of type
$\cal E$ and hence by Theorem~\ref{solvable}
$x$ is exponential in $H$. Therefore $H$ is an exponential Lie group. \qed
\section{Proof of Theorem~\ref{thm}}
We shall now deduce Theorem~\ref{thm}; we follow the notations $G$, $R$, and $S$ as in the statement
of the theorem in \S\,1. Since $S$ satisfies condition $\frak U$ there
exists a unipotent one-parameter subgroup $U$ of $S$ such that the centralizer
of $U$ in $S$ contains no compact subgroup of positive dimension. We note the
following.
\begin{lemma}\label{lem}
Let $S$ and $U$ be as above. Let $u$ be a nontrivial element of $U$. Then $U$ is
the only one-parameter subgroup of $S$ containing $u$.
\end{lemma}
\proof We may assume $U=\{u_t\}$ and $u=u_1$. It suffices to show that if $\{x_t\}$ is any one-parameter
subgroup of $S$ such that $x_1=u$ then $x_t=u_t$ for all $t$. Let $s\in \R$ be arbitrary.
Since $x_s$ commutes
with $x_1=u$ and $\{u_t\}$ is a unipotent one-parameter subgroup it follows that $x_s$
commutes with $u_t$ for all $t\in \R$. Since this holds for all $s\in \R$ it follows that
$\{x_{-t}u_t\} $ is a one-parameter subgroup. Moreover, since $x_{-1}u_1=u^{-1}u =e$,
the identity element, $\{x_{-t}u_t\}$ is a compact subgroup. Since it is contained in
the centraliser of $U$, the condition on $U$ implies that the subgroup is trivial, namely
$x_t=u_t$ for all $t$. This proves the Lemma. \qed
Now let
$\eta: G\to S$
be the canonical quotient map of $G$ onto $S$, and let $H=\eta^{-1}(U)$.
We note that $U$ is a closed connected subgroup of $S$ and hence $H$
is a connected Lie group.
Since $N$ is the nilradical of $G$, $G/N$ is reductive,
and in particular it follows that $H/N$ is abelian; in particular $H$ is a solvable Lie
group.
We now first prove the following.
\begin{proposition}\label{H-exp}
Any $h\in H$ which is not contained in $R$ is exponential in $H$.
\end{proposition}
\proof Let $h\in H\backslash R$ be given. Since $G$ is
exponential there exists a one-parameter subgroup $P$ of $G$
containing $h$.
Consider the one-parameter subgroup $\eta (P)$.
We note that $\eta (h)$ is a nontrivial element of $U$. Hence by Lemma~\ref{lem} we
have $\eta (P)=U$. Thus $P$ is contained in $\eta^{-1}(U)=H$. Therefore
$h$ is exponential in~$H$. \qed
\medskip
\noindent{\it Proof of Theorem~\ref{thm}}:
We first note that in proving the theorem $N$ may be assumed to be simply connected:
Let $C$ be maximal compact subgroup of $N$. Then $C$ is a connected subgroup
contained in the centre
of $G$. We note that the hypothesis of the theorem holds
for $G/C$, and upholding that the desired conclusion for $G/C$ (namely showing
that $R/C$ is exponential) implies that $R$ is exponential, as desired. We
may therefore assume without loss of generality that $C$ is trivial.
Equivalently this means that $N$ is simply connected.
Now let $x\in R$ be given. Let $H$ be
the solvable group introduced above and $y=ux \in H$, where $u$ is a nontrivial element of $U$.
We note that since $H/R$
is topologically isomorphic to $\R$, to prove that $x$ is exponential in $R$ it suffices
to prove that it is exponential in $H$. Now
let $A=H/N$ and $a, a' $
be the elements $a=xN$ and $a'=yN$.
\iffalse
A simple argument comparing the adjoint actions
of $R$ and $H$ on the Lie algebra of $N$ and using that the action of $U$ is unipotent
shows that for any irreducible subquotient $(M,M')$, $M,M' \in \frak N(R)$, $M'\subset M$,
(for the action of $A'$) there exists an irreducible subquotient $(\bar M, \bar M') \in \frak N (H)$,
$\bar M,\bar M' \in \frak N(H)$, $\bar M'\subset \bar M$, (for the action of $A$) such that
for any $\alpha \in A'$ the action of $\alpha $ on $M/M'$ is trivial if and only if $\bar M/\bar M'$ is
trivial.
\fi
By Proposition~\ref{H-exp}
$yn$ is exponential in $H$ for all $n\in N$. Hence by Theorem~\ref{solvable},
there exists a one-parameter subgroup $B'$ of $A$ containing $a'$ such that
for any irreducible subquotient
$(M,M')$, $M, M'\in \frak N(H)$, for which the action of $a'$ on $M/M'$ is trivial,
the action of $B'$ on $M/M'$ is also trivial. Let $B'=\{b'_t\}$ and $U=\{u_t\}$, with $b'_1=a'$ and
$u_1=u$. Let $B=\{b_t\}$ be the one-parameter subgroup of $A$ defined by $b_t=(u_{-t}N)b'_t$
for all $t\in R$; in particular $b_1=u^{-1}a'=a$.
We note that as $U$ is a unipotent one-parameter subgroup of $S$, the action of $U$
on any irreducible subquotient $M/M'$ as above is trivial. Hence on any
irreducible subquotient the actions of $b_t$ and $b'_t$ coincide for each $t\in \R$.
Therefore we get that for any irreducible subquotient
$(M,M')$, $M, M'\in \frak N(H)$, for which the action of $a$ on $M/M'$ is trivial,
the action of $B$ on $M/M'$ is trivial. Thus condition~(i)
of Theorem~\ref{solvable} holds for $x$, and the theorem implies that $xn$ is
exponential for all $n\in N$; in particular $x$ is exponential in $H$, and hence, as seen above,
also in $R$. This proves the theorem. \qed
\begin{corollary}
Let $G$ be a Lie group as in Theorem~\ref{thm} and $R$ be its radical. Let $Z(G)$
and $Z(R)$ be the centers of $G$ and $R$ respectively. Then $Z(R)$ and $Z(G)\cap R$
are connected.
\end{corollary}
\proof By Corollary~\ref{cor} $Z (R)$ is connected. Clearly $Z(G)\cap R$ is the set of fixed
points of the canonical action of $G/R$ on $Z(R)$, induced by the conjugation action of $G$. As $Z(R)$ is abelian and $G/R$ is connected it follows that the set of fixed points is a connected subgroup. \qed
\section{Groups satisfying condition $\frak U$}
In this section we consider semisimple Lie groups $S$ satisfying condition $\frak U$ and
prove Theorem~\ref{ss}. We
shall follow the notation as in \S\,1; we recall in particular that $M$ denotes the subgroup
consisting of all elements in a maximal compact subgroup $K$ of $S$, commuting with all
elements in a maximal subgroup $\Ad$-diagonalisable over $\R$.
\iffalse
We begin by noting the following consequence of some
results in \cite{A}.
\begin{proposition}\label{Andre}
Let $S$ be a connected semisimple Lie group and the notations $K,A,N,M, \Delta$
and $N'_\alpha, \alpha \in \Delta$ be as in \S 1. Let $P=MAN$. Let $x\in N$ be
an $\R$-regular element (viz. $x\notin N'_\alpha$ for any $\alpha \in \Delta$). Then
the centraliser of $x$
in $S$ is contained in $P$.
\end{proposition}
\proof We first show that it suffices to prove the proposition in the case when $S$ has
trivial center. Let $S^*$ be the adjoint group of $S$ and $\Ad :S\to S^*$ be the adjoint
homomorphism. The restriction of $\Ad$ to $AN$ is an isomorphism of Lie groups
and hence it follows that the subgroups $N'_\alpha$, $\alpha \in \Delta$ defined for
the two groups $S$ and $S^*$ correspond to each other under the isomorphism.
Now let $x$ be a $\R$-regular element in $S$. The preceding observation then
implies that $\Ad x$ is $\R$-regular in $S^*$.
Also, $\Ad P$ is the subgroup of $S^*$ corresponding to $P$ as above. Thus if the
proposition is proved for $S^*$, we get that
$\Ad x$ is contained in $\Ad P$. But $P$ contains the center of $S$, and hence
this implies that $x\in P$ as sought to be proved.
Now suppose that $S$ has trivial center. It can therefore be realised as the connected
component of the identity in the group of $\R$-points of a semisimple algebraic group
defined over $\R$. Now as in \S 1 let $\frak A$ and $\frak N$
be the Lie subalgebras corresponding to $A$ and $N$ respectively and let
$\frak N=\sum_{\varphi \in \Phi} \frak N_\varphi$ be the decomposition of $\frak N$
into the root spaces. Let $\Delta $ be the system of simple roots in $\Phi$,
say $\Delta =\{\varphi_1, \dots, \varphi_r\}$; here $r$ is the $\R$-rank of $S$.
For any $j=1, \dots r$ let $E_j=\{\exp X\mid X\in \frak N_{\varphi_j}, X\neq 0\}$. Then
any element $x$ of the form $x_1x_2\cdots x_ry$ with $x_j\in E_j$ and $y\in [N,N]$
is an $\R$-regular
unipotent element of $S$ in \cite{A} (see \cite{A}, Proposition~10), and one consequence of this is that
the centraliser of $x$ in $S$ is contained in the minimal parabolic subgroup $P=MAN$ (see \cite{A}, Corollary~5).
It is straightforward to see that $x\in N$ satisfies the condition as above if and only
if $x\notin N'_\alpha$ for any $\alpha \in \Delta$, and hence when this holds the centraliser
of $x$ in $S$ is contained in $P$. This proves the proposition. \qed
\fi
\begin{remark}\label{semireg}
{\rm We note that when $M$ is abelian there exists a unipotent one-parameter subgroup
whose centraliser is contained in the center of $S$. It suffices
to see this in the case when $S$ has trivial center, so $S$ may be taken to be the group
of $\R$-elements of a semisimple algebraic group defined over $\R$. In this case $M$ being abelian
is equivalent to the group being quasi-split, namely that there exists a Borel subgroup
defined over $\R$. Under that condition it follows from \cite{R}, Proposition~5.1 (see also \cite{C} for a generalisation of the result to non-semisimple
algebraic groups) and \cite{A} and Corollary~5 (page 114) that there exists $u\in N$ such that
the centraliser of $u$ in $S$ is contained in $P$. On the other
the conjugation action of $M$ action on $N$ has no fixed points, so the centraliser of $u$, and hence
of any one-parameter subgroup $U$ containing $u$, has no compact subgroup of positive dimension.
Hence $S$ satisfies condition $\frak U$.
}
\end{remark}
\medskip
\noindent {\it Proof of Theorem~\ref{ss}}: Since $M$ is abelian by Remark~\ref{semireg} there exists a unipotent one-parameter subgroup, say $U$, in $S$ whose
centraliser is contained in the (discrete) center of $S$; in particular the centraliser of
$U$ in $S$ contains no subgroup of positive dimension. Hence condition~$\frak U$ is satisfied for $S$.
\iffalse
Let $W=N/[N,N]$. Since $N$ is simply connected $W$ is a vector space over $\R$.
The conjugation action of $P=MAN$ on $N$ factors to a linear action on $W$. We note
that if $w\in W$ is fixed under the action of a compact subgroup of $P$ then it is
fixed by a compact subgroup of positive dimension in $M$, since the stabiliser
of any element under the action contains
$N$ and in $P/N\approx MA$ all compact subgroups are contained in $M$.
which is not fixed by any one-parameter subgroup of $S$. From the identification of
$W$ with $\frak N_\Delta$ as above we get that there exists $Y\in \frak N_\Delta$, of
that is not fixed by any one-parameter subgroup of $M$, under the Adjoint action of $M$.
Let
$Y=\sum_{j=1}^r Y_j$ with $Y_j\in \frak N_{\varphi_j}$. Let $X=\sum_{j=1}^r X_j$, $
X_j\in \frak N_{\varphi_j}$ be such that $X_j=Y_j$ if $Y_j\neq 0$ and $X_j\neq 0$
for all $j$. Since all $\frak N_\varphi$ are $M$-invariant, it follows that $X$ is also
not fixed by any one-parameter subgroup of $M$.
Now let $x_j=\exp X_j$ for all $j=1,\dots, r$.
Then $x_j\in E_j$ for all $j$ (in the notation introduced above). Let $x=x_1x_2\cdots x_r$.
We shall show that the centraliser of $x$ does
not contain any compact subgroup of positive dimension; the desired assertion
would then follow for the (unique) one-parameter subgroup containing~$x$.
As recalled above $x$ is a $\R$-regular unipotent element in $S$ and its centraliser is
contained in $P$. Suppose there exists a compact subgroup $C$ of positive
dimension, contained in $P$ that
centralises $x$. Then $x[N,N]$ is fixed under action of $C$ on $P/[N,N]$, induced
by the conjugation action. Using the natural correspondence of the action of
$C$ on $N/[N,N]$ with the Adjoint action on $\frak N_\Delta$, we see that each $X_j$,
and hence $X$, is fixed under the action of $C$.
Since $M$ is a maximal compact subgroup of $MN$, by the conjugacy of maximal
compact subgroups, there exists a $n\in N$ such that $nCn^{-1} $ is contained in
$M$. Since $X$ is fixed under the Ad action of $C$ we get that $\Ad \,n (X)$
is fixed under the Ad action of $nCn^{-1}$. But $\Ad \,n(X)\in X+\frak N_\Psi$ and,
each $\frak N_\varphi$ being $M$-invariant, we get that $X$ is fixed under the action of
$nCn^{-1}$. Recall that $X$ is not fixed under the action of any nontrivial one-parameter
subgroup of $M$, and hence $nCn^{-1}$, and in turn $C$, must be trivial.
But this contradicts $C$ being of positive dimension. This shows that the centraliser
of $x$ does not contain any compact subgroup of positive dimension, so
condition $\frak U$ is satisfied. This proves~(i).
ii)
Since $M$ is given to be abelian, $S$ is a semisimple group with no nontrivial compact
simple factor. We note that in this case the action of any closed positive
dimensional subgroup $C$ of $M$ on $W$ is nontrivial: If the action of $C$ on $W$ is
trivial then it follows that its action on $N$, by conjugation, is trivial and by symmetry this also implies
that the action of $C$ on the nilradical $N^-$ of the opposite parabolic subgroup (which
also contains $C$) is trivial. Since $S$ has no compact simple factors $N$ and $N^-$
generate $S$ and we get that $C$ is contained in the center of $S$, which is not possible.
Hence the $C$-action on $W$ is nontrivial.
Since $M$ is abelian it has only countably many closed positive-dimensional
subgroups and, as noted above, for each of them the set of fixed points under the action on $W$
is a proper subspace of $W$. Hence we can find a $w\in W$ which is not fixed
under the action of any positive-dimensional closed subgroup of $M$, and moreover
$w$ may also be chosen so that it is not contained in $N'_\alpha/[N,N]$, for any $\alpha
\in \Delta$ (with $N'_\alpha$ as in the notation as in the definition of $\R$-regular
elements, in \S 1).
Now let $n\in N$ be such that $w=n[N,N]$.
Suppose if possible there exists a compact subgroup $C$ of positive dimension centralising $n$. By its choice as above $n$ is a $\R$-regular
element and hence by Proposition~\ref{Andre} its centraliser is contained in $P$.
Thus $C$ is contained in $P$, and the action of $C$ fixes $w$. As noted above this implies
that there is compact subgroup of positive dimension contained in $M$ whose action
fixes $w$. But this contradicts our choice of $w$ as an element not fixed any
subgroup of $M$ of positive dimension. This shows that $n$ is not fixed by any
compact subgroup of positive dimension, and the same applies to the one-parameter
subgroup of $N$ containing $n$. Hence $S$ satisfies condition~$\frak U$. This proves~(ii).
\fi
For split semisimple Lie groups the subgroup $M$ as above is trivial, and for complex Lie groups
it is abelian. It therefore follows from the above results the condition $\frak U$ holds in
these cases. \qed
\iffalse
Using Theorem~\ref{ss} we show also that the special unitary group $SU (p,q)$, consisting of
$n\times n$ matrices,
where $n=p+q$, with entries in $\C$ and determinant~$1$, leaving invariant the Hermitian
form $z_1\bar z_1 + \cdots z_p\bar z_p -z_{p+1}\bar z_{p+1} \cdots -z_n\bar z_n $
satisfies condition $\frak U$ when $q\leq p\leq 2q$; we note that it is a semisimple Lie group
in which the subgroup $M$ as above is nonabelian for $q\geq 2$. It may also be recalled
here that $SU(p,q)$ is an exponential Lie group (see \cite{DN} and \cite{W2}); it
can therefore occur as the semisimple quotient of an exponential Lie group, and Theorem~\ref{thm}
the radical is exponential in these instances. (The argument below also shows that
the special orthogonal groups $SO (p,q)$ also satisfy condition $\frak U$; however they
are not exponential Lie groups and hence not of interest in the present context).
\begin{corollary}
The group $SU(p,q)$ satisfies condition $\frak U$ when $q\leq p\leq 2q$.
\end{corollary}
\proof Let $p,q$ with $q\leq p\leq 2 q$ be given, and $n=p+q$. For any matrix $W$ we denote
by $W^*$ the Hermitian transpose of $W$. We denote by $0$ the zero matrix of any size,
and similarly by $I$ the identity (square) matrix of any
size, the sizes being determined by the context. Let $G=SU(p,q)$ and
$\frak G$ be the Lie algebra of $G$. We realise $\frak G$ as the space of $n\times n$ complex valued matrices of the form \\
\noindent where $A,B,C$ are skew Hermitian square matrices of sizes $(p-q)$, $q$
and $q$ respectively such that $\trace A +\trace B +\trace C=0$, and $X,Y,Z$ are (arbitrary, rectangular) matrices of sizes $(p-q)\times q$,
$(p-q)\times q$ and $q\times q$ respectively. Let
$$\frak K= \{\xi \in \frak G\mid X(\xi)=0, Y(\xi)=0 \hbox{ \rm and } Z(\xi)=0\},$$
and $$\frak A= \{\xi \in \frak G \mid A(\xi)=0, B(\xi)=C(\xi)=0, X(\xi)=0, Y(\xi)=0 \hbox{ \rm and } Z(\xi)\in \frak D\},$$
where $\frak D$ denotes the space of diagonal (square) matrices of size $q$ with entries in~$\R$.
Then $\frak K$ is the Lie algebra of a maximal compact subgroup, say $K$, of $G$ and $\frak A$
is the Lie algebra of a maximal Lie subalgebra whose action is diagonalisable over $\R$.
For $1\leq k,l \leq q$ let $E_{kl}$ denote the square matrix of size $q$ in which the $(k,l)$th entry
is $1$ and all other entries are $0$. Let
$$\frak S=\{\xi \in \frak G \mid A(\xi)=0, X(\xi)=0=Y(\xi)\}.$$
Let $W$ be the space of $q\times q$ skew Hermitian matrices and $W_0$ be the
subspace of $W$ consisting the elements
in which the diagonal entries are $0$.
For $B=\sum_{k,l} z_{kl}E_{kl}\in W$, let
$\theta (B)$ denote the Hermitian matrix $\sum_{k<l}z_{kl}E_{kl}+\sum_{k>l}\bar z_{kl}E_{kl}$.
Let $$\frak N_1=\{\xi \in \frak S\mid B(\xi) \in W_0, Z(\xi) =\theta (B(\xi))
\hbox{ \rm and } C(\xi)=B(\xi)\}$$
and
$$\frak N_2=\{\xi \in \frak S\mid B(\xi) \in W, Z(\xi)=-B(\xi)=C(\xi)\}.$$
Also let
$$\frak N_3=\{\xi \in \frak G\mid A(\xi)=0, B(\xi)=C(\xi)=0, Y(\xi) = -X(\xi), Z(\xi)=0\}$$
and $$\frak N=\frak N_1+\frak N_2+\frak N_3.$$
Note that $\xi \in \frak A$ may be identified canonically with $Z(\xi)=\sum_{k=1}^q d_kE_{kk}$
and the correspondence $\xi \to (d_1,\dots , d_q)$ sets a canonical bijection of $\frak A$ with $\R^q$.
Let $\chi_k:\frak A \to \R$ be the linear forms, defined via the correspondence, by setting
$\chi_k(\xi)=d_k-d_{k+1}$ for
$k=1,\dots, q-1$ and $\chi_q(\xi)=d_q$. Then $\chi_1, \dots, \chi_q$ form a system of
simple roots for $\frak G$ with respect to $\frak A$, and it can further be verified that $\frak N$
is the sum of all root spaces corresponding to the system of positive
roots, viz. nonnegative integral combinations of $\chi_1, \dots, \chi_q$. Then
$\frak N$ is a maximal nilpotent Lie subalgebra of $\frak G$ normalised
by $\frak A$, as may be verified either by direct computation or using
structure theory for semisimple Lie algebras (see for example~\cite{Hel}).
We shall show that for an open dense set of elements $\nu$ in $\frak N$ there is no
compact subgroup of $G$ of positive dimension centralising $\nu$, namely consisting
of $g$ such that $\Ad (g) (\nu)=\nu$.
Firstly we assume $\nu$ to be such that its component
in each positive root space is non-zero, a condition satisfied by elements in an open
dense subset. Under this condition $\exp \nu$ is a $\R$-regular element in the sense of
\cite{A} and hence it follows that $\exp \nu$ is contained in a unique minimal parabolic
subgroup and the latter also contains the centraliser of $\exp \nu$ in $G$. We shall
show that under some further conditions, also satisfied by elements in open dense
subsets of $\frak N$, the centralser does not contain any compact subgroup of positive
dimension.
With the above definitions $\frak G=\frak K + \frak A +\frak N$ is an Iwasawa decomposition of $\frak G$
and if $K$, $A$ and $N$ are the connected Lie subgroups corresponding to $\frak K, \frak A$ and
$\frak N$ respectively then $G=KAN$ is an Iwasawa decomposition of $G$. Let $\Delta $
denote the space of diagonal matrices of size $q$, with purely imaginary entries.
The Lie subalgebra
corresponding to $M$, namely the centraliser of $A$ in $K$, is seen to be given by
$$\frak M=\{\xi \in \frak G\mid X(\xi)=Y(\xi)=0, Z(\xi)=0, B(\xi)\in \Delta, C(\xi)=B(\xi)\}.$$
Now let $\mu \in \frak M$, $\nu =\nu_1+\nu_2+\nu_3 \in \frak N$, with $\nu_k\in \frak N_k$
for $k=1,2,3$, and suppose that $\mu$ and $\nu$ commute, namely $[\mu,\nu]=0$. Let
$\frak U$ be the Lie subalgebra of $\frak G$ consisting of $\xi$ for which all components other
than $A(\xi)$ are $0$. Then we have $\mu=\mu_1+\mu_2$, expressed uniquely, with $\mu_1 \in \frak U$,
$\mu_2\in \frak S$ and moreover $B(\mu_2)=C(\mu_2)\in \Delta$. Let $B(\mu_2)=\delta$.
A direct matrix computation shows that when $[\mu,\nu]=0$ we have $[\delta, B(\nu_1)]=0$ and
$[\delta, \theta B((\nu_1))]=0$.
\fi
| 194,763
|
\begin{document}
\title{Dynamics of $\cB$-free sets: a view through the window}
\author[1]{Stanis\l aw Kasjan$^{\ast\dagger}$}
\author[2]{Gerhard Keller$^\dagger$}
\author[1]{Mariusz Lema\'nczyk\thanks{Research supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736.}\thanks{Research supported by the special program of the semester
``Ergodic Theory and Dynamical Sytems in their Interactions with Arithmetic and Combinatorics'', Chair Jean Morlet, 1.08.2016-30.01.2017.}}
\affil[1]{\small Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toru\'n, Poland}
\affil[2]{Department of Mathematics, University of Erlangen-N\"urnberg, Germany}
\date{Version of \today}
\maketitle
{\begin{abstract}
Let $\cB$ be an infinite subset of $\{1,2,\dots\}$.
We characterize arithmetic and dynamical properties of the $\cB$-free set $\cF_\cB$ through
group theoretical, topological and measure theoretic properties of a set $W$ (called the \emph{window}) associated with $\cB$. This point of view stems from the interpretation of the set $\cF_\cB$ as a weak model set. Our main results are: $\cB$ is taut if and only if the window is Haar regular; the dynamical system associated to $\cF_\cB$ is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to $\cF_\cB$ is proximal if and only if the window has empty interior; and the dynamical system associated to $\cF_\cB$ has the ``na\"ively expected'' maximal equicontinuous factor if and only if the interior of the window is aperiodic.
\end{abstract}}
\blfootnote{\emph{MSC 2010 clasification:} 37A35, 37A45, 37B05.}
\blfootnote{\emph{Keywords:} $\cB$-free dynamics, sets of multiples, maximal equicontinuous factor.}
\section{Introduction and main results}
For any given set $\cB\subseteq\N=\{1,2,\dots\}$ one can define its \emph{set of multiples}
\begin{equation*}
\cM_\cB:=\bigcup_{b\in\cB}b\Z
\end{equation*}
and the set of \emph{$\cB$-free numbers}
\begin{equation*}
\cF_\cB:=\Z\setminus\cM_\cB\ .
\end{equation*}
The investigation of structural properties of $\cM_\cB$ or, equivalently, of $\cF_\cB$ has a long history (see the monograph \cite{hall-book} and the recent paper \cite{BKKL2015} for references), and dynamical systems theory provides some useful tools for this. Namely, denote by $\eta\in\{0,1\}^\Z$ the characteristic function of $\cF_\cB$, i.e. $\eta(n)=1$ if and only if $n\in\cF_\cB$, and consider the
orbit closure $X_\eta$ of $\eta$ in the shift dynamical system
$(\{0,1\}^\Z,\sigma)$, where $\sigma$ stands for the left shift. Then topological dynamics and ergodic theory provide a wealth of concepts to describe various aspects of the structure of $\eta$, see \cite{Sa} which originated
this point of view by studying the set of square-free numbers, and also
\cite{Ab-Le-Ru}, \cite{BKKL2015} which continued this line of research.
In this paper we continue to provide a dictionary that characterizes arithmetic properties of $\cB$ in terms of dynamical properties of $X_\eta$, and, as an intermediate step, also in terms of topological and measure theoretic properties of a pair $(H,W)$ associated with the passage from $\cB$ to
$X_\eta$, where $H$ is a compact abelian group and $W$ a compact subset of $H$. This latter point of view is borrowed from the theory of weak model sets, which applies here, because $\cF_\cB$ is a particular example of such a set, see e.g. \cite{BHS2015,KR2015}.
Finally the Chinese Remainder Theorem allows
us to interpret our dynamical results combinatorially.
In order to formulate our main results, we need to
recall some notions from the theory of sets of multiples \cite{hall-book}
and also to
introduce some further notation.
Let $\cB$ be a non-empty subset of $\N$.
\begin{itemize}
\item $\cB$ is \emph{primitive}, if there are no $b,b'\in\cB$ with $b\mid b'$.
From any set $\cB\subseteq\N$ one can remove all multiples of other numbers in $\cB$, which results in the set
\begin{equation}\label{eq:Bprim}
\prim{\cB}:=\cB\setminus \bigcup_{b\in\cB}b\cdot(\N\setminus\{1\})\ .
\end{equation}
$\prim{\cB}$ is primitive by construction, and $\cM_\cB=\cM_{\prim{\cB}}$.
\item $\cB$ is \emph{taut}, if $\ddelta(\cM_{\cB\setminus\{b\}})<\ddelta(\cM_\cB)$ for each $b\in\cB$, where $\ddelta(\cM_\cB):=\lim_{n\to\infty}\frac{1}{\log n}\sum_{k\leqslant n,k\in\cM_\cB}k^{-1}$
denotes the logarithmic density of this set, which is known to exist by the Theorem of Davenport and Erd\"os \cite{DE1936,DE1951}.
So a set is taut, if removing any single point from it changes its set of multiples drastically and not only by ``a few points''.
\item $\tilde{H}:=\prod_{b\in\cB}\Z/b\Z$ and $\Delta:\Z\to\tilde{H}$, $\Delta(n)=(n,n,\dots)$ -- the canonical diagonal embedding.
\item $H:=\overline{\Delta(\Z)}$ is a compact abelian group, and we denote by $m_H$ its normalised Haar measure.
\item $R_{\Delta(1)}: H\to H$ denotes the rotation by $\Delta(1)$, i.e. $(R_{\Delta(1)}h)_b=(h_b+1)$ mod $b$ for all $b\in\cB$.
\item The \emph{window} is defined as
\begin{equation}\label{eq:W}
W:=\{h\in H: h_b\neq0\ (\forall b\in\cB)\}.
\end{equation}
\item $\varphi:H\to\{0,1\}^\Z$ is the coding function: $\varphi(h)(n)=1$, if and only if $R_{\Delta(1)}^nh\in W$, equivalently, if and only if $h_b+n\neq0$ mod $b$ for all $b\in\cB$.
\item By $S,S'\subset\cB$ we always mean \emph{finite} subsets.
\item The topology on $H$ is generated by the (open and closed) cylinder sets
\begin{equation*}
U_S(h):=\{h'\in H: \forall b\in S: h_b=h'_b\},\text{ defined for finite }S\subset\cB\text{ and }h\in H\ .
\end{equation*}
\end{itemize}
A recurring theme of the main results in this paper is to characterize arithmetic and dynamical properties of a $\cB$-free set $\cF_\cB$ through
group theoretical, topological and measure theoretic properties of the window $W$ defined above.
\begin{remark}
With the notation introduced above, we can write
\begin{equation*}
X_\eta=\overline{\varphi(\Delta(\Z))}\ .
\end{equation*}
This is certainly a subset of $X_\varphi:=\overline{\varphi(H)}$, the set studied in \cite{KR2015} under the name $\MWG$. In Proposition~\ref{prop:Mariusz4.2} we show that $X_\eta=X_\varphi$ when $\cB$ has \emph{light tails} (see Subsection~\ref{subsec:light-tails} for a definition), but we do not know whether also tautness of $\cB$ suffices (see also Subsection~\ref{subsec:light-tails}).
\end{remark}
\subsection{Tautness as a measure theoretic property}
\begin{theorem}\label{theo:taut-regular}
\footnote{The authors are indebted to J. Ku\l{}aga-Przymus for pointing out the relevance of \cite[Lemma 1.17]{hall-book} for the proof of this theorem.}
Suppose that the set $\cB$ is primitive.
Then the following are equivalent:
\begin{compactenum}[(i)]
\item $\cB$ is taut.
\item The window $W$ is \emph{Haar regular}, i.e. $\supp(m_H|_W)=W$.
\end{compactenum}
Moreover, these properties imply
\begin{compactenum}[(i)]
\addtocounter{enumi}{2}
\item $\overline{\Delta(\Z)\cap W}=W$.
\end{compactenum}
\end{theorem}
\noindent
The proof of the theorem is provided in Section~\ref{sec:proof-taut-regular}.
The concept of a \emph{Haar regular window} was introduced in \cite{KR2016} in the context of general weak model sets.
\\[0mm]
Given a set $\cB\subset\N$, one says that $h:=(h_b)_{b\in\cB}\in\Z^{\cB}$ satisfies the CRT
(Chinese Remainder Theorem) if for each finite $S\subset\cB$ there exists $n\in\Z$ such that
\begin{equation}\label{freeCRT}
h_b=n\text{ mod }b\;\text{ for each }b\in S.
\end{equation}
Clearly,
\begin{equation*}
h\text{ satisfies the CRT\; iff\;} h\in H.
\end{equation*}
We are looking for solutions of~\eqref{freeCRT} with $n\in\cF_\cB$. If for $h$ as above we can solve~\eqref{freeCRT} with $n=n_S\in\cF_\cB$ for all finite $S\subset \cB$, then we say that $h$ satisfies the $\cB$-free CRT.
A necessary condition for $h=(h_b)_{b\in\cB}$ to satisfy the $\cB$-free CRT is, of course, that $h_b\neq0\mod b$ for each $b\in\cB$, and a moment's reflection shows that
\begin{equation}\label{eq:B-free-CRT}
h\text{ satisfies the $\cB$-free CRT\; iff\;} h\in \overline{\Delta(\Z)\cap W}\ .
\end{equation}
Therefore the implication $(i)$ $\Rightarrow$ $(iii)$ of Theorem~\ref{theo:taut-regular} is an immediate consequence of the following proposition.
\begin{proposition}\label{prop:mariusz-stronger}
Assume that $\cB$ is taut. Let $h\in W$ and $S\subset\cB$ finite.
Then the set of
\emph{$\cB$-free} integers $n$ that solve $n = h_b$ mod $b$ for $b\in S$ has asymptotic density
$m_H(U_S(h)\cap W)>0$.
\end{proposition}
In Subsection~\ref{subsec:property(iii)} we provide a sequence $\cB$, which is not taut, but for which $\overline{\Delta(\Z)\cap W}=W$ (Example~\ref{example:non(iii)to(i)}). Hence $(iii)$ of Theorem~\ref{theo:taut-regular} is not equivalent to $(i)$ and $(ii)$. Here we provide two simpler examples which throw some light on property $(iii)$. Denote by $\cP\subseteq\N$ the set of all \emph{prime numbers}.
\begin{example} If $\cB=\cP$ then $H=\prod_{p\in\cP}\Z/p\Z$, $W$ is uncountable (although of Haar measure zero) and
$\overline{\Delta(\Z)\cap W}\neq W$,
since for each $n$ we find $p\in\cP$ such that $p\mid n$, so $n=0$ mod~$p$.
\end{example}
\begin{example} If $\cB\subset{\cP}$ is \emph{thin}, i.e. if $\sum_{p\in\cB}1/p<+\infty$, then $\overline{\Delta(\Z)\cap W}=W$ in view of \eqref{eq:B-free-CRT},
because each $h\in H$ satisfies the $\cB$-free CRT. Indeed, if $S\subset\cB$ is finite and
$n=h_b\mod b$ for $b\in S$, then $n+\lcm(S)\Z$ is the set of all solutions to this system of congruences. Moreover, if $h\in W$, then $\gcd(n,b)=1$ for all $b\in S$. We only need to find $r\in\Z$ so that $n+r\lcm(S)$ is a prime number which is not in $\cB$. The latter follows from Dirichlet's theorem:
The set of prime numbers contained in $n+\lcm(S)\Z$ is not thin. Of course this is a special case of Theorem~\ref{theo:taut-regular}.
\end{example}
\begin{remark}
Denote by $\nu_\eta:=m_H\circ\varphi^{-1}$ the \emph{Mirsky measure} on $X_\eta$.
There are two independent proofs of the fact that the two equivalent conditions from Theorem~\ref{theo:taut-regular} imply that
the measure preserving dynamical system $(X_\eta,\sigma,\nu_\eta)$ is isomorphic to the group rotation $(H,R_{\Delta(1)},m_H)$:
In \cite[Theorem F]{BKKL2015} it is proved that this is implied by $(i)$. That it is also a direct consequence of $(ii)$ follows - in the more general context of model sets - from \cite{KR2016}. The proof uses our observation that $W$ is aperiodic (see Proposition~\ref{p:ape3}). To see this, denote by $H_W:=\{h\in H: W+h=W\}$ the period group of $W$ and by $H_W^{Haar}:=\{h\in H: m_H((W+h)\triangle W)=0\}$ its group of \emph{Haar periods}. It is easily seen that
$H_W=H_W^{Haar}$ for Haar regular $W$, in particular whenever the sequence $\cB$ is taut. Hence, if $W$ is aperiodic, it is also Haar aperiodic, and this is what is needed to apply the general theorem from \cite{KR2016} to the present context.
A word of caution is in order at this point: Althoug, in the $\cB$-free context, the window $W$ is always aperiodic (Proposition~\ref{p:ape3}), this is not necessarily true for its Haar regularization $W_{reg}:=\supp(m_H|_W)$, because that window is not of the same arithmetic type as $W$. On the other hand, as proved in \cite[Theorem C]{BKKL2015}, each non-taut set $\cB$ can be modified into a taut set $\cB'$ whose corresponding Mirsky measure $\nu_{\eta'}$ coincides with $\nu_\eta$ (as a measure on $\{0,1\}^\Z$).
The (arithmetic!) window $W'\subseteq H'$ defined by $\cB'$ is then aperiodic and Haar regular, and we suspect that it to be
closely related to $W_{reg}\subseteq H$.
\end{remark}
\subsection{The proximal and the Toeplitz case}
From \cite[Theorem A]{BKKL2015} we know that $X_\eta$ has a unique minimal subset $M$.
In Lemma~\ref{lemma:C_varphi}, we prove that $M=\overline{\varphi(C_\varphi)}$, where $C_\varphi$ denotes the set of continuity points of $\varphi:H\to\{0,1\}^\Z$, see also \cite[Lemma 6.3]{KR2015}.
$M$ is degenerate to a singleton, namely to $M= \{(\dots,0,0,0,\dots)\}$, if and only if $\inn(W)=\emptyset$ \cite{KR2015}, and we collect a number of equivalent characterizations of this extreme case in Theorem~\ref{theo:proximal} below.
Assuming primitivity of $\cB$ and property $(iii)$ of Theorem~\ref{theo:taut-regular}, we prove the following equivalent characterizations of minimality of $(X_\eta,\sigma)$, i.e. of $M=X_\eta$, in Subsection~\ref{subsec:proof-theo-minimality}.
For $S\subset\cB$ let
\begin{equation}\label{eq:A_S-def}
\cA_S:=\{\gcd(b,\lcm(S)): b\in\cB\},
\end{equation}
and note that $\cF_{\cA_S}\subseteq\cF_\cB$, because
$b\mid m$ for some $b\in\cB$ implies $\gcd(b,\lcm(S))\mid m$ for any $S\subset\cB$.
Let
\begin{equation}\label{def:Cinf}
\Ainf:=\{n\in\N: \forall_{S\subset\cB}\ \exists_{S': S\subseteq S'}: n\in\cA_{S'}\setminus S'\}.
\end{equation}
In Lemma~\ref{lemma:ASk-C} we prove: If $(S_k)_k$ is a filtration of $\cB$ with finite sets, then
\begin{equation}
\limsup_{k\to\infty}\left(\cA_{S_k}\setminus S_k\right)=\Ainf\ .
\end{equation}
\begin{theorem}\label{theo:minimality}
Suppose that $\cB$ is primitive.
Consider the following list of properties:
\begin{compactenum}
\myitem The window $W$ is topologically regular, i.e. $\overline{\inn(W)}=W$.
\myitem $\cF_\cB=\bigcup_{S\subset\cB\text{ finite}}\cF_{\cA_S}$.
\myitem $\Ainf=\emptyset$.
\myitem There are no $d\in\N$ and no infinite pairwise
coprime set $\cA\subseteq\N\setminus\{1\}$ such that $d\,\cA\subseteq\cB$.
\myitem $\eta=\varphi(0)$ is a Toeplitz sequence (see \cite{Do}, \cite{JK1969} for the
definition) different from $(\dots,0,0,0,\dots)$.
\myitem $0\in C_\varphi$ and
$\varphi(0)\neq(\dots,0,0,0,\dots)$.
\myitem $\eta\in M$ and $\eta\neq(\dots,0,0,0,\dots)$.
\myitem $X_\eta$ is minimal., i.e. $X_\eta=M$, and $\card(X_\eta)>1$.
\myitem The dynamics on $X_\eta$
is a minimal almost \oneone extension of $(H,R_{\Delta(1)})$,
the rotation by $\Delta(1)$ on $H$.
\end{compactenum}
\begin{compactenum}[a)]
\item (B1) - (B6) are all equivalent,
and each of these conditions implies that $\cB$ is taut.
\item (B7) and (B8) are equivalent.
\item Each of (B1) - (B6) implies (B9).
\item (B9) implies (B7) and (B8).
\item If $\overline{\Delta(\Z)\cap W}=W$ (in particular if $\cB$ is taut), then (B1) - (B9) are all equivalent.
\end{compactenum}
\end{theorem}
\begin{remark}\label{remark:window-toeplitz}
One ingredient of the proof of Theorem~\ref{theo:minimality} is the observation that the set $\cB$ is taut whenever $\eta$ is a Toeplitz sequence. This was pointed out to us by
A. Bartnicka who also gave a proof of it, which we recall in Lemma~\ref{lemma:Aurelia} below.
Moreover, we can interpret the result purely arithmetically as follows: If $\cB$ is primitive and satisfies (B4) then the set of elements for which the $\cB$-free CRT holds is topologically regular, i.e. it contains a dense subset of points for which all sufficiently close points satisfying the CRT satisfy also the $\cB$-free CRT.
\end{remark}
The following characterization of regular Toeplitz sequences is included in Proposition~\ref{prop:toeplitz} in
Subsection~\ref{subsec:Toeplitz}, where also the precise definition of regularity
of a Toeplitz sequence is recalled.
\begin{proposition}\label{prop:Toeplitz-regular}
Assume that $\overline{\inn(W)}=W$. Then the Toeplitz sequence
$\eta$ is regular, if and only if $m_H(\partial W)=0$.
\end{proposition}
In Subsection~\ref{subsec:Toeplitz} we also provide examples of sets $\cB$ that give rise to regular Toeplitz sequences and others giving rise to irregular Toeplitz sequences.
Note also that
$m_H(\partial W)=0$ if and only if
$\inf_{S\subset\cB}\bar d(\cM_{\cA_S}\setminus\cM_\cB)=0$, see
Lemma~\ref{lemma:Haar-boundary}, and observe that $m_H(\partial W)=0$ implies unique ergodicity of the dynamics on $X_\eta$ \cite[Theorem 2c]{KR2015}.
\quad\\
The next theorem is complementary to Theorem~\ref{theo:minimality}.
Most of its equivalences follow from results in \cite{BKKL2015} and \cite{KR2015} and are proved in Subsection~\ref{subsec:proof-theo-proximal}.
They do not rely on the more advanced arithmetic concept of tautness.
\begin{theorem}\label{theo:proximal}
The following are equivalent:
\begin{compactenum}
\myitem $\inn(W)=\emptyset$
\myitem $\bigcup_{S\subset\cB\text{ finite}}\cF_{\cA_S}=\emptyset$,
i.e. $\cF_{\cA_S}=\emptyset$ for all finite $S\subset\cB$.
\myitem $\forall S\subset\cB: 1\in\cA_S$.
\myitem $\cB$ contains an infinite pairwise coprime subset.
\myitem If $\cC\subseteq\N$ is finite and if $\cB\subseteq\cM_\cC$, then $1\in\cC$.
\myitem $M=\{(\dots,0,0,0,\dots)\}$.
\myitem The dynamics on $X_\eta$ are proximal.
\end{compactenum}
\end{theorem}
\begin{remark}
Under the conditions
of Theorem \ref{theo:proximal} no element of $W$ is
stable, that is, for each $h$ staisfying the $\cB$-free CRT there is an element
$h'\in\Z^{\cB}$ arbitrarily close to $h$ which satisfies the CRT but not the $\cB$-free
CRT.\end{remark}
\subsection{The maximal equicontinuous factor}
We finish with a result that identifies the maximal equicontinuous factor of the dynamics on $X_\eta$ and answers Question 3.14 in \cite{BKKL2015}.
Given a subset $A\subseteq H$, denote by
\begin{equation*}
H_A:=\left\{h\in H: A+h=A\right\}
\end{equation*}
the \emph{period group} of $A$. The set $A\subseteq H$ is \emph{topologically aperiodic}, if $H_A=\{0\}$.
Observe also that $H_{\inn(A)}$ is a closed subgroup of $H$, whenever $A$ is closed \cite[Lemma 6.1]{KR2016}.
In Proposition~\ref{p:ape3} we prove that $H_W=\{0\}$ whenever $\cB$ is primitive.
If $\inn(W)=\emptyset$, then of course $H_{\inn(W)}=H$. If $\inn(W)\neq\emptyset$,
the situation is more complicated: $H_{\inn(W)}$ is obviously always a strict subgroup of $H$, and very often $H_{\inn(W)}=\{0\}$, but
there are examples where $H_{\inn(W)}$ is a non-trivial strict subgroup of $H$, see Subsection~\ref{subsec:examples}. In any case, however, $H_{\inn(W)}$ determines the maximal equicontinuous factor. The following is proved in \cite[Theorem A2]{KR2016}:
\begin{equation}
\begin{minipage}{0.9\textwidth}
\textbf{Theorem}\; \emph{The translation by $\Delta(1)+H_{\inn(W)}$ on $H/H_{\inn(W)}$ is the maximal equicontinuous factor of the dynamics on $X_\eta$.}
\end{minipage}
\end{equation}
Let $S_1\subset S_2\subset\dots$ be any filtration of $\cB$ by finite sets.
In Subsection~\ref{subsec:period-group} we define divisors $d_k$ of $\lcm(S_k)$:
\begin{equation}
d_k:=\lim_{j\to\infty}\gcd(s_k,c_{k+j})\,,\;\text{ where }\;
s_k:=\lcm(S_k)\;\text{ and }\;
c_l:=\text{minimal period of }\cM_{\cA_{S_l}}\ .
\end{equation}
By Remark \ref{remark:factors} we have $\frac{s_k}{d_k}\mid\frac{s_{k+1}}{d_{k+1}}$ for any $k$.
The sequences $(s_k)$, $(d_k)$ and $(c_k)$ determine $H_{\inn(W)}$ in the following way:
\begin{proposition}\label{prop:period}
\begin{compactenum}[a)]
\item
\begin{equation*}
0\rightarrow H_{\inn(W)}\rightarrow H\cong\lim\limits_{\leftarrow}\Z/{s_k}\Z\rightarrow \lim\limits_{\leftarrow}\Z/{d_k}\Z\rightarrow 0
\end{equation*}
is an exact sequence.\footnote{A sequence of abelian groups and homomorphisms $...\mapr{}{} M_{k-1}\mapr{f_{k-1}}{} M_{k}\mapr{f_k}{} M_{k+1}\rightarrow...$ is called {\em exact}
if the kernel of $f_k$ is equal to the image of $f_{k-1}$ for any $k$. In particular, a sequence
$$
0\rightarrow M' \mapr{f}{} M\mapr{g}{} M''\rightarrow 0
$$
is exact, when $f$ is injective, the kernel of $g$ equals the image of $f$ and $g$ is surjective. We say that it is a "short exact sequence". In particular, the homomorphism $g$ induces an isomorphism $M''\cong M/f(M')$ in this case.
}
\item $H_{\inn(W)}\cong \lim\limits_{\leftarrow}\Z/\frac{s_k}{d_k}\Z$.
\item $H/H_{\inn(W)}\cong \lim\limits_{\leftarrow}\Z/{d_k}\Z$.
\item $H_{\inn(W)}=\{0\}$ if and only if $s_k=d_k$ for each $k\in\N$,
equivalently if for each $b\in\cB$ there is $n>0$ such that $b$ divides $c_{n}$.
\end{compactenum}
\end{proposition}
\begin{theorem}\label{theo:MEF}
\begin{compactenum}[a)]
\item The translation by $(1,1,\dots)$ on $H/H_{\inn(W)}\cong\lim\limits_{\leftarrow}\Z/{d_k}\Z$ is the
maximal equicontinuous factor of the dynamics on $X_\eta$.
\item In case d) of Proposition~\ref{prop:period},
the translation by $\Delta(1)$ on $H\cong \lim\limits_{\leftarrow}\Z/{s_k}\Z$ is the maximal equicontinuous factor of the dynamics on $X_\eta$.
\end{compactenum}
\end{theorem}
In Subsection~\ref{subsec:examples} we provide a number of examples illustrating this theorem.
\begin{remark}\label{rem:Y}
In \cite{BKKL2015}, the following set $Y$ is defined: \footnote{Versions of this set occur also in \cite{Peckner2012} and \cite{BaakeHuck14}.}
\begin{equation*}
Y:=\left\{x\in\{0,1\}^\Z: \card(\supp(x)\mod b)=b-1\;\forall b\in\cB\right\}.
\end{equation*}
Observe that $\card(\supp(x)\mod b)\leqslant b-1$ for all $x\in X_\eta$ and $b\in\cB$.
\footnote{Indeed, if $\card(\supp(x)\mod b)= b$ for some $x\in X_\eta$ and $b\in\cB$,
then this happens on some integer interval $[-M,M]$, and hence
$\card(\supp(\eta)\mod b)= b$, which contradicts the fact that $\supp(\eta)\subseteq\cF_\cB$.\label{foot:b-1}}
Proposition 3.27 of \cite{BKKL2015} asserts that
$(H,R_{\Delta(1)})$ is the maximal equicontinuous factor of $(X_\eta,S)$,
whenever $X_\eta\subseteq Y$. Hence, in that case, $H_{\overline{\inn W}}= H_{\inn W}=\{0\}=H_W$ by Theorem~\ref{theo:MEF} and Proposition~\ref{p:ape3}. This is the second one of the following two implications:
\begin{equation}
W=\overline{\inn W}\quad\Rightarrow\quad X_\eta\subseteq Y\quad\Rightarrow\quad
H_W=H_{\overline{\inn W}}\ .
\end{equation}
The first one is proved in Proposition~\ref{eta_in_Y}.
\end{remark}
\section{Tautness of $\cB$ and Haar regularity of $W$}\label{sec:proof-taut-regular}
\subsection{Arithmetic of $\cB$ and topology of $W$, part I}
\begin{definition}
Let $\cM\subseteq\N$.
\begin{compactenum}[a)]
\item The \emph{upper} resp. \emph{lower density} of $\cM$ is
\begin{equation*}
\overline{d}(\cM)=\limsup_{N\to\infty}
\frac1N\card\left(\cM\cap\{1,\dots,N\}\right)
\text{ resp. }
\underline{d}(\cM)=\liminf_{N\to\infty}
\frac1N\card\left(\cM\cap\{1,\dots,N\}\right)
\end{equation*}
If the limit exists, we write $d(\cM)$.
\item The \emph{logarithmic density} of $\cM$ is
\begin{equation*}
\ddelta(\cM)=\lim_{N\to\infty}\frac{1}{\log N}\sum_{n\in\cM\cap\{1,\dots,N\}}\frac{1}{n}
\end{equation*}
whenever the limit exists.
\end{compactenum}
\end{definition}
The theorem of Davenport and Erd\"os \cite{DE1936,DE1951} asserts that $\ddelta(\cM_\cB)=\underline{d}(\cM_\cB)$ exists for any subset $\cB\subseteq\N$.
\begin{definition}
$\cB\subseteq\N\setminus\{1\}$ is a \emph{Behrend sequence}, if $\ddelta(\cM_\cB)=1$.
\end{definition}
Recall that $\cB$ is {taut}, if $\ddelta(\cM_{\cB\setminus\{b\}})<\ddelta(\cM_\cB)$ for each $b\in\cB$.
The following is a corollary to a theorem of Behrend \cite{Behrend1948}:
\begin{proposition}
A set $\cB\subseteq\N$ is taut, if and only if it is primitive and there are no $q\in\N$ and no
Behrend set $\cA\subseteq \N\setminus\{1\}$ such that
$q\,\cA\subseteq\cB$ \cite[Corollary 0.19]{hall-book}.
\end{proposition}
This motivates the next definition:
\begin{definition}\label{def:pre-taut}
A set $\cB\subseteq\N$ is \emph{pre-taut}, if there are no $ q\in\N$ and Behrend set $\cA\subseteq\N\setminus\{1\}$ such that $q\,\cA\subseteq\cB$.
\end{definition}
\begin{lemma}\label{lemma:taut-facts}
Let $\cB\subseteq\N$ and $c\in\N$.
\begin{compactenum}[a)]
\item If $c\,\cB$ is pre-taut, then also $\cB$ is pre-taut. Moreover,
$\cB$ is
taut if and only if $c\,\cB$ is taut.
\item Each subset of a (pre-)taut set is (pre-)taut.
\item A finite union of pre-taut sets is pre-taut.
\item If $\cB$ is taut, then $\cB=\{1\}$ or $d(\cM_\cB)\neq1$ (possibly non-existing). Equivalently, if $d(\cM_\cB)=1$, then $\cB=\{1\}$ or $\cB$ is not taut.
\item If $\cB$ is pre-taut, then $1\in\cB$ or $d(\cM_\cB)\neq1$ (possibly non-existing). Equivalently, if $d(\cM_\cB)=1$, then $1\in\cB$ or $\cB$ is not pre-taut.
\end{compactenum}
\end{lemma}
\begin{proof}
a)\;The first implication is obvious. It is also clear that $\cB$ is primitive if and only if $c\cB$ is primitive. Moreover,
\begin{equation*}
\begin{split}
\cB\text{ is taut}
&\Leftrightarrow
\forall b\in\cB: \underline{d}(\cM_\cB)>\underline{d}(\cM_{\cB\setminus\{b\}})
\Leftrightarrow
\forall b\in\cB: c^{-1}\underline{d}(\cM_\cB)>c^{-1}\underline{d}(\cM_{\cB\setminus\{b\}})\\
&\Leftrightarrow
\forall b\in\cB: \underline{d}(\cM_{c\cB})>\underline{d}(\cM_{c(\cB\setminus\{b\})})
=\underline{d}(\cM_{c\,\cB\setminus\{cb\}})\\
&\Leftrightarrow
\forall b'\in c\,\cB: \underline{d}(\cM_{c\cB})>\underline{d}(\cM_{c\,\cB\setminus\{b'\}})\\
&\Leftrightarrow
c\,\cB\text{ is taut.}
\end{split}
\end{equation*}
{b) is obvious (see Remark 2.1)}.
\\
{c) follows from \cite[Corollary 0.14]{hall-book}, see also \cite[Proposition 2.33]{BKKL2015}}.
\\
d) Suppose that $\cB$ is taut. Then $\cB$ is primitive, and $d(\cM_\cB)\neq1$ unless $1\in\cB$ by \cite[Corollary 0.19]{hall-book}.
Hence $d(\cM_\cB)\neq1$ or $\cB=\{1\}$.\\
e) follows directly from Definition~\ref{def:pre-taut} \footnote{{Note that d) follows from e) and Remark 2.2}.}.
\end{proof}
\begin{remark}
$\cB$ is taut if and only if it is pre-taut and primitive. If $\cB$ is pre-taut, then $\prim{\cB}$ is taut in view of Lemma~\ref{lemma:taut-facts}b.
\end{remark}
For $q\in\N$ and $\cB\subseteq\Z$ let
\begin{equation*}
\cB'(q)=\left\{\frac{b}{\gcd(b,q)}:b\in\cB\right\},
\end{equation*}
and note that $1\in\cB'(q)$ if and only if $q\in\cM_\cB$.
\begin{lemma}\label{lemma:introductory}
Let $q\in\N$, $\cB,\cC\subseteq\Z$, and $q\,\cC\subseteq\cM_\cB$. Then
$\cM_\cC\subseteq\cM_{\cB'(q)}$.
\end{lemma}
\begin{proof}
Let $c\in \cC$. There are $\ell\in\Z$ and $b\in\cB$ such that $qc=\ell b$.
Since $q\mid\ell b$, it follows that $q\mid\ell \gcd(b,q)$, thus $k=\frac{\ell \gcd(b,q)}{q}$ is an integer. We have
\begin{equation*}
c=
\frac{\ell b}{q}
=
k\cdot\frac{b}{\gcd(b,q)}
\in\cM_{\cB'(q)}\ .
\end{equation*}
This shows that $\cC\subseteq\cM_{\cB'(q)}$ and hence also $\cM_\cC\subseteq\cM_{\cB'(q)}$.
\end{proof}
\begin{lemma}\label{lemma:q-taut}
Let $\cB\subseteq\N$ and $q\in\N$.
\begin{compactenum}[a)]
\item If $\cB$ is pre-taut, then $\cB'(q)$ is pre-taut.
\item If $\cB$ is taut, then $\cB'(q)$ is a finite disjoint union of taut sets $\cB_i'$ defined below in the proof {of a)}.
\item If $d(\cM_\cB)=1$, then $d(\cM_{\cB'(q)})=1$.
\item If $\cB=\bigcup_{i=1}^N\cC_i$ and if $d(\cM_\cB)=1$, then $d(\cM_{\cC_i})=1$ for at least one $i\in\{1,\dots,N\}$.
\end{compactenum}
\end{lemma}
\begin{proof}
Let $I:=\left\{\frac{q}{\gcd(b,q)}:b\in\cB\right\}$. For $i\in I$
denote $\cB_i:=\left\{b\in\cB:\frac{q}{\gcd(b,q)}=i\right\}$ and
$\cB_i':=\left\{\frac{b}{\gcd(b,q)}: b\in\cB_i\right\}$. Then $I$ is finite, $\cB=\bigcup_{i\in I}\cB_i$, and $\cB'(q)=\bigcup_{i\in I}\cB_i'$. Moreover,
$\cB_i=\left\{\frac{q}{i}b':b'\in\cB_i'\right\}=\frac{q}{i}\cB_i'$.\\
a)\; If $\cB$ is pre-taut, then all $\cB_i$ are pre-taut (Lemma~\ref{lemma:taut-facts}b), then all
$\cB_i'$ are pre-taut (Lemma~\ref{lemma:taut-facts}a), and then $\cB'(q)$ is pre-taut (Lemma~\ref{lemma:taut-facts}c).\\
b)\;If $\cB$ is taut, then all $\cB_i$ are taut (Lemma~\ref{lemma:taut-facts}b), and then all
$\cB_i'$ are taut (Lemma~\ref{lemma:taut-facts}a).\\
c)\;As $\cB\subseteq\cM_{\cB'(q)}$, we have also $\cM_\cB\subseteq\cM_{\cB'(q)}$.\\
d)\;If $\cB$ is Behrend, then at least one of the sets $\cC_i$ is Behrend \cite[Corollary 0.14]{hall-book}, and so ${d(\cM_{\cC_i})=1}$.
Otherwise $1\in\cB$, so that $1\in\cC_i$ for some $i$, whence $\cM_{\cC_i}=\Z$.
\end{proof}
\begin{lemma}\label{lemma:gen_of_4.25}(compare \cite[Proposition 4.25]{BKKL2015})
Assume that $\cB\subseteq\N$ is taut and $d(\cM_\cC)=1$ for some $\cC\subseteq\Z$. If
$q\,\cC\subseteq\cM_\cB$ for some $q\geqslant1$,
then $b\mid q$ for some $b\in\cB$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma:introductory}, $\cM_\cC\subseteq\cM_{\cB'(q)}$, so that $d(\cM_{\cB'(q)})=1$.
Then $\cB'(q)=\{1\}$ or $\cB'(q)$ is not taut (Lemma~\ref{lemma:taut-facts}d). If $\cB'(q)=\{1\}$, then $\gcd(b,q)=b$ for all $b\in\cB$, i.e $b\mid q$ for all $b\in\cB$, which is impossible because $\cB$ is infinite. Hence $\cB'(q)$ is not taut.
On the other hand, as $\cB$ is taut by assumption, $\cB'(q)$ is a finite union of taut sets $\cB_i'$ (Lemma~\ref{lemma:q-taut}b). As $d(\cM_{\cB'(q)})=1$, also $d(\cM_{\cB_i'})=1$ for at least one of the sets $\cB_i'$ (Lemma~\ref{lemma:q-taut}d), so that $\cB_i'=\{1\}$ for this set (Lemma~\ref{lemma:taut-facts}d). This implies $q\in\cM_\cB$.
\end{proof}
Recall that the topology on $H$ is generated by the (open and closed) cylinder sets
\begin{equation*}
U_S(h):=\{h'\in H: \forall b\in S: h_b=h'_b\},\text{ defined for finite }S\subset\cB\text{ and }h\in H\ ,
\end{equation*}
and recall also the definition of $\cA_S:=\{\gcd(b,\lcm(S)):b\in\cB\}$. Note that $\cA_S$ is finite and $S\subseteq\cA_\cS$.
\begin{lemma}\label{lemma:easy}
Let $U=U_S(\Delta(n))$ for some $S\subset\cB$ and $n\in\Z$.
\begin{compactenum}[a)]
\item If $n\in\cM_S$, then $U\cap W=\emptyset$.
\item If $U\cap W=\emptyset$, then $n+\lcm(S)\cdot\Z\subseteq \cM_{\cB\cap\cA_S}$.
\item There is a filtration of $\cB$ by finite sets $S$ for which $\cB\cap\cA_S= S$.
\item If $\cB\cap\cA_S= S$, then $n\in\cM_S$ iff $U\cap W=\emptyset$
iff $n+\lcm(S)\cdot\Z\subseteq\cM_S$.
\end{compactenum}
\end{lemma}
\begin{proof}
a)\; This follows immediately from the definitions of $U_S(\Delta(n))$ and $W$.\\
b)\; For each $h\in U$ there is $b\in\cB$ such that $h_b=0$.
As $U$ is compact, the Heine-Borel argument produces
a finite set $S'\subset\cB$ such that for each $h\in U$
there is $b\in S'$ such that $h_b=0$.
Let $s=\lcm(S)$. This observation applies in particular to all $h\in\Delta(n+s\Z)\subseteq U_{S}(\Delta(n))=U$. That means, for each $k\in\Z$ there is $b_k\in S'$ such that
$b_k\mid n+sk$. In other words: $n+s\Z\subset\cM_{S'}$.
The set $S'$ need not be primitive automatically, but we can replace it w.l.o.g. by a primitive subset without changing its set of multiples. Then,
as $S'$ is finite, it is taut. Denote $q=\gcd(n,s)$ and $\cC=\frac{n}{q}+\frac{s}{q}\cdot\Z$. Then $q\,\cC=n+s\Z\subseteq\cM_{S'}$, and as
$\gcd(n/q,s/q)=1$, $d(\cM_\cC)=1$ (Dirichlet, see \cite[Corollary 4.24]{BKKL2015}).
Now Lemma~\ref{lemma:gen_of_4.25} shows that $b\mid q=\gcd(n,s)$ for some $b\in S'$. In particular, $n\in b\Z$
and $b\mid s=\lcm(S)$ for that $b\in S'$, so that $b=\gcd(b,s)\in\cB\cap\cA_S$ and
$n+\lcm(S)\cdot\Z\subseteq b\Z\subseteq\cM_{\cB\cap\cA_S}$.\\
c)\; It suffices to prove that for any finite $S\subset\cB$ there exists a finite $S'\subset\cB$ with $\cB\cap\cA_{S'}=S'$. So let $S\subset\cB$ and $S':=\cB\cap\cA_S$. $S'$ is finite, because $\cA_S$ is finite, and obviously $S\subseteq S'\subseteq\cB\cap\cA_{S'}$. As each $b'\in S'\subseteq\cA_S$ divides $\lcm(S)$, also $\lcm(S')$ divides $\lcm(S)$. Therefore $\lcm(S')=\lcm(S)$, so that $\cA_{S'}=\cA_S$.
Hence $S'=\cB\cap\cA_{S'}$.
\\
d)\; This follows from a) and b).
\end{proof}
\subsection{Proof of Theorem~\ref{theo:taut-regular}}
Let $\cB=\{b_1,b_2,\dots\}$ be primitive, and denote $S_1\subset S_2\subset\dots\subset\cB$ a filtration of $\cB$ by finite sets $S_k$.
Let $s_k=\lcm(S_k)$.
We can assume without loss of generality that $b\mid s_k\Rightarrow b\in S_k$ holds for all $b\in\cB$ and all $k\in\N$.
For each $k\in\N$, the collection of all cylinder sets $U_{S_k}(h)$, $h\in H$, can be written explicitly as
\begin{equation*}
\cZ_k:=\left\{U_{S_k}(\Delta(n)): n=1,\dots,s_k\right\}.
\end{equation*}
Suppose first that $\cB$ is not taut. Then it contains a scaled copy $c\cA$ of a Behrend set $\cA\subseteq\{2,3,\dots\}$.
Enlarging $\cA$, if necessary, we can assume that $c\cA=\cB\cap c\Z$. (As $\cB$ is primitive, also the enlarged $\cA$ does not contain the number $1$.)
Let $a_0>1$ be the smallest element of $\cA$ and denote $b_0=ca_0$. Let $H_0=\{h\in H:h_{b_0}\in c\Z\}$. Then $H_0$ is open and closed, and we will show that $H_0\cap W\neq\emptyset$ but $m_H(H_0\cap W)=0$, so that $W$ is not Haar regular.
First observe that $(\Delta(c))_{b_0}=c\in c\Z$, so that $\Delta(c)\in H_0$. Suppose for a contradiction that $H_0\cap W=\emptyset$. Then $\Delta(c)\not\in W$, i.e. there is $b\in\cB$ such that $c\in b\Z$. Hence $c\cA\subseteq b\Z$, so that $c\cA=\{b\}$, because $b\in\cB$ and $\cB$ is primitive. Hence $b=ca_0=b_0$, so that $\cA=\{a_0\}$, a contradiction, as $\cA$ is Behrend.
We turn to the proof of $m_H(H_0\cap W)=0$.
Let $\cH_W^\ell=\{n\in\{0,\dots,s_\ell-1\}: U_{S_\ell}(\Delta(n))\cap H_0\cap W\neq\emptyset\}$. It suffices to show that $\sum_{n\in\cH_W^\ell}m_H(U_{S_\ell}(\Delta(n))\to0$ as $\ell\to\infty$. As all cylinder sets $U_{S_\ell}(\Delta(n))$ have identical Haar measure $s_\ell^{-1}$, this is equivalent to $\#\cH_W^\ell/s_\ell\to0$ as $\ell\to\infty$. So let $\ell$ be so large that $b_0\in S_\ell$.
Denote $\cA^\ell=\{a\in\cA:ca\mid s_\ell\}$.
As $c\cA\subseteq\cB$, the sequence $(\cA^\ell)_\ell$ is increasing and exhausts the set $\cA$.
If $n\in\cH_W^\ell$, then $n\in c\Z$ and, by Lemma~\ref{lemma:easy}a, $n\in\cF_{S_\ell}$. Hence $n=cn'\in\cF_{S_\ell}$ for some $n'\in\Z$. Suppose for a contradiction that $n'\in\cM_{\cA^\ell}$, i.e. there are $k\in\Z$ and $a\in\cA_\ell$ such that $n'=ka$. Then $n=kca$, where $ca\in\cB$ and $ca\mid s_\ell$, so that $ca\in S_\ell$, which contradicts $n\in\cF_{S_\ell}$. Hence $n'\in\cF_{\cA^\ell}$ so that $n\in c\cF_{\cA^\ell}=c\,(\Z\setminus\cM_{\cA^\ell})$. As $\cA$ is Behrend, $\bar{d}(\Z\setminus\cM_{\cA^\ell})\to0$ as $\ell\to\infty$. Hence
\begin{equation*}
\#\cH_W^\ell/s_\ell
\leqslant
\#\left(c(\Z\setminus\cM_{\cA^\ell})\cap[0,s_l)\right)/s_\ell
\le
\#\left((\Z\setminus\cM_{\cA^\ell})\cap[0,s_l)\right)/s_\ell
=
d(\Z\setminus\cM_{\cA^\ell})
\to0\ .
\end{equation*}
Suppose now that $\cB$ is taut.
We must show that for any $k\in\N$ and $U\in\cZ_k$
\begin{equation*}
U\cap W=\emptyset\quad\text{or}\quad m_H(U\cap W)>0\ .
\end{equation*}
So fix some $U=U_{S_k}(\Delta(n))$ such that $m_H(U\cap W)=0$. We have to show that $U\cap W=\emptyset$. Observe first that $U_{S_k}(\Delta(m))=U$ if and only if $m\in s_k\Z+n$.
For $\ell>k$ let
\begin{equation*}
\cG_\ell:=(s_k\Z+n)\cap
\left\{m\in\Z:
U_{S_\ell}(\Delta(m))\cap W=\emptyset\right\}
=
(s_k\Z+n)\cap \cM_{S_\ell},
\end{equation*}
where we used Lemma~\ref{lemma:easy}c for the last equality.
Observe that
\begin{equation*}
\cG_\ell
=
\cG_\ell+s_\ell\Z
=
\left(\cG_\ell\cap[0,s_\ell)\right)+s_\ell\Z\ .
\end{equation*}
Hence,
for each $\ell>k$,
\begin{equation*}
\begin{split}
\underline{d}\left((s_k\Z+n)\cap\cM_\cB\right)
&=
\liminf_{t\to\infty}\frac{\# \left(
(s_k\Z+n)\cap \cM_\cB\cap[0,t)
\right)}{t}
\geqslant
\liminf_{t\to\infty}\frac{\# \left(
(s_k\Z+n)\cap \cM_{S_\ell}\cap[0,t)
\right)}{t}\\
&=
\liminf_{t\to\infty}\frac{\# \left(\cG_\ell\cap[0,t)\right)}{t}
=
\frac{\# \left(\cG_\ell\cap[0,s_\ell)\right)}{s_\ell}.
\end{split}
\end{equation*}
As all $U'\in\cZ_\ell$ have identical Haar measure $m_H(U')=s_\ell^{-1}$ and as $m_H(U\setminus W)=m_H(U)$ by assumption, it follows that
\begin{equation*}
\begin{split}
\underline{d}\left((s_k\Z+n)\cap\cM_\cB\right)
&\geqslant
\limsup_{\ell\to\infty}\frac{\# \left(\cG_\ell\cap[0,s_\ell)\right)}{s_\ell}
=
\limsup_{\ell\to\infty}m_H\left(\bigcup_{U'\in\cZ_\ell,U'\subseteq U\setminus W}U'\right)\\
&=
m_H(U\setminus W)
=m_H(U)
=s_k^{-1}
=
d(s_k\Z+n)\ ,
\end{split}
\end{equation*}
so that
\begin{equation*}
d\left((s_k\Z+n)\cap\cM_\cB\right)=d(s_k\Z+n)\ .
\end{equation*}
Let $q=\gcd(s_k,n)$, $a'=s_k/q$ and $r'=n/q$. Then $\gcd(a',r')=1$ and
$q\Z\cap\cM_\cB=q\Z\cap\cM_{q\cdot\cB'(q)}$, in particular $(s_k\Z+n)\cap\cM_\cB=(s_k\Z+n)\cap\cM_{q\cdot\cB'(q)}$. Hence
\begin{equation*}
\begin{split}
d\left((a'\Z+r')\cap\cM_{\cB'(q)}\right)
&=
q\cdot d\left(q\cdot\left((a'\Z+r')\cap\cM_{\cB'(q)}\right)\right)
=
q\cdot d\left((s_k\Z+n)\cap \cM_{q\cdot\cB'(q)}\right)\\
&=
q\cdot d\left((s_k\Z+n)\cap \cM_{\cB}\right)
=
q\cdot d(s_k\Z+n)
=
q\cdot d\left(q(a'\Z+r')\right)\\
&=
d(a'\Z+r')
=1/a'\ .
\end{split}
\end{equation*}
In view of Lemma 1.17 in \cite{hall-book}, this suffices to conclude that
$\cB'(q)$ is Behrend.
On the other hand, as $\cB$ is taut,
$\cB'(q)$ is pre-taut (Lemma~\ref{lemma:q-taut}), so that $1\in\cB'(q)$ or $\cB'(q)$ is not Behrend (Lemma~\ref{lemma:taut-facts}e). Hence $1\in\cB'(q)$.
This implies
$q\in\cM_\cB$, which in turn implies
$U\cap W=U_{S_k}(\Delta(n))\cap W=\emptyset$ (the property to be proved):
Indeed, if $q\in\cM_\cB$, then there is some $b\in\cB$ with $b\mid q$, and as $q\mid s_k$, this implies $b\mid s_k$, so that $b\in S_k$. From $b\mid q\mid n$ we then conclude that $n\in\cM_{S_k}$, and Lemma~\ref{lemma:easy}a
implies $U_{S_k}(\Delta(n))\cap W=\emptyset$.
It remains to show that the implication $(i)$ $\Rightarrow$ $(iii)$ follows from Proposition~\ref{prop:mariusz-stronger}, which will be proved in the next subsection. So let $h\in W$.
By the proposition there exists $n\in\cF_\cB$ such that $\Delta(n)\in U_S(h)$, hence $\Delta(n)\in U_S(h)\cap(\Delta(\Z)\cap W)$. As this holds for all finite $S\subset\cB$, this proves the claim.
\subsection{Tautification of the set $\cB$ and regularization of the window $W$}
In \cite[Section 4.2]{BKKL2015} the authors provide a construction that associates to each (non-taut) set $\cB$ a taut set $\cB'$ such that
$\cF_{\cB'}\subseteq\cF_\cB$ but $\overline{d}(\cF_\cB\setminus\cF_{\cB'})=0$, and such that the two Mirsky measures
$\nu_\eta$ and $\nu_{\eta'}$
determined by $\cB$ and $\cB'$ coincide.
$\cB$ and $\cB'$ determine groups $H$ resp. $H'$ with windows $W$ resp. $W'$, and while the window $W$ is not Haar regular (if $\cB$ is non-taut), the window $W'$ is Haar regular because of Theorem~\ref{theo:taut-regular}.
On the abstract level one can also pass
from the window $W\subseteq H$ to its \emph{Haar regularization} $W_{reg}:=\supp(m_H|_W)$ (introduced in \cite{KR2016}), which also determines the same Mirsky measure on $\{0,1\}^\Z$. However, $W_{reg}$ will not be a window of the particular arithmetic type defined in \eqref{eq:W}, in particular it need not be aperiodic. The construction of $\cB'$ given $\cB$ in \cite{BKKL2015} suggests an obvious factor map $f:H\to H'$, and we expect that also $f(W_{reg})=W'$, so that in this sense the regularization of $W$ and the tautification of $\cB$ are two sides of the same medal.
The following example illustrates this discussion.
\begin{example}\label{ex:ape1}
Let $\cP=\{p_1,p_2,\ldots\}$ denote the set of primes. Let $\cB:=\bigcup_{i\geq1}p_i^2(\cP\setminus\{p_i\})$. Note that $\cB$ is primitive. It is not taut, because it contains rescalings of Behrend sets. The corresponding taut set is $\cB'=\{p_i^2:\:i\geq1\}$, which generates the square-free system.~\footnote{Note that $\eta(n)=1$ at all square-free numbers and also at $p_i^k$ for $i\geq1$ and $k\geq2$.}
\end{example}
\subsection{The property $\overline{\Delta(\Z)\cap W}=W$}\label{subsec:property(iii)}
\begin{proof}[Proof of Proposition~\ref{prop:mariusz-stronger}]
Given $h\in W$, we need to show that for each finite $S\subset\cB$
the set
\begin{equation*}
\cL_S(h)
:=
\left\{n\in\cF_\cB:\text{ $h_b=n$ mod~$b$ for each $b\in S$}\right\}
\end{equation*}
has asymptotic density $m_H(U_S(h)\cap W)>0$.
By Theorem~\ref{theo:taut-regular}, the tautness assumption on $\cB$ implies that $W$ is Haar regular, so that indeed
\begin{equation*}
m_H(U_S(h)\cap W)>0\ .
\end{equation*}
Let $\cB=\{b_1,b_2,\ldots\}$ and, for $K\geq 1$, $W_K:=\{g\in H:\:g_i\neq0\text{ for }i=1,\ldots,K\}$. Then $W_K$ is clopen and $W\subseteq W_K$. Moreover, $W_{K}\supseteq W_{K+1}$ and $\bigcap_KW_K=W$. Fix $\varepsilon>0$. We now choose $K\geq1$ so that
\begin{equation}\label{tcrt1}
m_H(W_K\setminus W)<\varepsilon\ .
\end{equation}
Since $U_S(h)\cap W_K$ is clopen (and $T$ is strictly ergodic)
\begin{equation}\label{tcrt2}
\left|\frac1N\sum_{n\leq N}\1_{U_S(h)\cap W_K}(T^n0)-m_H\left(U_S(h)\cap W_K\right)\right|<\varepsilon
\end{equation}
for all $N\geq N_0$. Moreover, we can choose $N_1$ so that for $N\geq N_1$, we also have
\begin{equation}\label{tcrt3}
\left|\frac1N\sum_{n\leq N}\1_{U_S(h)\cap W_K}(T^n0)-\frac1N\sum_{n\leq N}\1_{U_S(h)\cap W}(T^n0)\right|<\varepsilon\ .
\end{equation}
Indeed, if
\begin{equation*}
T^n0=\Delta(n)\in \left(U_S(h)\cap W_K\right)\setminus \left(U_S(h)\cap W\right)\subset W_K\setminus W\ ,
\end{equation*}
then (by setting $\cB_K=\{b_1,\ldots,b_K\}$), we have
\begin{equation*}
n\in \cF_{\cB_K}\cap \cM_{\cB}=\cM_{\cB}\setminus\cM_{\cB_K}\ .
\end{equation*}
Therefore, by the Davenport-Erd\"os theorem \cite[Eq.~(0.67)]{hall-book}, we can choose first $K\geq1$ sufficiently large so that $\overline{d}(\cM_{\cB}\setminus\cM_{\cB_K})<\varepsilon$ and then $N_1$ so that
\begin{equation*}
\frac1N\sum_{n\leq N}\1_{W_K\setminus W}(T^n0)
=
\frac1N\sum_{n\leq N}\1_{\cM_{\cB}\setminus\cM_{\cB_K}}(n)<\varepsilon
\end{equation*}
for all $N\geq N_1$, so in particular \eqref{tcrt3} holds.
In view of \eqref{tcrt1}, \eqref{tcrt2} and \eqref{tcrt3}, it follows that
\begin{equation*}
\lim_{N\to\infty}\frac1N\sum_{n\leq N}\1_{U_S(h)\cap W}(T^n0)=m_H(U_S(h)\cap W)\ .
\end{equation*}
As $T^n0=\Delta(n)\in U_S(h)\cap W$ if and only if $n\in\cL_S(h)$, this finishes the proof.
\end{proof}
\begin{example}
($\overline{\Delta(\Z)\cap W}=W$ does not imply tautness)\\
\label{example:non(iii)to(i)}
Suppose that $(m_k,r_k)$, ${k\in\N}$, is an enumeration of all coprime pairs of natural numbers. For any $k$ choose a prime $p_k\in r_k+m_k\Z$ such that $p_k>2^{k+1}$. Let $\cB=\cP\setminus\{p_k:k\in\N\}$. Clearly $\cB$ is primitive, and $\cM_{\{p_k:k\in\N\} }$ has upper density less than or equal to $\sum_{k=1}^{\infty}1/2^{k+1}=1/2$. Thus $d(\cM_{\cB})=1$ and $\cB$ is not taut \cite[Corollary 0.14]{hall-book}.
But $\overline{\Delta(\Z)\cap W}=W$. Indeed, let $h=(h_b)_{b\in\cB}\in W$ and take any finite set $S\subset \cB$. We are going to show that $U_S(h)\cap W\cap \Delta(\Z)\neq \emptyset$. Let $n\in\Z$ be such that $n= h_b$ mod $b$ for $b\in S$. Since $h\in W$, $b$ does not divide $n$ for any $b\in S$, i.e. $\lcm(S)$ and $n$ are coprime. Then $(\lcm(S),n)=(m_k,r_k)$ for some $k$, and the prime number $p_k$ belongs to arithmetic progression $r_k+m_k\Z=n+\lcm(S)\Z$, in other words $\Delta(p_k)\in U_S(\Delta(n))=U_S(h)$.
Finally, $\Delta(p_k)\in W$, because the prime number $p_k$ does not belong to $\cB$ and hence also not to $\cM_\cB$.
\end{example}
\subsection{$X_\eta$ and $X_\varphi$}
\label{subsec:light-tails}
The set $\cB\subseteq\N$ has \emph{light tails}, if
\begin{equation}
\lim_{K\to\infty}\overline{d}\left(\cM_{\{b\in\cB:b>K\}}\right)=0\ .
\end{equation}
If $\cB$ has light tails, then $\cB$ is taut, but the converse doses not hold \cite[Section 4.3]{BKKL2015}. Here we prove:
\begin{proposition}\label{prop:Mariusz4.2} If $\cB$ has light tails, then $X_\eta= X_\varphi$.
\end{proposition}
\begin{proof} Let $H=(h_k)\in H$ and $n\in\N$. We are going to show that $\varphi(h)[-n,n]=\eta[l+1,l+2n+1]$ for some $l\in\Z$. We know that $\varphi(h)(i)=1$ if and only if $h_j+i$ is not a multiple of $b_j$ for any $j\in\N$. For any $i\in[-n,n]$ such that $\varphi(h)(i)=0$ let $k_i$ be such that $b_{k_i}|h_{k_i}+i$.
Let $K\in\N$ be such that the set $\mathscr{A}:=\{b_1,\ldots,b_K\}$ contains $b_{k_i}$, for $i\in[-n,n]$ and any $b_k$ with $k>K$ has a prime factor $p>2n+1$.
Since $h\in H$, there exists $m\in Z$ such that
\begin{equation}\label{d1}
m=h_k \mod b_k
\end{equation}
for all $k\le K$. It follows that
\begin{equation*}
(\supp \varphi(h)\cap [-n,n])+m=[-n+m,n+m]\cap \cF_{\cA}
\end{equation*}
Indeed, if $i\in \supp \varphi(h)\cap [-n,n]$, then $h_k+i$ is not a multiple of $b_k$ for any $k\in \N$. By (\ref{d1}) we get that $m+i$ is not a multiple of $b_k$ for any $k\le K$, that is, $m+i\in \cF_{\cA}$. On the other hand, if $i\notin {\rm supp}
\varphi(h)\cap [-n,n]$, then $b_{k_i}|h_{k_i}+i$. Since $k_i\le K$, again by (\ref{d1}), we obtain $b_{k_i}|m+i$, that is $m+i\notin \cF_{\cA}$.
By \cite[Proposition 5.11]{BKKL2015} \footnote{Assume that $\cB\subset \N$ has light tails and $\cB^{(n)}\subset \cA\subset \cB$.
Suppose that
\begin{equation}\label{dziedz:as1}
\{k+1,\ldots,k+n\}\cap {\cal M}_{\cA}=\{k+i_0,k+i_1,\ldots,k+i_r\}
\end{equation}
for some $1\le i_0,\ldots,i_r\le n$, $r<n$. Then the density of $k'\in\N$ such that
$$
\{k'+1,\ldots,k'+n\}\cap {\cal M}_{\cB}=\{k'+i_0,k'+i_1,\ldots,k'+i_r\}
$$
is positive. (Here $
\cB^{(n)}:=\{b\in \cB : p\le n \text{ for any }p\in{\rm Spec}(b)\}
$. If $\cB$ is primitive, then $\cB^{(n)}$ is finite.)}
there exists $l\in\Z$ such that
\begin{equation*}
([-n+m,n+m]\cap \cF_{\cA})+l+n+1-m=[l+1,l+2n+1]\cap \cF_{\cB}
\end{equation*}
It follows that $\varphi(h)[-n,n]=\eta[l+1,l+2n+1]$.
\end{proof}
We now present a Behrend set (hence a non-taut set), for which $X_\eta$ is a strict subeset of $X_\varphi$.
\begin{example} Let $\cB=\{p_2,p_3,\ldots\}=\{3,5,7,11,\ldots\}$ - the set of all odd prime numbers. Since we are in the coprime case,
$$
H=\prod_{k=2}^\infty\Z/p_k\Z.$$
Now, $\eta=\varphi(\Delta(0))$ is the characteristic function of the ${\cB}$-free set $\{\pm 2^m:\:m\geq0\}$. We compute an initial block of $\varphi(h)$ for
$$
h=(0,1,0,0,\ldots)\in H.$$
We have $\varphi(h)(0)=0$,
$\varphi(h)(1)=1$, $\varphi(h)(2)=1$, $\varphi(h)(3)=0$, $\varphi(h)(4)=0$\footnote{If we add 4 to each coordinate of $h$, we obtain the sequence $(1,0,4,4,\ldots)$, whence $\varphi(h)(4)=0$.}, $\varphi(h)(5)=1$, $\varphi(h)(6)=0$, $\varphi(h)(7)=0$ and $\varphi(h)(8)=1$. It follows that the block $11001001$ appears on $\varphi(h)$. But there is no block $\underline{a}$ of length 8 appearing on $\eta$ and such that $11001001\le\underline{a}$. Indeed, the two neighboring 1's at the beginning of $\underline{a}$ could only appear at the positions 1,2 or -2,-1 in $\eta$. In the both cases this would force $\eta(5)=1$, which is not true. This shows that $\varphi(h)\not\in X_\eta$, although it belongs to $X_\varphi$.\footnote{Indeed, $\varphi(h)$ does not even belong to $\widetilde X_\eta$, the hereditary closure of $X_\eta$, see \cite{BKKL2015}.}
\end{example}
\begin{question} If $\cB$ is taut, is then $X_\eta=X_\varphi$?~\footnote{We recall that in case of $\cB$ taut, the Mirsky measure is supported on $X_\eta$.}
\end{question}
\section{Minimality/proximality of $X_\eta$ and topological properties of $W$}\label{sec:proof-theo-minimal}
Throughout this section we assume that $\cB$ is primitive.
\subsection{Arithmetic of $\cB$ and topology of $W$, part II}
\label{subsec:arithmetics-window}
Recall from \eqref{eq:A_S-def} that $\cA_S:=\{\gcd(b,\lcm(S)): b\in\cB\}$
and $\cF_{\cA_S}\subseteq\cF_\cB$
for $S\subset\cB$. If $S\subseteq S'\subset\cB$, then the following inclusions and implications are obvious:
\begin{equation}\label{eq:inclusions}
S\subseteq S'\subseteq\cA_{S'}\subseteq\cM_{\cA_S}
\Rightarrow\cM_S\subseteq\cM_{S'}\subseteq \cM_{\cA_{S'}}\subseteq\cM_{\cA_S}
\Rightarrow \cF_{\cA_S}\subseteq\cF_{\cA_{S'}}\subseteq\cF_{S'}\subseteq\cF_S\ .
\end{equation}
Let $\cE:=\bigcup_{S\subset\cB}\cF_{\cA_S}$ and observe that $\cE\subseteq\cF_\cB$.
\begin{lemma}\label{lemma:interiorW}
\begin{compactenum}[a)]
\item For all $S\subset\cB$ and $n\in\Z$ we have:
$U_S(\Delta(n))\subseteq W\Leftrightarrow n\in\cF_{\cA_S}$.
\item {If $(S_k)_k$ is a filtration of $\cB$ with finite sets and $\lim_k\Delta(n_{S_k})=h$ (see Remark \ref{limits}), then $h\in \inn(W)$ if and only if $n_{S_k}\in\cF_{\cA_{S_k}}$ for some $k$.}
\item For all $n\in\Z$ we have:
$\Delta(n)\in{\inn(W)}\Leftrightarrow n\in\cE$\ .
\item $\inn(W)=\emptyset\Leftrightarrow \cE=\emptyset
\Leftrightarrow \forall S\subset\cB: \cF_{\cA_S}=\emptyset
\Leftrightarrow \forall S\subset\cB: 1\in\cA_S$
\end{compactenum}
\end{lemma}
\begin{proof}
\begin{compactenum}[a)]
\item
As $U_S(\Delta(n))$ is clopen,
\begin{equation*}
\begin{split}
U_S(\Delta(n))\not\subseteq W
&\Leftrightarrow
\exists m\in n+\lcm(S)\cdot\Z\ \exists c\in\cB: c\mid m\\
&\Leftrightarrow
\exists c\in\cB\
\exists k\in\Z: c\mid n+k\cdot\lcm(S)\\
&\Rightarrow
\exists c\in\cB:\ \gcd(c,\lcm(S))\mid n\\
&\Leftrightarrow
\exists k\in\cA_S: k\mid n\\
&\Leftrightarrow
n\not\in\cF_{\cA_S}\ .
\end{split}
\end{equation*}
{That the only implication is also an equivalence is a consequence of the CRT. Indeed, if $\gcd(c,\lcm(S))\mid n$,
then there exist $k,l\in\Z$ such that $l\cdot c-k\cdot\lcm(S)=n$, thus $c\mid n+k\cdot\lcm(S)$. }
\item {Assume that $h\in \inn(W)$, that is $U_S(h)\subseteq W$ for some $S$. Then, for $k$ such that $S\subseteq S_k$, we have
$U_{S_k}(\Delta(n_{S_k}))=U_{S_k}(h)\subseteq W$, which is equivalent to $n_{S_k}\in\cF_{\cA_{S_k}}$ by a). Conversely, if $n_{S_k}\in\cF_{\cA_{S_k}}$ then, again by a), $U_{S_k}(\Delta(n_{S_k}))=U_{S_k}(h)\subseteq W$ and $h\in\inn(W)$. }
\item Follows from a).
\item Follows from c).
\end{compactenum}
\end{proof}
Recall from \eqref{def:Cinf} that
$\Ainf:=\{n\in\N: \forall_{S\subset\cB}\ \exists_{S': S\subseteq S'}: n\in\cA_{S'}\setminus S'\}$.
\begin{lemma}\label{lemma:ASk-C}
\begin{compactenum}[a)]
\item If $(S_k)_k$ is a filtration of $\cB$ with finite sets, then
\begin{equation*}
\limsup_{k\to\infty}\cA_{S_k}\setminus S_k=\Ainf\ .
\end{equation*}
\item For each $n\in\Ainf$ there is
a filtration $(S_k)_k$ of $\cB$ with finite sets such that
\begin{equation*}
n\in\bigcap_{k\in\N}\cA_{S_k}\setminus S_k\ .
\end{equation*}
\end{compactenum}
\end{lemma}
\begin{proof}
a) Assume that $n\in\cA_{S_k}\setminus S_k$ for infinitely many $k$, and let $S\subset\cB$.
Then there is $k$ such that $S\subseteq S_k$ and $n\in\cA_{S_k}\setminus S_k$. Hence $n\in\Ainf$.
Conversely, let $n\in\cC_{\infty}$. There is a finite set $S_1$ such that $n\in\cA_{S_1}\setminus S_1$. Assume that we have constructed sets $S_1\subset S_2\subset \ldots \subset S_k$ with the property that $n\in\cA_{S_i}\setminus S_i$ for $i=1,\ldots,k$ and
$\{1,\ldots,k\}\cap\cB\subset S_k$. Then there is a set $S_{k+1}$ containing $S_{k}\cup\{k+1\}$ and such that $n\in\cA_{S_{k+1}}\setminus S_{k+1}$. In this inductive way we construct a filtration $(S_k)_k$ as required.
\\
b) follows from a).
\end{proof}
\begin{lemma}\label{lemma:E=FBC}
The sets $\cE$ and $\Ainf$ are related by the identity
$$
\cE=\cF_{\cB\cup{\Ainf}}=\cF_\cB\cap\cF_{\Ainf}.
$$
\end{lemma}
\begin{proof}
Let $n\in\cE$ and chose $S$ such that $n\in\cF_{\cA_S}$. Take arbitrary $b\in\cB$ and $c\in \Ainf$. There exists a finite set $S'$ such that
$S\cup\{b\}\subseteq S'$ and $c\in\cA_{S'}\setminus S'$. Since $\cF_{\cA_S}\subseteq \cF_{\cA_{S'}}$, $n\in \cF_{\cA_{S'}}$, hence neither $b$ nor $c$ divides $n$.
We have proved that $\cE\subseteq \cF_{\cB\cup\Ainf}$.
In order to prove the other inclusion assume that $n\in \N$ and that for any $S$ there exists $c_S\in\cA_S$ dividing $n$.
As $n$ has only finitely many divisors, it has a divisor $c$ such that there exists a filtration $(S_k)_k$ of $\cB$ such that $c\in\cA_{S_k}$ for any $k\in \N$. If $c\notin\cB$, then $c\in\cA_{S_k}\setminus S_k$ for any $k\in \N$. This proves $n\notin\cF_{\cB\cup\Ainf}$.
\end{proof}
\begin{lemma}\label{lemma:E=FB}
$\Ainf=\emptyset$ if and only if $\cE=\cF_\cB$.
\end{lemma}
\begin{proof}
If $\Ainf=\emptyset$, then $\cE=\cF_\cB$ by Lemma~\ref{lemma:E=FBC}. Conversely, assume that
$\cE=\cF_\cB$. Then $\cF_\cB\subseteq\cF_{\Ainf}$ by Lemma~\ref{lemma:E=FBC}, so that
$\Ainf\subseteq\cM_{\Ainf}\subseteq\cM_\cB$. Suppose for a contradiction that there exists some $n\in\Ainf$. Then there is $b\in\cB$ such that $n\in b\Z$, i.e. $b\mid n$, and there is
a finite set $S=S_k\subset\cB$ such that
$n\in\cA_{S}\setminus S$, see Lemma~\ref{lemma:ASk-C}b. Hence there exists $b'\in\cB$ such that $n=\gcd(\lcm(S),b')$. It follows that $b\mid n\mid b'$, which is impossible, because $\cB$ is assumed to be primitive.
\end{proof}
\begin{proposition}\label{prop:filtation}
The following conditions are equivalent:
\begin{compactenum}[(i)]
\item $W\neq\overline{\inn(W)}$
\item For any filtration $S_0\subset S_1\subset\ldots \subset \cB$ of $\cB$ with finite subsets $S_k$, there exists a number $d$ such that $d\in \cA_{S_k}\setminus S_k$, for infinitely many $k\in\N$.
\item There exists a filtration $S_0\subset S_1\subset\ldots \subset \cB$ of $\cB$ with finite subsets $S_k$ and there exists a number $d$ such that $d\in \cA_{S_k}\setminus S_k$, for every $k\in\N$.
\item There are $d\in\N$ and an infinite pairwise
coprime set $\cA\subseteq\N\setminus\{1\}$ such that $d\,\cA\subseteq\cB$.
\end{compactenum}
\end{proposition}
\begin{proof}
$(i)\Rightarrow (ii)$: Let $h=(h_b)\in W\setminus \overline{\inn(W)}$. There exists $S$ such that $U_S(h)\cap \inn(W)=\emptyset$. We can assume that any $b\in \cB$ such that $b|\lcm(S)$, belongs to $S$.\footnote{Otherwise we can incorporate all such $b$'s into $S$, there are finitely many of them.} Let $n$ be a number such that
\begin{equation}\label{f1}
n= h_b \mod b
\end{equation}
for $b\in S$.
Then $\Delta(n+k\lcm(S))\in U_S(h)$, hence $\Delta(n+k\lcm(S))\notin \inn(W)$ for any $k\in\Z$. This means (see Lemma \ref{lemma:interiorW}) that for any finite set $T$, in particular for any $T=S_k$, the arithmetic progression $n+\lcm(S)\Z$ is contained in $\cM_{\cA_T}$. Since the set $\cA_T$ is finite, it follows that $\cA_T$ contains a divisor of $\gcd(n,\lcm(S))$\footnote{Apply Dirichlet theorem on primes in arithmetic progressions.}. There is only finitely many divisors of $\gcd(n,\lcm(S))$, hence one of them, denote it by $d$, appears in $\cA_{S_k}$ for infinitely many $k$. To finish the proof it is enough to observe that $d\notin\cB$ (consequently, $d\notin S_k$, for any $k$). Indeed, otherwise $d\in S$, by our assumption on $S$. Moreover, $d|n$ and then, by (\ref{f1}), $d|h_b$, where $b=d$, which leads to a contradiction with the assumption $h\in W$.
$(ii)\Rightarrow (iii)$: obvious
$(iii)\Rightarrow (i)$: Assume that $d\in\cA_{S_k}\setminus S_k$ for any $k$. Then $d\notin \cM_{\cB}$\footnote{Otherwise $d$ is divisible by some $b\in\cB$. On the other hand, $d$ divides some $b'\in\cB$ as a member of $\cA_{S_k}$, which in view of the fact that $\cB$ is primitive, leads to the conclusion that $d=b=b'\in\cB$. But it is not true, since $d\notin S_k$ for any $k$.}, hence $\Delta(d)\in W$. We prove that $\Delta(d)\notin\overline{\inn(W)}$. It is enough to show that $U_{S_0}(\Delta(d))\cap \inn(W)=\emptyset$.
Assume that $h=(h_b)\in U_{S_0}(\Delta(d))\cap \inn(W)$. It means that
\begin{equation}\label{f2}
d=h_b \mod b\; \mbox{\rm for any}\;b\in S_0\ ,
\end{equation}
and there exists a finite set $T\subset\cB$ such that $U_T(h)\subset W$. We can assume that $T=S_k$ for some $k$.
Let $m\in\Z$ be such that
\begin{equation}\label{f3}
m=h_b \mod b\; \mbox{\rm for any}\;b\in S_k\ .
\end{equation}
Let $c\in \cB$ be such that $\gcd(c,\lcm(S_k))=d$. {Clearly, $c\notin S_k$, since $d\notin \cB$.} Since $U_{S_k}(h)\subset W$, it follows
that there exists $b\in S_k$ such that
\begin{equation}\label{f4}
\gcd(c,b)\; \mbox{\rm does not divide}\; h_b \ ,
\end{equation}
{Indeed, otherwise there would exist $l\in\Z$ such that $l\equiv h_b$ mod $b$ for $b\in S_k$ and $l= 0$ mod $c$, hence $\Delta(l)\in U_{S_k}(h)$, but $\Delta(l)\notin W$, a contradiction.}
Thus, in view of (\ref{f3}),
\begin{equation}\label{f4a}
\gcd(c,b) \; \mbox{\rm does not divide}\; m\ .
\end{equation}
On the other hand,
\begin{equation}\label{f5}
\gcd(c,b)|\gcd(c,\lcm(S_k))=d\ .
\end{equation}
Since $d\in\cA_{S_0}\setminus S_0$ we get
\begin{equation}\label{f6}
d|\lcm(S_0)\ .
\end{equation}
By (\ref{f2}) and (\ref{f3}),
\begin{equation}\label{f7}
\lcm(S_0)|m-d\ .
\end{equation}
Now, (\ref{f5}), (\ref{f6}) and (\ref{f7}) imply $\gcd(c,b)|m$, a contradiction with (\ref{f4a}).
$(iii)\Rightarrow (iv)$: Assume that $d\in\cA_{S_k}\setminus S_k$ for any $k$. Then
\begin{equation}\label{eq:search-for-coprime}
\forall k\in\N\ \exists b_k\in\cB\setminus S_k:\ d=\gcd(b_k,\lcm(S_k))\ .
\end{equation}
As $d\not\in S_k$, we have $b_k\neq d$ for all $k$.
We choose a subsequence $b_{k_1},b_{k_2},\dots$ of $(b_k)_k$ in the following way: Let $k_1=1$, and given $k_1,\dots,k_j$, let
\begin{equation*}
k_{j+1}:=\min\left\{k\in\N: b_{k_1},\dots,b_{k_j}\in S_{k_{j+1}}\right\}.
\end{equation*}
Let $a_j=b_{k_j}/d$ for all $j\in\N$ and denote $\cA=\{a_j:j\in\N\}$.
Then $\cA\subseteq\N$ and $d\,\cA\subseteq \cB$ by construction.
Suppose that $1\in\cA$. Then $d\in\cB$, a contradiction to
\eqref{eq:search-for-coprime}, as $\cB$ is primitive. Hence $\cA\subseteq\N\setminus\{1\}$.
It remains to prove that
$\cA$ is pairwise coprime.
Suppose for a contradiction that there is a prime number $p$ dividing some $a_i$ and $a_j$, $i<j$. Then $pd\mid b_{k_i}$ and $pd\mid b_{k_j}$.
As $b_{k_i}\in S_{k_j}$, it follows that $pd\mid\lcm(S_{k_j})$, so that $pd\mid\gcd(b_{k_j},\lcm(S_{k_j}))=d$ (see \eqref{eq:search-for-coprime}), which is impossible.
$(iv)\Rightarrow (iii)$: Let $d\in\N$ and $\cA=\{a_1<a_2<\dots\}$ be as in $(iv)$. Then $d\not\in\cB$, because $\cB$ is primitive.
For $k\in\N$ let $S_k=\cB\cap\{1,\dots,k\}\cup\{da_k\}$. As all $a_j$ are pairwise coprime, there are $j_1<j_2<\dots\in\N$ such that $a_{j_k}$ is coprime to $\lcm(S_k)$. On the other hand, $d\mid\lcm(S_k)$. Hence $d=\gcd(da_{j_k},\lcm(S_k))\in\cA_{S_k}$. As $d\not\in\cB$, we see that $d\in\cA_{S_k}\setminus S_k$ for all $k\in\N$.
\end{proof}
\begin{proposition}\label{prop:Staszek-prop-equiv}
The following conditions are equivalent:
\begin{compactenum}[(i)]
\item $W$ is topologically regular, i.e. $W=\overline{\inn(W)}$.
\item There are no $d\in\N$ and no infinite pairwise
coprime set $\cA\subseteq\N\setminus\{1\}$ such that $d\,\cA\subseteq\cB$.
\item $\Ainf=\emptyset$.
\item $\cE=\cF_\cB$.
\end{compactenum}
\end{proposition}
\begin{proof}
The equivalence of $(i)$ and $(ii)$ follows from Proposition~\ref{prop:filtation},
that of $(iii)$ and $(iv)$ from Lemma~\ref{lemma:E=FB}..
In view of Lemma~\ref{lemma:ASk-C},
Proposition~\ref{prop:filtation} finally implies the equivalence of $(i)$ and $(iii)$, too.
\end{proof}
\begin{lemma}\label{lemma:DeltaZ-1}
$\Delta(\Z)\cap\left(\overline{\inn(W)}\setminus\inn(W)\right)=\emptyset$.
\end{lemma}
\begin{proof}
Assume $\Delta(m)\in \overline{\inn(W)}\setminus\inn(W)$. {Then for any $S\subset \cB$ there exists $n_S\in\Z$ such that $\Delta(n_S)\in U_S(\Delta(m))\cap \inn(W)$.} It means that for any $S$ there exist: a finite set $T_S\subset \cB$, (we can assume that $S\subset T_S$), $b_S\in\cB$ and $n_S\in\Z$ such that (see Lemma \ref{lemma:interiorW} c)):
\begin{compactenum}[$\bullet$]
\item $\lcm(S)|m-n_s$ (that is, $\Delta(n_s)\in U_S(\Delta(m))$)
\item $\gcd(b_S,\lcm(T_S))$ does not divide $n_S$ ($\Delta(n_S)$ is chosen to be an element of $U_S(\Delta(m))\cap \inn(W)$)
\item $\gcd(b_S,\lcm(T_S))|m$ (since $\Delta(m)\notin \inn(W)$)
\end{compactenum}
Then $\gcd(b_S,\lcm(T_S))$ does not divide $\lcm(S)$.
Let us iterate: $S_0$ is arbitrary and $S_{k+1}:=T_{S_k}$, $c_k:=\lcm(S_{k+1})$, $d_k:=\gcd(b_{S_k},\lcm(S_{k+1}))$.
We have:
\begin{compactenum}[$\bullet$]
\item $c_k|c_{k+1}$
\item $d_k|m$
\item $d_k|c_k$
\item $d_k$ does not divide $c_{k-1}$
\end{compactenum}
{ Since $d_k|m$ for every $k$, the sequence $(\lcm(d_1,\ldots d_k))_k$ stabilizes on $\lcm(d_1,\ldots d_{k_0})$ for some $k_0$, which means $d_l$ divides $\lcm(d_1,\ldots d_{k_0})$, and consequently $d_l$ divides $\lcm(c_1,\ldots,c_{k_0})=c_{k_0}$, for any $l$, a contradiction.}
\end{proof}
For $x\in\{0,1\}^\Z$ denote $\supp x:=\{n\in\Z: x(n)=1\}$. Following \cite{BKKL2015} we consider the set
\begin{equation*}
\begin{split}
Y
:=&
\left\{x\in\{0,1\}^\Z: |\supp x\text{ mod }b|=b-1\text{ for all }b\in\cB\right\}\\
=&
\left\{x\in\{0,1\}^\Z: \text{for all }b\in\cB\text{ there is exactly one }r\in\{0,\dots,b-1\}\text{ with }\supp x\cap(b\Z+r)=\emptyset\right\}.
\end{split}
\end{equation*}
As $\supp\eta=\cF_\cB$ is disjoint from $b\Z$ for all $b\in\cB$, we have
\begin{equation}\label{eq:eta-in-Y}
\eta\in Y
\Leftrightarrow
\forall b\in\cB\ \forall r\in\{1,\dots,b-1\}: \cF_\cB\cap (b\Z+r)\neq\emptyset\ .
\end{equation}
\begin{lemma}\label{lemma:Y-eta}
If $\overline{\Delta(\Z)\cap W}=W$, then $\eta\in Y$.
\end{lemma}
\begin{proof}
For $b\in\cB$ and $r\in\{0,\dots,b-1\}$ let $V_b(r):=\{h\in H: h_b=r\}$ and observe that these sets are open and closed in $H$.
Hence $\overline{\Delta(\Z)\cap V_b(r)\cap W}=V_b(r)\cap W$, because $\overline{\Delta(\Z)\cap W}=W$.
Suppose for a contradiction that $\eta\not\in Y$. Then (\ref{eq:eta-in-Y}) implies
that there are $b\in\cB$ and $r\in\{1,\dots,b-1\}$ such that
\begin{equation*}
\Delta(\Z)\cap V_b(r)\cap W=\emptyset\ ,
\end{equation*}
which implies that also $V_b(r)\cap W=\emptyset$. Hence
\begin{equation*}
V_b(r)\subseteq
W^c=
\bigcup_{b'\in\cB}V_{b'}(0)\ ,
\end{equation*}
and as $V_b(r)$ is compact and the $V_{b'}(0)$ are open, there is a finite $S\subset\cB$ such that
\begin{equation*}
V_b(r)\subseteq
\bigcup_{b'\in S}V_{b'}(0)\ ,
\end{equation*}
In other words, whenever $h_b=r$ for some $h\in H$, then $h_{b'}=0$ for some $b'\in S$.
Applied to any $h=\Delta(n)$ this yields:
\begin{equation*}
n\in b\Z+r\;\Rightarrow\;n\in\bigcup_{b'\in S}b'\Z\ .
\end{equation*}
{Since $r$ is not divisible by $b$, we can assume that $b\notin S$.}
Let $q=\gcd(b,r)$, $\tilde{b}=b/q$, $\tilde{r}=r/q$. Then $q\,(\tilde{b}\Z+\tilde{r})=b\Z+r\subseteq\cM_S$, so that $\cM_{\tilde{b}\Z+\tilde{r}}\subseteq\cM_{S'(q)}$ by Lemma~\ref{lemma:introductory}. But $d(\cM_{\tilde{b}\Z+\tilde{r}})=1$ by Dirichlet's theorem, whereas $d(\cM_{S'(q)})<1$, because $S'(q)\subseteq\{1,\dots,\max S\}$ is finite {and $1\notin S'(q)$} \footnote{{As $q|b$ and $\cB$ is primitive, $q\notin S$, thus $1\notin S'(q)$.}}. This is a contradiction.
\end{proof}
\begin{remark}
Together with Theorem~\ref{theo:taut-regular} this shows that
$\eta\in Y$ whenever $\cB$ is taut. This implication was proved previously in \cite[Corollary 4.27]{BKKL2015}.
\end{remark}
Lemma~\ref{lemma:Y-eta} provides the implication
\begin{equation*}
\overline{\Delta(\Z)\cap W}=W\Rightarrow \eta\in Y.
\end{equation*}
The reverse implication does not hold, as is shown by the next example.
\begin{example} Observe that for every $k\in\Z$ there exists a prime divisor $p_k$ of $5+12k$ such that
\begin{equation}\label{ex_1}
p_k\neq 1\mod 12 \;\;\text{and}\;\;p_k\neq -1\mod 12
\end{equation}
Let $$\cB=\{4,6\}\cup\{p_k:k\in\Z\}$$
Let us enumerate the elements of $\cB$ as $b_0,b_1,b_2,\ldots$ and $b_0=4, b_1=6$.
Observe that
\begin{equation}\label{ex_2}
5+12\Z\subset\cM_{\cB}
\end{equation}
Since niether 2 nor 3 divides an element of the progression $5+12\Z$, in view of (\ref{ex_1}) we see that $1,2,3,11,22\in\cF_{\cB}$. It follows that
\begin{equation}\label{ex_3}
|\supp\cF_{\cB} \mod 4|=3\;\;\text{and} \;\; |\supp\cF_{\cB} \mod 6|=5
\end{equation}
We claim that
\begin{equation}\label{ex_4}
|\supp\cF_{\cB} \mod b_k|=b_k-1\;\text{for any} \; k\ge 2
\end{equation}
It is clear that $\gcd(12,b_k)=1$ for any $k\ge 2$. Let $k\ge 2$ and take arbitrary $r\in\{1,\ldots,b_k-1\}$. There exists $r'\in\Z$ such that
\begin{equation}\label{ex_5}
\left\{\begin{array}{c}
r'\equiv r\mod b_k\\
r'\equiv 1 \mod 12
\end{array}\right.
\end{equation}
Then $\gcd(12b_k,r')=1$ and, by Dirichlet Theorem, there exists a prime number $q$ of the form $q=12b_kl+r'$ for some $l\in\Z$. Since, by (\ref{ex_5}), $q\equiv 1\mod 12$, $q\in\cF_{\cB}$ by (\ref{ex_1}). Moreover, $q\equiv r\mod b_k$ by (\ref{ex_5}).
Thus the claim (\ref{ex_4}) follows. Clearly, (\ref{ex_4}) and (\ref{ex_3}) yield $\eta\in Y$.
We shall construct $h\in W$ such that $h\notin\overline{\Delta(\Z)\cap W}$. We denote $S_k=\{b_0,b_1,\ldots b_k\}$. Inductively we construct a sequence $(n_{S_k})$ of integers satisfying:
\begin{compactenum}[a)]
\item $n_{S_1}=5$
\item $\lcm(S_k)|n_{S_{k+1}}-n_{S_k}$ for $k=1,2,\ldots$
\item $n_{S_k}\in\cF_{S_k}$ for $k=1,2,\ldots$
\end{compactenum}
Assume that $n_{S_1},\ldots,n_{S_k}$ have been constructed. If $b_{k+1}$ does not divide $n_{S_k}$, we set $n_{S_{k+1}}=n_{S_k}$. Otherwise we set
$n_{S_{k+1}}=n_{S_k}+\lcm(S_k)$. The conditions a), b), c) follow easily by induction.
Let
$$
h=\lim_k\Delta(n_{S_k})
$$
Thanks to c), $h\in W$.
But
$$
U_{S_1}(h)\cap\Delta(\Z)\cap W=U_{S_1}(\Delta(5))\cap\Delta(\Z)\cap W=\Delta(5+\12\Z)\cap W=\emptyset
$$
the last equality by (\ref{ex_2}). (Clearly, $d(\cM_{\cB})=1$ and $\cB$ is not taut.)
\end{example}
\subsection{Proof of Theorem~\ref{theo:minimality}}\label{subsec:proof-theo-minimality}
\begin{lemma}\label{lemma:Aurelia}
\footnote{The authors are indebted to A.~Bartnicka for pointing out and proving this lemma.}
If $\cB$ is primitive and $\eta$ is a Toeplitz sequence, then $\cB$ is taut.
\end{lemma}
\begin{proof}
Suppose that $\cB$ is not taut. Then there are $c\in\N$ and a Behrend set $\cA$ such that $c\cA\subseteq \cB$. Hence
\begin{equation}\label{eq:Aurelia}
d(\cM_\cB\cap c\Z)=c^{-1}\ ,
\end{equation}
because $\cM_\cA$ has density one. As $\cB$ is primitive, $c$ must be $\cB$-free.
So $\eta(c)=1$, and (since $\eta$ is Toeplitz) there exists $m\in\N$ such that $c+m\Z\subseteq \cF_\cB$.
But then
\begin{equation*}
\underline{d}(\cF_\cB\cap c\Z)
\geqslant
\underline{d}((c+m\Z)\cap c\Z)
=
d(\lcm(c,m)\Z)
=\lcm(c,m)^{-1}>0,
\end{equation*}
which contradics \eqref{eq:Aurelia}.
\end{proof}
\begin{lemma}\label{lemma:almost-periodic-Y}
Assume that $\eta\in Y$. If $\eta=\1_{\cF_\cB}$ is almost periodic (i.e. if the orbit closure of $\eta$ is minimal), then
$X_\eta\subseteq Y$.
\end{lemma}
\begin{proof}
Fix $k\geq1$. Since $\eta\in Y$, the support of $\eta$ taken mod $b_k$ misses exactly one residue class mod $b_k$ (that is, it misses zero). Let $B$ be a block on $\eta$ such that its support mod $b_k$ misses exactly one residue class mod $b_k$. Since $\eta$ is almost periodic, the block $B$ appears on $\eta$ with bounded gaps. It follows that if $C$ is any sufficiently long block that appears on $\eta$, its support misses exactly one residue class. Clearly this property passes to limits in the product topology, so each $y=\lim S^{m_i}\eta$ is also in $Y$.
\end{proof}
In general, we can define a map $\theta:Y\to\prod_{k\geq1}\Z/b_k\Z$ by setting
\begin{equation*}
\theta(y)=g=(g_k)_{k\geq1}\text{ iff } \supp y\cap(b_k\Z-g_k)=\emptyset\text{ for all }k\geq1.
\end{equation*}
Remark 2.51 in \cite{BKKL2015} tells us that
\begin{equation*}
\theta(Y\cap X_\eta)\subset H\ ,
\end{equation*}
while Remark 2.52 says that $\theta$ is continuous.
\begin{corollary}\label{coro:theta}
By the definitions of $\varphi$ and $\theta$, we have $\theta\circ\varphi(h)=h$ provided $\varphi(h)\in Y$. In particular, $\theta(\eta)=0$ and $\theta$ is continuous at $\eta$. Moreover, $\theta$ is equivariant.
\end{corollary}
For any map $\psi:X\to Y$ denote by $C_\psi\subseteq X$ the set of continuity points of this map.
\begin{lemma}\label{lemma:minimality}
Let $(X,S)$ and $(Y,T)$ be compact dynamical systems and assume that $(X,S)$ is minimal. Let $\psi:X\to Y$ be a map satisfying $\psi\circ S=T\circ\psi$. Then $\overline{\psi(C_\psi)}$ is a minimal subset of $Y$.
\end{lemma}
\begin{proof}
Denote by $Z:=\overline{\{(x,\psi(x)): x\in X\}}$ the closure of the graph of $\psi$ and note that a fibre $Z_x=\{(x,y):y\in Z\}$ is a singleton, if and only if $x\in C_\psi$. Let $Z_0:=\overline{\{(x,\psi(x)): x\in C_\psi\}}$. We claim that $Z_0\subseteq A$ whenever $A$ is a non-empty closed $S\times T$-invariant subset of $Z$. Indeed, $\pi_X(A)$ is a non-empty closed $S$-invariant subset of $X$, so $\pi_X(A)=X$ by minimality of $(X,S)$. In particular, $C_\psi\subseteq\pi_X(A)$. As all $A_x\subseteq Z_x$ with $x\in C_\psi$ are singletons, $\{(x,\psi(x)):x\in C_\psi\}\subseteq A$. Hence also $Z_0\subseteq A$.
This shows that $Z_0$ is a minimal subset of $X\times Y$ (and, by the way, that it is the only minimal subset of $Z$). It follows that $\pi_Y(Z_0)$ is a minimal subset of $Y$, and so it remains to show that $\psi(C_\psi)\subseteq \pi_Y(Z_0)$. But, for $x\in C_\psi$, $(x,\psi(x))\in Z_0$, and so $\psi(x)\in\pi_Y(Z_0)$.
\end{proof}
Denote by $C_\varphi$ the set of all points in $H$ at which $\varphi:H\to\{0,1\}^\Z$ is continuous.
\begin{lemma}\label{lemma:C_varphi}
\begin{compactenum}[a)]
\item $C_\varphi=\left\{h\in H:\ (h+\Delta(\Z))\cap\partial W=\emptyset\right\}$.
\item $C_\varphi+\Delta(1)=C_\varphi$.
\item $\overline{\varphi(C_\varphi)}$ is the unique minimal subset $M$.
\end{compactenum}
\end{lemma}
\begin{proof}
a) This is proved by direct inspection, see e.g. \cite[Lemma 6.1]{KR2015}.\\
b) This is obvious.\\
c) This follows from Lemma~\ref{lemma:minimality}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:minimality}]
We start with a list of implications, which, when suitably combined, prove the assertions a) - e) of Theorem~\ref{theo:minimality}.
Most of these implications can be proved without assuming that $\cB$ is primitive and that $\overline{\Delta(\Z)\cap W}=W$.
Therefore we indicate explicitly, for which implications we use these extra assumptions.
\\[2mm]
\emph{Proof of the equivalence of B1 -- B4:}
These equivalences follow from Proposition~\ref{prop:Staszek-prop-equiv}.
\\[2mm]
\emph{Proof of B1 $\Rightarrow$ B6:}
Observe first that $0\in H$ belongs to $C_\varphi$ if and only if $\Delta(\Z)\cap\partial W=\emptyset$, see Lemma~\ref{lemma:C_varphi}. But $\Delta(\Z)\cap\partial W=\Delta(\Z)\cap\left(\overline{\inn(W)}\setminus\inn(W)\right)$ in view of B1, and this intersection is empty by Lemma~\ref{lemma:DeltaZ-1}. As $\inn(W)\neq\emptyset$ and as $H=\overline{\Delta(\Z)}$, $\Delta(\Z)\cap W\neq\emptyset$ and hence $\varphi(0)\neq(\dots,0,0,0,\dots)$.
\\[2mm]
\emph{Proof of B6 $\Rightarrow$ B5:}
Let $\cB=\{b_1,b_2,\ldots\}$ and assume (B6) that $0\in C_\varphi$,
i.e. $\Delta(\Z)\cap\partial W=\emptyset$, and $\eta\neq(\dots,0,0,0,\dots)$.
Now, take $n\in\Z$. Either $n\in\cM_{\cB}$ - then $\eta(n)=0$, so $b_s\mid n$ for some $s\geq1$ and $\eta(n+jb_s)=0$ for each $j\in\Z$. Or
$n\in \cF_{\cB}$, i.e. $\Delta(n)\in W$. As $\Delta(\Z)\cap\partial W=\emptyset$ by assumption, this implies $\Delta(n)\in\inn(W)$, so that $n\in\cE=\bigcup_{S\subset\cB}\cF_{\cA_S}$ by Lemma~\ref{lemma:interiorW}. Hence there is a finite subset $S\subset\cB$ such that $n\in\cF_{\cA_S}$. As $\lcm(\cA_S)=\lcm(S)$, this implies
\begin{equation*}
n+\lcm(S)\,\Z\subseteq\cF_{\cA_S}\subseteq\cE\subseteq\cF_\cB\ .
\end{equation*}
Hence $\eta(n+j\lcm(S))=1$ for each $j\in\Z$. This proves that $\eta$ is a Toeplitz sequence
different from $(\dots,0,0,0,\dots)$.
\\[2mm]
\emph{Proof of B5 $\Rightarrow$ B1:}
Assume that $\eta$ is a Toeplitz sequence. Then $\cB$ is taut by Lemma~\ref{lemma:Aurelia}, hence $\overline{\Delta(\Z)\cap W}=W$ by Theorem~\ref{theo:taut-regular}. Now B1 follows from the chain of the next three implications.
\\[2mm]
\emph{Proof of B5 $\Rightarrow$ B8:}
Each Toeplitz sequence is almost periodic \cite{Do}, \cite[Theorem 4]{JK1969}, i.e. its orbit closure is minimal.
\\[2mm]
\emph{Proof of B8 $\Rightarrow$ B7:}
If $X_\eta=M$, then $\eta\in M$, and $\eta\neq(\dots,0,0,0,\dots)$, because otherwise the minimality of $X_\eta$ implies $X_\eta=\{(\dots,0,0,0,\dots)\}$, contradicting $\card(X_\eta)>1$.
\\[2mm]
\emph{Proof of B7 $\Rightarrow$ B1 (assuming that $\overline{\Delta(\Z)\cap W}=W$):}
Assume that $(\dots,0,0,0,\dots)\neq\eta\in M=\overline{\varphi(C_\varphi)}$.
Then $M=X_\eta\subseteq Y$ by Lemma~\ref{lemma:almost-periodic-Y}, and
there is a sequence $h_1,h_2,\dots\in C_\varphi$ such that $\eta=\lim_{i\to\infty}\varphi(h_i)$. Consider $n\in\Z$ with
$\Delta(n)\in W$, i.e. such that $\eta(n)=1$. In particular $\eta=\varphi(0)\neq(\dots,0,0,0,\dots)$. Corollary~\ref{coro:theta} implies
$\lim_{i\to\infty}h_i=\lim_{i\to\infty}\theta(\varphi(h_i))=\theta(\eta)=0$. Then
$1=\eta(n)=\lim_{i\to\infty}\varphi(h_i)(n)$, i.e. $h_i+\Delta(n)\in W$ for all sufficiently large $i$. As $h_i\in C_\varphi$, we have $h_i+\Delta(\Z)\cap\partial W=\emptyset$ (Lemma~\ref{lemma:C_varphi}). Hence
$h_i+\Delta(n)\in \inn(W)$ for all sufficiently large $i$, what implies
that $\Delta(n)=\lim_{i\to\infty}h_i+\Delta(n)\in\overline{\inn(W)}$.
This proves that $\Delta(\Z)\cap W\subseteq\overline{\inn(W)}$.
Hence
$W=\overline{\Delta(\Z)\cap W}\subseteq\overline{\inn(W)}$, i.e. $W$ is topologically regular.
\\[2mm]
\emph{Proof of B7 $\Rightarrow$ B8:}
As $\eta\in M$, also $X_\eta\subseteq M$, and hence $X_\eta= M$. As $\eta\neq(\dots,0,0,0,\dots)$, $X_\eta$ contains no fixed point. Hence $\card(X_\eta)>1$.
\\[2mm]
\emph{Proof of B1 $\Rightarrow$ B9 (assuming that $\cB$ is primitive):}
The window $W$ is aperiodic because of Proposition~\ref{p:ape3}, and it is topologically regular by B1. As B1 $\Rightarrow$ B8, $X_\eta$ is minimal. Therefore Corollary 1a) of \cite{KR2015}, together with Lemmas 4.5 and 4.6 of the same reference, implies B9.
\\[2mm]
\emph{Proof of B9 $\Rightarrow$ B8:}
This is trivial.
\end{proof}
\begin{proposition}\label{eta_in_Y}
Assume that the window $W$ is topologically regular. Then $X_\eta\subseteq Y$.
\end{proposition}
\begin{proof}
We start proving that $\eta\in Y$. Assume
the contrary, that is, there are $b_0\in\cB$ and $r\in\{1,\ldots,b_0-1\}$ such that
\begin{equation}\label{eta_in_Y1}
r+b_0\Z\subset\cM_{\cB}.
\end{equation}
Let $a=\gcd(r,b_0)$ and $r'=r/a$, $b_0'=b_0/a$.
(\ref{eta_in_Y1}) yields that for any $k\in \N$ there exists $b_k\in\cB$ such that
\begin{equation*}
b_k\mid a(r'+kb_0').
\end{equation*}
Let $J=\{k\in\N:r'+kb_0'\;\text{is prime}\}$. By Dirichlet Theorem the set $J$ is infinite. As $\cB$ is primitive, $b_k$ does not divide $a=\gcd(r,b_0)$. Hence
$$
b_k=\gcd(a,b_k)\,(r'+kb_0')
$$
for any $k\in J$.
Since $a$ has only finitely many divisors, there exists a divisor $a'$ such that
$$
b_k=a'(r'+kb_0')
$$
for infinitely many $k\in J$. Thus we obtain a contradiction with the condition (B4) of Theorem~\ref{theo:minimality}, which is equivalent to (B1) $W=\overline{\inn W}$. Thus $\eta\in Y$.
Assume now that $x\in X_\eta$ and let $b\in\cB$.
As $\eta\in Y$, there is $N_b\in\N$ such that $\card\left(\supp(\eta|_{[0:N_b]})\mod b\right)=b-1$. As $X_\eta$ is minimal by (B8) of Theorem~\ref{theo:minimality}, there is
$n\in\N$ such that $\supp(x|_{[n:n+N_b]})=\supp(\eta|_{[0:N_b]})$. Hence
\begin{displaymath}
\card\left(\supp(x)\mod b\right)\geqslant \card\left(\supp(x|_{[n:n+N_b]})\mod b\right)
\geqslant\card\left(\supp(\eta|_{[0:N_b]})\mod b\right)=
b-1\ ,
\end{displaymath}
so that $x\in Y$, because $\card\left(\supp(x)\mod b\right)\leqslant b-1$ for all $x\in X_\eta$, see Footnote~\ref{foot:b-1} to Remark~\ref{rem:Y}.
\end{proof}
\subsection{Proof of Theorem~\ref{theo:proximal}}\label{subsec:proof-theo-proximal}
The equivalence of C1, C2 and C3 follows from
Lemma~\ref{lemma:interiorW}.
If C1 holds, i.e. if $\inn(W)=\emptyset$, then $\varphi(C_\varphi)=\{(\dots,0,0,0,\dots)\}$ is a shift invariant set \cite[Proposition 3.3d with Remark 3.2b]{KR2015}, so that $M=\overline{\varphi(C_\varphi)}=\{(\dots,0,0,0,\dots)\}$. This is C6, and Theorem 3.8 in \cite{BKKL2015} shows that C4, C5, C6 and C7 are all equivalent.
We finish by proving C5 $\Rightarrow$ C3: Consider any finite $S\subset \cB$. As $\cB\subseteq\cM_{\cA_S}$ by definition of the set $\cA_S$, C5 implies that $1\in\cA_S$.
\section{The sequence $\cB$ and Haar measure}\label{sec:B-Haar}
\subsection{Measure and density}\label{subsec:measure-density}
\begin{lemma}
$m_H(W)=1-\underline d(\cM_\cB)=\bar d(\cF_\cB)$.
\end{lemma}
\begin{proof}
For $S\subset\cB$ denote by
$\cU_S$
the family of all sets $U_S(\Delta(n))$ that are contained in $W^c$ and by $\cup\cU_S$ the union of these sets. Then
\begin{equation*}
m_H(W^c)
=
\sup_S m_H(\cup\cU_S)
=
\sup_S
\frac{\#\cU_S}{\lcm(S)}
\geqslant
\sup_S \frac{\#(\cM_S\cap\{1,\dots,\lcm(S)\})}{\lcm(S)}
=
\sup_S d(\cM_S)
=\underline d(\cM_\cB)
\end{equation*}
by Lemma~\ref{lemma:easy}a,
and similarly
\begin{equation*}
m_H(W^c)
\leqslant
\sup_S \frac{\#(\cM_{\cB\cap\cA_S}\cap\{1,\dots,\lcm(S)\})}{\lcm(S)}
=
\sup_S d(\cM_{\cB\cap\cA_S})
\leqslant
\underline d(\cM_\cB)
\end{equation*}
by Lemma~\ref{lemma:easy}b.
\end{proof}
\begin{corollary}\textbf{\cite[Theorem 4.1]{BKKL2015}}
$\cB$ is a Besicovich sequence
if and only if $\cF_\cB$ is generic for the Mirsky measure.
(As $n\in\cF_\cB$ iff $\Delta(n)\in W$, it would be more precise to say that the sequence $(\Delta(n))_n$ is generic for the Mirsky measure.)
\end{corollary}
\begin{proof}
If $\cB$ is Besicovich, then $d(\cF_\cB)=m_H(W)$, so that $\cF_\cB$ has maximal density. Hence it is generic for the Mirsky measure, see \cite[Theorem 5b]{KR2015}. On the other hand, if $\cF_\cB$ is generic for (any) measure, then its frequency of ones converges in particular, which means that its asymptotic density exists.
\end{proof}
\begin{lemma}
$m_H(\inn(W))
=
\sup_S d(\cF_{\cA_S})
\leqslant
\underline d(\cE)$
\end{lemma}
\begin{proof}
For $S\subset\cB$ denote by
$\cU^o_S$
the family of all sets $U_S(\Delta(n))$ that are contained in $\inn(W)$ and by $\cup\cU^o_S$ the union of these sets.
Recall from Lemma~\ref{lemma:interiorW}a that
$\#\cU^o_S=\#(\cF_{\cA_S}\cap\{1,\dots,\lcm(S)\})$.
Then
\begin{equation*}
m_H(\inn(W))
=
\sup_S m_H(\cup\cU^o_S)
=
\sup_S
\frac{\#\cU^o_S}{\lcm(S)}
=
\sup_S \frac{\#(\cF_{\cA_S}\cap\{1,\dots,\lcm(S)\})}{\lcm(S)}
=
\sup_S d(\cF_{\cA_S})
\end{equation*}
\end{proof}
\begin{lemma}\label{lemma:Haar-boundary}
$m_H(\partial W)=\inf_S\bar d(\cM_{\cA_S}\setminus\cM_\cB)
\leqslant \inf_S d(\cM_{\cA_S\setminus\cB})$.
\end{lemma}
\begin{proof}
\begin{equation*}
\begin{split}
m_H(\partial W)
&=
m_H(W)-m_H(\inn(W))
=
\bar d(\cF_\cB)-\sup_S d(\cF_{\cA_S})
=\inf_S\left(\bar d(\cF_\cB)-d(\cF_{\cA_S})\right)\\
&=
\inf_S\bar d(\cM_{\cA_S}\setminus\cM_\cB)
\end{split}
\end{equation*}
\end{proof}
\subsection{Regular Toeplitz sequences}\label{subsec:Toeplitz}
Let $\cB=\{b_1,b_2,\ldots\}$. For each $k\geq1$, consider the sequence
$$
b_1,\ldots,b_k,c^{(k)}_{k+1},c^{(k)}_{k+2},\ldots,$$
where
$$
c^{(k)}_{k+i}:={\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+i}),\;i\geq1.$$
Then:
\begin{equation*}\label{cirm1}
c^{(k)}_i|{\rm lcm}(b_1,\ldots,b_k), \mbox{whence $\{c^{(k)}_{k+i}:\:i\geq1\}$ is finite}.\end{equation*}
\begin{equation*}\label{cirm2}
\mathscr{A}_{\{b_1,\ldots,b_k\}}=\{b_1,\ldots,b_k\}\cup\{c^{(k)}_{k+i}:\:i\geq1\}.\end{equation*}
\begin{equation*}\label{cirm3}
c^{(k)}_{k+1}|b_{k+1}.\end{equation*}
\begin{equation*}\label{cirm4}
c^{(k)}_{k+1+i}|c^{(k+1)}_{k+1+i},\;\text{for each }i\geq1.
\end{equation*}
Moreover, following Lemma~\ref{lemma:easy}c, there is an increasing sequence $(k_n)$ such that
\begin{equation}\label{cirm0}
\cB\cap \mathscr{A}_{\{b_1,\ldots,b_{k_n}\}}=\{b_1,\ldots,b_{k_n}\}.
\end{equation}
We assume that $W\subset H$ is topologically regular, so by Remark~\ref{remark:window-toeplitz}, $\eta=\1_{\cF_{\cB}}$ is a Toeplitz sequence. We set $s_k:={\rm lcm}(b_1,\ldots,b_k)$ and would like now to examine the sequence $(s_k)$ as a periodic structure of $\eta$. More precisely, we would like to see for how many $n\in[1,s_k]$, we have $\eta(n)=\eta(n+js_k)$ for each $j\in\Z$. We call any such $n$ to be ``good''. Now, if $n\in \cF_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$, then
$n+s_k\Z\subset \cF_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$, so $n$ is good. Otherwise, $n\in \cM_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$. Then either
$n\in \cM_{b_1,\ldots,b_k}$ and then clearly $\eta(n+js_k)=0$ for each $j\in\Z$, so again $n$ is good, or
$$n\in \cM_{\{c^{(k)}_{K+i}:\:i\geq1\}}\setminus\cM_{\{b_1,\ldots,b_k\}}.$$
Only for such $n$, we are not sure that $n$ is good. Moreover, note that in view of~\eqref{cirm4}, we have
\begin{equation*}\label{cirm5}
\cM_{\{c^{(k+1)}_{k+1+i}:\:i\geq1\}}\subset \cM_{\{c^{(k)}_{k+i}:\:i\geq1\}},\end{equation*}
so the sequence $(d(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}}))_k$ is decreasing, and so is the sequence $(\overline{d}(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}}\setminus\cM_{\cB}))$.
Therefore, by taking into account~\eqref{cirm0}, the infimum of this sequence is equal to the liminf, in fact to the limit and we have
\begin{equation}\label{cirm6}
\inf_{S\subset \cB}\overline{d}(\cM_{\mathscr{A}_S})\setminus \cM_{\cB})=\liminf_{k\to\infty}\overline{d}(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}\setminus\cM_{\{b_1,\ldots,b_k\}}})\ .
\end{equation}
\begin{definition}
Let
$\eta=\1_{\cF_{\cB}}$ be a Toeplitz sequence.
It is a \emph{regular Toeplitz sequence} for the periodic structure $(s_k)$, $s_k={\rm lcm}(b_1,\ldots,b_k)$, if the $\liminf$ in \eqref{cirm6} is zero.
\end{definition}
\noindent
Now, using Lemma~\ref{lemma:Haar-boundary}, the identity in \eqref{cirm6} shows the following result.
\begin{proposition}\label{prop:toeplitz} If $W$ is topologically regular, then $\eta=\1_{\cF_{\cB}}$ is a regular Toeplitz sequence for the periodic structure $(s_k)$, $s_k={\rm lcm}(b_1,\ldots,b_k)$, if and only if $m_H(\partial W)=0$.
\end{proposition}
\begin{example} Assume that $\{b'_k:\:k\geq1\}$ is a coprime set of odd numbers and let $b_k=2^kb'_k$. Then $c^{(k)}_{k+i}=2^k$ for each $i\geq1$.
Hence, we have even $d(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}})\to 0$,
in particular $\eta$ is a regular Toeplitz sequence for the periodic structure $(s_k)$ with
$s_k=2^kb_1'\cdots b_k'$.
This example comes from \cite{BKKL2015}.
\end{example}
We will now show that we can obtain Toeplitz sequences also in case $m_H(\partial W)>0$.
\begin{example}
We will construct $\cB=\{b_1,b_2,\ldots\}$ such that this set is thin (hence taut) and
that
$\lim_{k\to\infty}\inf(\{c^{(k)}_{k+i}:\:i\geq1\}\setminus\{b_1,\ldots,b_k\})=\infty$, which, by Proposition~\ref{prop:Staszek-prop-equiv}, implies that $W$ is topologically regular (and hence $\eta$ is Toeplitz by Remark~\ref{remark:window-toeplitz}). Let $\delta_k>0$ and $\sum_{k\geq1}\delta_k<1/16$.
We start with $b_1=2^3$ and set $c^{(1)}_{1+i}=2$ for each $i\geq1$. Suppose that
a sequence
$$
b_1,\ldots, b_k, c^{(k)}_{k+1},c^{(k)}_{k+2},\ldots$$
has been defined. We require that this sequence satisfies:
\begin{equation*}\label{cirm7} c^{(k)}_{k+i}|{\rm lcm}(b_1,\ldots,b_k),\;i\geq1,\end{equation*}
\begin{equation*}\label{cirm8} c^{(k)}_{k+i}\notin \{b_1,\ldots,b_k\}, \;i\geq1,\end{equation*}
\begin{equation*}\label{cirm9} \mbox{for each $i\geq1$, $|\{j\geq1:\:c^{(k)}_{k+j}=c^{(k)}_{k+i}\}|=+\infty$}.\end{equation*}
We will now show how to define $c^{(k+1)}_{k+2}$, $c^{(k+1)}_{k+3}$, $\ldots$ and then $b_{k+1}$.
Recall an elementary lemma.
\begin{lemma}\label{l:cirm1} Let $F_1, F_2$ be finite sets of natural numbers such that ${\rm gcd}(f_1,f_2)=1$ for each $f_i\in F_i$, $i=1,2$.
Then $d(\cM_{F_1\cdot F_2})=d(\cM_{F_1}\cap\cM_{F_2})=d(\cM_{F_1})d(\cM_{F_2})$.
\end{lemma}
Choose $P\subset \mathscr{P}\setminus{\rm spec}\,\{b_1,\ldots,b_k\}$, so that (by Lemma~\ref{l:cirm1})
\begin{equation*}\label{cirm10}
d(\cM_{P\cdot\{c^{(k)}_{k+i}:\:i\geq1\}})\geq d(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}})-\delta_k.\end{equation*}
In view of \eqref{cirm9}, $$
\{i\geq2:\:c^{(k)}_{k+i}=c^{(k)}_{k+1}\}=\{r_1,r_2,\ldots\}$$
Let $P=\{q_1,\ldots,q_t\}$. Then set
$$
c^{(k+1)}_{k+r_1+tj}:=c^{(k)}_{k+1}q_1,c^{(k+1)}_{k+r_2+tj}:=c^{(k)}_{k+1}q_2,\ldots, c^{(k+1)}_{k+r_t+tj}:=c^{(k)}_{k+1}q_t$$
for each $j=0,1,\ldots$
If $2\notin \{i\geq2:\:c^{(k)}_{k+i}=c^{(k)}_{k+1}\}$ then repeat the same construction with the set
$\{i\geq2:\: c^{(k)}_{k+i}=c^{(k)}_{k+2}\}$. Since (by~\eqref{cirm7}) the set $\{c^{(k)}_{k+i}:\:i\geq1\}$ is finite, our construction of the sequence $(c^{(k+1)}_{k+1+i})_i$ is done in finitely many steps. Finally, we set $b_{k+1}:=c^{(k)}_{k+1}\prod_{q\in P}q$ (or, if needed, $b_{k+1}:=c^{(k)}_{k+1}\prod_{q\in P}q^{\alpha_{k+1}}$ for any $\alpha_{k+1}\in\N$). Note that
$$
{\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+1})=c^{(k)}_{k+1}$$
since $P\cap\{b_1,\ldots,b_k\}=\emptyset$. More than that, by the construction, we also have
$$
{\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+i})=c^{(k)}_{k+i}\text{ for each }i\geq1.$$
Moreover, it is not hard to see that the new sequence
$$
b_1,\ldots,b_k,b_{k+1},c^{(k+1)}_{k+2},c^{(k+1)}_{k+3},\ldots$$
satisfies \eqref{cirm7}-\eqref{cirm9}. Furthermore, $\cB=\{b_1,b_2,\ldots\}$ satisfies the other requirements mentioned at the beginning of the construction so that $\eta$ is a Toeplitz sequence and $W$ is topologically regular. Note that in our construction
$d(\cM_{\{c^{(1)}_{1+i}:\:i\geq1\}})=1/2$. Moreover, by \eqref{cirm10}
$$
d(\cM_{\{c^{(k)}_{k+i}:\:i\geq1\}})\geq d(\cM_{\{c^{(1)}_{1+i}:\:i\geq1\}})-\sum_{j=1}^k\delta_k\geq \frac14$$
for each $k\geq1$. Finally notice that $d(\cM_{\cB})\leq \sum_{k\geq1}1/b_k$, which (by construction) can be made smaller than 1/8.
It follows that $\lim_{k\to\infty}\overline{d}(\cM_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}\setminus \cM_{\cB})>0$, whence $m_H(\partial W)>0$.
\end{example}
\section{The maximal equicontinuous factor of $X_\eta$}\label{sec:mef}
\subsection{The period groups of $W$ and of $\inn(W)$}\label{subsec:period-group}
Given a subset $A\subseteq H$, denote by
\begin{equation*}
H_A:=\left\{h\in H: W+h=W\right\}
\end{equation*}
the \emph{period group} of $A$. The set $A\subseteq H$ is \emph{topologically aperiodic}, if $H_A=\{0\}$.
The following simple observations are proved in \cite[Lemma 6.1]{KR2016}:
\begin{itemize}
\item $H_A\subseteq H_{\bar A}\cap H_{\inn(A)}$.
\item If $A$ is closed, then $H_{\inn(A)}=H_{\overline{\inn(A)}}$ is closed.
\end{itemize}
\begin{proposition}\label{p:ape3}
Assume that $\cB$ is primitive. Then the window $W$ is topologically aperiodic.
\end{proposition}
\begin{proof} Suppose that $h=(h_b)_{b\in\cB}\neq0$ and
\begin{equation}\label{ape4}
W+h=W\ .
\end{equation}
Since $h\neq0$, there is $b\in\cB$ such that $b$ does not divide $h_b$. Let $n:=\gcd(b,h_b)$.
Then $n\in\cF_{\cB}$, as otherwise there exists $b'\in \cB$ such that $b'\mid n$; but then $b'\mid b$ a contradiction ($\cB$ is assumed to be primitive). Hence $\Delta(n)\in W$. There are $x,y\in\Z$ such that $n=xh_b+yb$, whence $b\mid n-xh_b$. It follows that
$\Delta(n)-xh\notin W$, a contradiction with~\eqref{ape4}.
\end{proof}
If $W$ is topologically regular, then clearly $\inn(W)$ is topologically aperiodic, as well. Otherwise $H_{\inn(W)}$ may be non-trivial, as we will see in the course of this section.
Recall from \eqref{eq:Bprim} that for any set $A\subseteq\N$,
\begin{equation*}
\cM_A = \cM_{\prim A}
\end{equation*}
If $\prim A$ is finite, then $\cM_A$ is a union of finitely many arithmetic progressions. Let $c_A$ denote the period of $\cM_A$, that is, the least natural number such that $c_A+\cM_A=\cM_A$.
\begin{lemma}\label{period-of-MA} Assume that $A,B\subset\N$ are finite
\begin{compactenum}[a)]
\item If $c+\cM_A=\cM_A$ for some $c\in\N$, then $\lcm(\prim A)\mid c$.
\item $c_A=\lcm(\prim A)$
\item if $A\subset\cM_B$ then $c_B\mid c_A$
\end{compactenum}
\end{lemma}
\begin{proof}
\begin{compactenum}[a)]
\item For any $a\in \prim A$ we have $a+c\Z\subseteq\cM_{\prim A}$, from which it follows that
there exists $a'\in \prim A$ such that $a'\mid \gcd(a,c)$. But, as $\prim A$ is primitive, that means that $a'=a$ and $a\mid c$. We conclude that $\lcm(\prim A)\mid c$.
\item Clearly $\cM_A+\lcm(\prim A)=\cM_A$, thus $c_A\mid \lcm(\prim A)$ and the assertion follows from a).
\item If $A\subset\cM_B$ then $\prim A\subset \cM_{\prim B}$, hence $\lcm(\prim B)\mid \lcm(\prim A)$, and we finish by b).
\end{compactenum}
\end{proof}
\begin{lemma}\label{A-goes-back}
Assume that $S\subseteq S'$ are finite subsets of $\cB$, then $\cA_S=\{\gcd(a,\lcm(S)):a\in\cA_{S'}\}$.
\end{lemma}
\begin{proof}
Since $\lcm(S)\mid \lcm(S')$, $\gcd(b,\lcm(S))=\gcd(\gcd(b,\lcm(S')),\lcm(S))$ for any $b\in\cB$ and the assertion follows.
\end{proof}
Let $S_1\subset S_2\subset\ldots\subset S_k\subset\ldots$ be a filtration of $\cB$ with finite sets and denote
\begin{equation*}
s_k:=\lcm(S_k),\; c_k:=c_{\cA_{S_k}}
\end{equation*}
By Lemma \ref{period-of-MA} c) we have $c_l\mid c_{l+1}$ for any $l$. It follows that, for any $k$, the sequence $(\gcd(s_k,c_l))_{l\ge 1}$ stabilizes on a
divisor $d_k$ of $s_k$.
Clearly, since $c_k\mid s_k$,
\begin{equation}\label{nowy_1G}
c_k\mid d_k\mid s_k\ .
\end{equation}
Observe that
\begin{equation}\label{nowy_2G}
d_k=\gcd(s_k,d_{k+1})\ .
\end{equation}
Indeed, there is $l_0\in\N$ such that $d_{k+1}=\gcd(s_{k+1},c_l)$ for all $l>l_0$. Since $s_k\mid s_{k+1}$,
we get
$$
\gcd(s_k,c_l)=\gcd(s_k,\gcd(s_{k+1},c_l))\ .
$$
It follows that $\gcd(s_k,c_l)=\gcd(s_k,d_{k+1})$ for $l>l_0$, and (\ref{nowy_2G}) follows.
By applying (\ref{nowy_2G}) we prove by induction that
\begin{equation}\label{nowy_3G}
d_k=\gcd(s_k,d_{k+j})\ .
\end{equation}
for $j\ge 0$.
\begin{lemma}\label{lemma:equivalent}
Let $(n_k)_{k\in\N}$ be a sequence of integers.
The following are equivalent:
\begin{equation}\label{eq:equiv1S}
\forall k\in\N:\ c_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k\ ,
\end{equation}
and
\begin{equation}\label{eq:equiv2}
\forall k\in\N:\ d_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k\ .
\end{equation}
\end{lemma}
\begin{proof}
As $c_{k}\mid d_{k}$, \eqref{eq:equiv2} implies \eqref{eq:equiv1S}. Conversely, assume that \eqref{eq:equiv1S} holds. We show inductively that for all $j\geqslant0$
\begin{equation}\label{eq:inductive}
\forall k\in\N:\ \gcd(s_{k},c_{k+j})\mid n_{k}\ ,
\end{equation}
and this implies \eqref{eq:equiv2} immediately.
For $j=0$, \eqref{eq:inductive} follows from \eqref{eq:equiv1S}, because $c_{k}\mid s_{k}$.
So suppose that \eqref{eq:inductive} holds for some $j\geqslant0$. Then
\begin{displaymath}
\begin{split}
n_{k+1}&=0\;\mod \gcd(s_{k+1},c_{k+1+j})\;\text{and}\\
n_{k+1}&=n_{k}\mod s_k\ .
\end{split}
\end{displaymath}
Hence $n_k=0\mod \gcd(s_k,s_{k+1},c_{k+1+j})=\gcd(s_k,c_{k+j+1})$ for all $k\in\N$,
i.e. \eqref{eq:inductive} for $j+1$.
\end{proof}
Recall that $H_{\inn(W)}=\{h\in H:\inn(W)+h=\inn(W)\}$ denotes the period group of $\inn(W)$.
\begin{proposition}\label{prop:periods}
\begin{compactenum}[a)]
\item $h\in H_{\inn(W)}$ if and only if $h=\lim_k\Delta(n_k)$ for some sequence
$(n_k)_k$ satisfying
\begin{equation}\label{eq:n_k}
\forall k\in\N:\ d_k\mid n_k \;\text{ and }\; s_k\mid n_{k+1}-n_k\ .
\end{equation}
Moreover, sequences $(n_k)_k$ can be defined inductively: For $n_1$ there are $s_1/d_1$ choices and, given $n_1,\dots,n_k$, there are precisely $s_{k+1}/\lcm(s_k,d_{k+1})$ many choices for $n_{k+1}$.
\item $H_{\inn(W)}=\{0\}$ if and only if $s_k=d_k$ for all $k\in\N$.
\end{compactenum}
\end{proposition}
\begin{remark}\label{remark:factors}
Observe that, in view of \eqref{nowy_2G},
\begin{equation*}
\frac{s_k}{d_k}\cdot \frac{s_{k+1}}{\lcm(s_k,d_{k+1})}
=
\frac{s_k\, s_{k+1}\gcd(s_k,d_{k+1})}{d_k\,s_k\,d_{k+1}}
=
\frac{s_k\, s_{k+1}\,d_k}{d_k\,s_k\,d_{k+1}}
=
\frac{s_{k+1}}{d_{k+1}},
\end{equation*}
so that
\begin{equation*}
\frac{s_k}{d_k}\mid \frac{s_{k+1}}{d_{k+1}}
\end{equation*}
\begin{equation*}
\frac{s_k}{d_k}
=
\frac{s_1}{d_1}\cdot\prod_{j=1}^{k-1}\frac{s_{j+1}}{\lcm(s_j,d_{j+1})}.
\end{equation*}
\end{remark}
\begin{proof}[Proof of Proposition~\ref{prop:periods}]
a) For each $S_k$ denote by $W_k:=\bigcup_{n\in\cF_{\cA_{S_k}}}U_{S_k}(\Delta(n))$. Then $\inn(W)$ is the increasing union of the sets $W_k$, see Lemma~\ref{lemma:interiorW},
and $U_{S_k}(\Delta(n))\subseteq W_k$ if and only if
$U_{S_k}(\Delta(n))\subseteq\inn(W)$.
Let $h=\lim_k\Delta(n_k)$, where $n_k$ stands for $n_{S_k}$, which was defined in Lemma~\ref{lemma:interiorW}b. Then
\begin{equation}
\forall k\in\N:\ s_k\mid n_{k+1}-n_k\ ,
\end{equation}
and
$h\in H_{\inn(W)}$, if and only if
\begin{equation}\label{eq:W_k-Delta_n_k}
\forall k\in\N:\ \cF_{\cA_{S_k}}+n_k=\cF_{\cA_{S_k}}\ .
\end{equation}
Indeed, let $k\in\N$, $m\in\cF_{\cA_{S_k}}$, and let $g=(g_b)_{b\in\cB}$ be any element from $U_{S_k}(\Delta(m))\subseteq \inn(W)$. Then $g_b=m\mod b$ for all $b\in S_k$. Assume now that $h\in H_{\inn(W)}$. Then $g+h\in\inn(W)$ and
$(g+h)_b=m+n_k\mod b$ for all $b\in S_k$, so that $g+h\in U_{S_k}(\Delta(m+n_k))$.
Hence $U_{S_k}(\Delta(m))+h\subseteq U_{S_k}(\Delta(m+n_k))=U_{S_k}(\Delta(m))+\Delta(n_k)$. In particular, $U_{S_k}(\Delta(m))$ and $U_{S_k}(\Delta(m+n_k))$ have identical Haar measure, and so do $U_{S_k}(\Delta(m))+h$ and $U_{S_k}(\Delta(m+n_k))$. As both are open sets and one is contained in the other, they must coincide.
Hence $U_{S_k}(\Delta(m+n_k))=U_{S_k}(\Delta(m))+h\subseteq\inn(W)+h=\inn(W)$, so that $m+n_k\in\cF_{\cA_{S_k}}$. This proves that $\cF_{\cA_{S_k}}+n_k\subseteq\cF_{\cA_{S_k}}$. As $\cA_{S_k}$ is a finite set, this implies $\cF_{\cA_{S_k}}+n_k=\cF_{\cA_{S_k}}$.
Conversely, assume that \eqref{eq:W_k-Delta_n_k}
holds, and let $U_{S_k}(\Delta(m))\subseteq\inn(W)$.
Recall that this implies $U_{S_k}(\Delta(m))\subseteq W_k$, i.e. $m\in\cF_{\cA_{S_k}}$.
Hence, by assumption, also $m+n_k\in\cF_{\cA_{S_k}}$, so that $U_{S_k}(\Delta(m+n_k))\subseteq W_k\subseteq\inn(W)$.
Let $g\in U_{S_k}(\Delta(m))$. Then $g_b=m\mod b$ for all $b\in S_k$, so that
$(g+h)_b=m+n_k\mod b$ for all $b\in S_k$, i.e. $g+h\in U_{S_k}(\Delta(m+n_k))$.
Hence $U_{S_k}(\Delta(m))+h\subseteq U_{S_k}(\Delta(m+n_k))\subseteq\inn(W)$.
As this argument applies to all $k$ and all $U_{S_k}(\Delta(m))\subseteq\inn(W)$,
it proves that $\inn(W)+h\subseteq\inn(W)$.
The same Haar measure argument as before, applied to the open set $\inn(W)$, shows that $\inn(W)+h=\inn(W)$, i.e. $h\in H_{\inn(W)}$.
Condition \eqref{eq:W_k-Delta_n_k} is equivalent to
\begin{equation}
\forall k\in\N:\ c_k=\lcm(\prim\cA_{S_k})\mid n_k\ .
\end{equation}
Invoking Lemma~\ref{lemma:equivalent}, we conclude
\begin{equation}
h\in H_{\inn(W)}
\quad\Leftrightarrow\quad
\forall k\in\N:\ d_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k\ .
\end{equation}
This proves the claimed equivalence.
Now we describe all sequences $(n_k)_{k\in\N}$ which satisfy\eqref{eq:n_k} and $n_k\in\{0,\dots,s_k-1\}$ for all $k$. Denote $q_k:=s_k/d_k$.
\begin{compactenum}[aaaaaa]
\item[$n_1$:] Let $n_1=m_1d_1$ for any $m_1\in\{0,\dots,q_1-1\}$.
\item[$n_2$:] $n_2$ must be chosen such that $n_2=0\mod d_2$ and $n_2=n_1\mod s_1$.
As $\gcd(s_1,d_2)=d_1\mid n_1$ in view of \eqref{nowy_2G}, the CRT guarantees the existence of at least one solution $n_2$, and if $n_2$ is one particular solution, then the set of all solutions is precisely $n_2+\lcm(s_1,d_2)\cdot \Z$. As $n_2$ is to be chosen in $\{0,\dots,s_2-1\}$, there are exactly $s_2/\lcm(s_1,d_2)$ possible choices for $n_2$.
\item[$\vdots$\quad]
\item[$n_{k+1}$:]
$n_{k+1}$ must be chosen such that $n_{k+1}=0\mod d_{k+1}$ and $n_{k+1}=n_k\mod s_k$.
As $\gcd(s_k,d_{k+1})=d_k\mid n_k$ in view of \eqref{nowy_2G}, the CRT guarantees the existence of at least one solution $n_{k+1}$, and if $n_{k+1}$ is one particular solution, then the set of all solutions is precisely $n_{k+1}+\lcm(s_k,d_{k+1})\cdot \Z$. As $n_{k+1}$ is to be chosen in $\{0,\dots,s_{k+1}-1\}$, there are exactly $s_{k+1}/\lcm(s_k,d_{k+1})$ possible choices for $n_{k+1}$.
\end{compactenum}
b) $H_{\inn(W)}=\{0\}$ $\Leftrightarrow$ there is unique choice of the numbers $n_k$ described in a) $\Leftrightarrow$ $s_1/d_1=1$ and $s_{k+1}/\lcm(s_k,d_{k+1})=1$ for any $k$ $\Leftrightarrow$ $d_{k}=s_{k}$ for any $k$, the last equivalence by Remark \ref{remark:factors}.
\end{proof}
\subsection{Proof of Theorem~\ref{theo:MEF}}
\begin{remark}\label{limits}
If $(S_k)_k$ is a filtration of $\cB$ by finite sets and if $h=(h_b)_{b\in\cB}\in H$, then we write $
\lim_k\Delta(n_{S_k})=h$, whenever $n_{S_k}\in\Z$ are numbers such that for every $k\in \N$:
$$
h_b= n_{S_k}\mod b\;\text{\rm for all}\;b\in S_k
$$
Let us denote $s_k=\lcm(S_k)$. There is an inverse system of groups
\begin{equation*}
\ldots \Z/s_{k+1}\Z\rightarrow \Z/s_{k}\Z \rightarrow \ldots\rightarrow \Z/s_1\Z
\end{equation*}
The homomorphisms are the canonical projections. Observe that $s_k|n_{S_{k+1}}-n_{S_k}$ for any $k$ and the sequence $(n_{S_k}+s_k\Z)_k$ is an element of the inverse limit
$\lim\limits_{\leftarrow}\Z/{s_k}\Z$. In this way we obtain an isomorphism of topological groups
\begin{equation}\label{sigma}
\sigma:\lim\limits_{\leftarrow}\Z/{s_k}\Z\cong H
\end{equation}
given by $(n_{S_k}+s_k\Z)_k\mapsto \lim_k\Delta(n_{S_k})$. Compare Remark 2.32 \cite{BKKL2015}. In particular, the inverse limit does not depend on the filtration $(S_k)_k$.\footnote{The last statement follows from a general property of inverse limits: the inverse limits of cofinal inverse systems are isomorphic, \cite[Chapter II, Section 12]{Fuchs}.}
\end{remark}
\begin{proof}[Proof of Proposition~\ref{prop:period}]
Let $\beta_k:\Z/s_k\Z\rightarrow \Z/d_k\Z$ be the map given by $n+s_{k}\Z\mapsto n+d_{k}\Z$, let $M_k$ be the kernel of $\beta_k$ and let $\alpha_k:M_k\rightarrow \Z/s_k\Z$ be the canonical embedding.
There is a commutative diagram of abelian groups
$$
\begin{array}{ccccccccc}
&&0&&0&&&&0\\
&&\mapdown{}&&\mapdown{}&&&&\mapdown{}\vspace{1ex}\\
\ldots&\mapr{f'_{k+1}}{}&M_{k}&\mapr{f'_{k}}{}&M_{k-1}&\mapr{f'_{k-1}}{}&\ldots&\mapr{f'_{2}}{}&M_{1}\vspace{1ex}\\
&&\mapdown{\alpha_{k}}&&\mapdown{\alpha_{k-1}}&&&&\mapdown{\alpha_{1}}\vspace{1ex}\\
\ldots&\mapr{f_{k+1}}{}&\Z/s_k\Z&\mapr{f_{k}}{}&\Z/s_{k-1}\Z&\mapr{f'_{k-1}}{}&\ldots&\mapr{f'_{2}}{}&\Z/s_{1}\Z\vspace{1ex}\\
&&\mapdown{\beta_{k}}&&\mapdown{\beta_{k-1}}&&&&\mapdown{\beta_{1}}\vspace{1ex}\\
\ldots&\mapr{f''_{k+1}}{}&\Z/d_k\Z&\mapr{f''_{k}}{}&\Z/d_{k-1}\Z&\mapr{f''_{k-1}}{}&\ldots&\mapr{f''_{2}}{}&\Z/d_{1}\Z\vspace{1ex}\\
&&\mapdown{}&&\mapdown{}&&&&\mapdown{}\vspace{1ex}\\
&&0&&0&&&&0
\end{array}
$$
where $f_k(n+s_k\Z)=n+s_{k-1}\Z$, $f'_k$ is the restriction of $f_k$ to $M_k$ and $f''_k(n+d_k\Z)=n+d_{k-1}\Z$.
The columns of the diagram are exact sequences of groups, in other words, the diagram can be interpreted as an exact sequence of inverse systems of abelian groups.
Since inverse limit is a left exact functor, see \cite[Chapter II, Theorem 12.3]{Fuchs}, we obtain an exact sequence
\begin{equation}\label{alg_2}
0\rightarrow \lim\limits_{\leftarrow}M_k\mapr{\alpha}{} \lim\limits_{\leftarrow}\Z/s_k\Z\mapr{\beta}{} \lim\limits_{\leftarrow}\Z/{d_k}\Z
\end{equation}
The condition \eqref{nowy_2G} yields that the homomorphism $\gamma$ in (\ref{alg_2}) is surjective, thus we have an exact sequence
\begin{equation}\label{alg_3}
0\rightarrow \lim\limits_{\leftarrow}M_k\mapr{\alpha}{} \lim\limits_{\leftarrow}\Z/{s_k}\Z\mapr{\beta}{} \lim\limits_{\leftarrow}\Z/{d_k}\Z\rightarrow 0
\end{equation}
Indeed, let $(n_k+d_k\Z)_k\in \lim\limits_{\leftarrow}\Z/{d_k}\Z$. By induction we construct the numbers $m_1,m_2,\ldots$ such that $d_k|m_k-n_k$ and $s_k|m_{k+1}-m_k$, for any $k$. Then $\beta((m_k+s_k\Z)_k)=(n_k+d_k\Z)_k$. We set $m_1=n_1$. Assume that $m_1,\ldots m_k$ have been defined. Since $d_k\mid n_{k+1}-n_k$, $d_k\mid m_{k}-n_k$ and $\gcd(d_{k+1},s_k)=d_k$, there exists integers $x,y$ such that $xd_{k+1}+ys_k=m_k-n_{k+1}$. We set $m_{k+1}=m_k-ys_k$.
There are group isomorphisms $g_k:\Z/{\frac{s_{k}}{d_{k}}}\Z \rightarrow M_k$ given by $g_k(n+\frac{s_{k}}{d_{k}}\Z)=d_kn+s_k\Z$ and making the following diagram commutative
$$
\begin{array}{ccccccccc}
\ldots&\rightarrow &\Z/{ \frac{s_{k+1}}{d_{k+1}}}\Z&\rightarrow& \Z/{\frac{s_{k}}{d_{k}}}\Z& \rightarrow&\ldots &\rightarrow& \Z/{\frac{s_{1}}{d_{1}}}\Z\vspace{1ex}\\
&&\mapdown{g_{k+1}}&&\mapdown{g_k}&&&&\mapdown{g_1}\vspace{1ex}\\
\ldots&\mapr{}{}&M_{k+1}&\mapr{f'_{k+1}}{}&M_{k}&\mapr{f'_{k}}{}&\ldots&\mapr{f'_{2}}{}&M_{1}\vspace{1ex}\\
\end{array}
$$
(the arrows in the upper row represent the canonical projections). It follows that there is an isomorphism
\begin{equation}\label{alg_4}
\lim\limits_{\leftarrow}M_k\cong \lim\limits_{\leftarrow}\Z/{\frac{s_{k}}{d_{k}}}\Z
\end{equation}
By Proposition~\ref{prop:periods} a) it follows that $\lim\limits_{\leftarrow}M_k$ is isomorphic to $H_{\inn(W)}$. There is an isomorphism given by $\sigma\alpha$, where $\sigma$ is the isomorphism defined in Remark~\ref{limits}.
Now a), b) and c) follow from (\ref{alg_3}), (\ref{alg_4}) and Remark~\ref{limits}. In order to prove d) it is enough to note that $s_k=d_k$ if and only if $s_k\mid c_{k+j}$ for some $j\ge 0$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo:MEF}]
This is an immediate corollary to Proposition~\ref{prop:period}.
\end{proof}
\subsection{Examples}\label{subsec:examples}
\begin{remark}\label{remark:limits}
Given a prime number $p$ and $m\in\Z$ we denote by $v_p(m)$ be the $p$-valuation of $m$, that is, if $m\neq 0$ then $v_p(m)$ is the maximal integer such that $p^{v_p(m)}\mid m$ and $v_p(0)=+\infty$. Assume that $t=(t_k)$ is a sequence of natural numbers such that $t_k\mid t_{k+1}$ for any $k$. Set $v_p(t)=\sup_kv_p(t_k)$. The sequence $t$ yields an inverse system of abelian groups
$$
\ldots\rightarrow Z/t_{k+1}\Z\rightarrow Z/t_{k}\Z \rightarrow\ldots \rightarrow Z/t_{1}\Z
$$
where the arrows represent the canonical projections $n+t_{k+1}\Z\mapsto n+t_{k}\Z$.
The inverse limit $\lim\limits_{\leftarrow}\Z/t_{k}\Z$ of this system is isomorphic to the group
$$
\prod\limits_{p\in\cP}G_p
$$
where $G_p=\Z/p^{v_p(t)}\Z$ if $v_p(t)<+\infty$ and $G_p=\widehat{\Z}_p$ (the group of $p$-adic numbers) otherwise, i.e. when $\lim_kv_p(t_k)=+\infty$.
\end{remark}
Recall from \eqref{def:Cinf} that
\begin{equation}
\Ainf=\{c\in\N: \forall_{S\subset\cB}\ \exists_{S': S\subseteq S'}: c\in\cA_{S'}\setminus S'\}.
\end{equation}
Our first exaxmple has a finite, non-trivial maximal equicontinuous factor and a finite set $\Ainf$.
\begin{example}
$\cB=\{36\}\cup \{2p_1,2p_2,\ldots\}\cup \{3q_1,3q_2,\ldots\}$, where $p_1,q_1,p_2,q_2,\ldots$ are pairwise different primes.
Let $S_k=\{36,2p_1,\dots,2p_k,3q_1,\dots,3q_k\}$. Then
\begin{equation*}
s_k=36p_1\cdots p_kq_1\cdots q_k,\; \cA_{S_k}=\{2,3\},\;c_k=d_k=6\ ,
\end{equation*}
so that
\begin{equation*}
\frac{s_k}{d_k}=6p_1\cdots p_kq_1\cdots q_k\ .
\end{equation*}
In particular, the maximal equicontinuous factor of $X_\eta$ is the translation by $1$ on $\Z/6\Z$. Moreover, $\Ainf=\{2,3\}$, so that $\emptyset\neq\overline{\inn(W)}\neq W$
by Theorems~\ref{theo:minimality} and~\ref{theo:proximal}.
\end{example}
Our next example has an infinite maximal equicontinuous factor different from $H$ and an infinite set $\Ainf$.
\begin{example}
Let $p_1,q_1,p_2,q_2,\ldots$ be pairwise different primes. Let
$$
\cB=\cB_1\cup\cB_2\cup\cB_3\ldots
$$
where
$$
\begin{array}{l}
\cB_1=\{p_1q_1\}\\
\cB_2=\{p_1p_2^2, p_1q_2^2, q_1q_2^2\}\\
\cB_3=\{p_1p_2p_3^2,p_1p_2q_3^2,p_1q_2q_3^2, q_1q_3^2\}\\
\cB_4=\{p_1p_2p_3p_4^2,p_1p_2p_3q_4^2, p_1p_2q_3q_4^2, p_1q_2q_4^2, q_1q_4^2\}\\
\cB_5=\{p_1p_2p_3p_4p_5^2,p_1p_2p_3p_4q_5^2,p_1p_2p_3q_4q_5^2,p_1p_2q_3q_5^2, p_1q_2q_5^2, q_1q_5^2\}\\
\ldots
\end{array}$$
That is,
$$
\cB_{k+1}=\{p_1\ldots p_k p_{k+1}^2,\;p_1\ldots p_k q_{k+1}^2,\;p_1\ldots p_{k-1} q_k q_{k+1}^2\}\cup\{\frac{b q_{k+1}^2}{q_k^2}:b\in\cB_k\setminus\{p_1\ldots p_{k-1}p_{k}^2,p_1\ldots p_{k-1 }q_{k}^2\}\}
$$
for $k\ge 2$.
Let $S_k=\cB_1\cup\ldots\cup\cB_k$. Then $s_k=\lcm(S_k)=p_1p_2^2\ldots p_k^2q_1q_2^2\ldots q_k^2$ and
$$
\cA_{S_k}=S_k\cup\{p_1\ldots p_k,\;p_1\ldots p_{k-1}q_k,\; p_1\ldots p_{k-2}q_{k-1},\;\ldots, p_1q_2,q_1\}\ ,
$$
so that
\begin{equation*}
\prim\cA_{S_k}
=
\{p_1\ldots p_k,\;p_1\ldots p_{k-1}q_k,\; p_1\ldots p_{k-2}q_{k-1},\;\ldots, p_1q_2,q_1\}\ .
\end{equation*}
Hence
\begin{equation*}
c_k=p_1\cdots p_k q_1\cdots q_k\quad\text{and}\quad d_k=\gcd(s_k,c_{k+j})=c_k\ ,
\end{equation*}
so that
\begin{equation*}
\frac{s_k}{d_k}=p_2\cdots p_kq_2\cdots q_k\ .
\end{equation*}
Hence $H_{\inn(W)}\cong\prod_{i=2}^{+\infty}\Z/p_iq_i\Z$ and $H/H_{\inn(W)}\cong\prod_{i=1}^{+\infty}\Z/p_iq_i\Z$ are infinite compact groups.
Moreover,
$$\Ainf=\limsup_{k\to\infty}\cA_{S_k}\setminus S_k=\{q_1,p_1q_2,p_1p_2q_3,p_1p_2p_3q_4,\ldots\}$$ is infinite and does not contain the number $1$, thus $\emptyset\neq\overline{\inn(W)}\neq W$ by Theorems~\ref{theo:minimality} and~\ref{theo:proximal}.
\end{example}
We end with a non-trivial example where the maximal equicontinuous factor equals $H$ and $\Ainf$ is an infinite set.
\begin{example}
Let $q, p_1,p_2,\ldots$ be pairwise different odd primes. Let
$$
\cB=\cB_1\cup\cB_2\cup\cB_3\ldots
$$
where
$$
\begin{array}{l}
\cB_1=\{p_1q\}\\
\cB_2=\{p_2q,p_1p_2\}\\
\cB_3=\{p_3q,p_1p_3, p_2p_3\}\\
\cB_4=\{p_4q,p_1p_4,p_2p_4,p_3p_4\}\\
\ldots
\end{array}$$
That is,
$$
\cB_{k}=\{p_kq,p_1p_k,\ldots,p_{k-1}p_k\}
$$
for $k\ge 1$.
Let $S_k=\cB_1\cup\ldots\cup\cB_k$. Then $s_k=\lcm(S_k)=qp_1\ldots p_k$ and
$$
\cA_{S_k}=S_k\cup\{p_1,\ldots, p_k\}\cup \{q\}
$$
hence $c_{\cA_{S_k}}=qp_1\ldots p_k=\lcm(S_k)$,
so that $s_k=c_k=d_k$ for all $k$. In particular
$\inn(W)$ is aperiodic by Proposition~\ref{prop:periods}.
Moreover, $$\Ainf=\limsup_{k\to\infty}\left(\cA_{S_k}\setminus S_k\right)=\{q,p_1,p_2,\ldots\}$$ is infinite and does not contain the number $1$, thus $\emptyset\neq\overline{\inn(W)}\neq W$
by Theorems~\ref{theo:minimality} and~\ref{theo:proximal}.
\end{example}
| 36,583
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Preparation time: 10 minutes
Cooking time: 20 minutes
Serves: 4 people
1 baby endive
3 bacon rashers, cut into 2.5cm (25 mm) long strips
5 oz (150 g) green beans, trimmed
2 tbsp (30 ml) white vinegar
4 eggs
1 shallot, finely chopped
1/4 cup (60 ml) Newman’s Own Balsamic Dressing*
Salt and pepper
1. Discard outer leaves of endive. Divide leaves and place in a large heatproof bowl. Set aside
2. Heat a large non-stick frying pan over a medium heat. Add bacon and cook, stirring occasionally for 8 minutes or until well done. Remove frying pan from heat, leaving bacon in pan.
3. Cook beans in boiling water for 3 minutes or until just tender. Drain and rinse under cold water to cool. Drain well and dry on kitchen paper. Set aside.
4. Have ready another frying pan filled with 2.5cm of warm water and set aside
5. Half fill a medium saucepan with water. Bring to the boil. Add vinegar then reduce heat to low. Break one egg into a small teacup. When the water surface is just quivering and there are a few small bubbles rising, stir the water with a wooden spoon in one direction to make a whirlpool effect. Stop stirring and immediately slide the egg into the center of the whirlpool, as close to the surface as possible. Do not stir after this. Poach the egg for 3 minutes for a soft yolk. Using a slotted spoon, transfer the poached egg to the warm water pan. Repeat with remaining eggs. Skim off any scum that may form on the top of the poaching liquid
6. Reheat the bacon in its frying pan over medium heat. Add eshallot and cook, stirring for 1 minute. Add Newman’s Own Balsamic Vinaigrette and boil 5 seconds. Immediately pour hot dressing over chicory and toss. Divide salad among four plates and top with well drained poached eggs and beans. Season eggs with salt and pepper and serve immediately.
Tip: If you prefer, use an egg poacher to cook the eggs.
* Newman's Own Light Balsamic Dressing can be used as well
| 109,302
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:
.”
This is.
This also gives Bioware the opportunity to make actual paid ME3 DLC down the line. After getting their free ending fix, fans will at least be somewhat more likely to buy future DLC packs. Bioware may have bought themselves some goodwill by doing this, but now fans have to hold their collective breath while asking themselves one question:
What if the new endings are terrible too?
They shouldn’t be, as now Bioware pretty much has an exact road map of what will make fans happy. There’s no reason that the new ending DLC shouldn’t please most fans, but A) you can’t satisfy everyone and B) if it does, it’s clear it should have been in the game in the first place.
Still, I’m rather astonished to see a fan outcry actually accomplish something of this magnitude. I’ve been cautioned against saying this is a “victory” for fans, as they’re waiting to see if what they get is what they want, but I don’t know how they can see it as anything but. The protesters need to be careful now. If they start saying things like “I wanted the ending CHANGED not EXPANDED,” they’re going to get slapped with the “entitled” and “whiny” labels they hate so much. And at this point, I would finally agree with that classification. This is a win for fans and a concession from Bioware, plain and simple.
For fans to dislike something in a beloved series so much, that they were able to convince the company to make additional free content for them is amazing and unprecedented. The scope of the mistake in this case was massive, and was matched only by how violently the fans reacted to it. A combination of both factors led to this unheard of resolution, and I’ve never seen anything like it in the industry before.
| 11,034
|
TITLE: what is the maximum number of roots of quadratic function with 3 variables?
QUESTION [1 upvotes]: Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$
I would like to ask what is the maximum number of roots of the provided equation.
Thanks in advance
It seems obvious from the first few comments that the set of solution which is the intersection between the two surface is infinity.
If adding 2 constraints: 1. the solutions are real numbers and 2. any 4 points in the set of solution can not lie on the same plane, what is maximum number of solutions?
REPLY [1 votes]: The question could be written more precisely: when you say "satisfies $x^2+y^2+z^2=1$", I assume you mean that you are looking for simulateneous solutions of both equations.
Anyway, the answer is that such a system of equations can have infinitely many solutions. Geometrically, each equation defines a surface in 3-dimensional space, and their common solutions are the intersection points of those surfaces.
If we are talking about real-number solutions, then your second equation defines the unit sphere, and it's not hard to visualise another surface (the solution set of the first equation) intersecting it in a curve, say.
REPLY [1 votes]: There could be infinitely many solutions to those two equations. Let $a=b=c=d=e=f=g=h=0$ and let $i=1$.
Then the first equation becomes $z=0$, which is the $xy$-plane, and that intersects the unit sphere in the unit circle.
| 120,551
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Definition of scungille in English:
scungille
(also scungile)
noun (plural scungilli /skʊnˈdʒiːli/)
A mollusc (especially with reference to its meat eaten as a delicacy): it was the only store in the city that carried scungilli
More example sentences
- At least we don't have organ grinders with monkeys dressed in red and white checked sailor suits crawling around Mary-Kate and Ashley's shoulders asking for scungili.
- They serve northern Italian fare including spaghetti carbonara, scungilli salad and shrimp Danielle.
Origin
From Italian dialect scunciglio, probably an alteration of Italian conchiglia 'seashell'.
For editors and proofreaders
Line breaks: scun|gille
Definition of scungille in:
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= trending
| 318,868
|
Willie Tuitama bounced back last weekend with a big performance against NAU. He talked to the media about that game as well as the UA's Saturday contest against New Mexico.
To watch the first half of the video, click here.
To watch the second half of the video, which is premium, click here.
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1. For you it’s no longer called Mercearia do Século but Fernanda. Fernanda is the gracious, courteous owner, who, no matter how busy she might be, always takes her time to greet you and make you feel like you are the only person in the room.
2. Every time you come back, you continue to be impressed and surprised by the food, which is due to the fact that Mercearia do Século has a small menu which changes almost daily. After about a year of regular visits, you have amassed a couple of culinary highlights, once eaten, never forgotten: the amazing Moroccan lamb with the lightest, fluffiest couscous, the pork cheeks cooked in red wine and a bunch of secret spices with a lovely puree of sweet potato and a perfect Gazpacho:
light, vibrant in colour and highly charged in taste of the original, fresh ingredients – apparently in the old days, Gazpacho was no more than an Alentejan farmer’s bread soup with garlic, water and vinegar to which tomatoes were only added occasionally. But whatever it is that you might get, it’s always going to be good, really fresh and well made: whether it’s Portuguese food with a sophisticated twist or Arab styled cooking, you really cannot go wrong here.
3. The desserts! The desserts! Pastries, cookies, cakes. It’s all good and most of it homemade. And almost all taste of wholesome butter. This is not the kind of place where you should care about that.
4. On a hot summer’s day, you make an attempt at imitating the fantastically refreshing ginger-mint lemonade Fernanda serves and end up longing for the real deal.
5. You have about 10 or so stories to tell about a space which is really not much bigger than 30 square meters. You have taken friends there, have taken friends’ moms there, have made friends there, have spent time there with Couchsurfers, it really is one of your go-to-places. Anyone you suspect of having a decent palate or at the very least a lust for life, you take to Fernanda. Be prepared to leave with a good story to tell the grandchildren.
6. You do not only go there for the food, but as well because it’s half-restaurant half-market with some specialty, high quality products. Maybe at first, Mercearia do Século seemed like it would be a hipster-tourist trap, but none of that. There are a lot of things to recommend or try here, but one of the highlights was the muxama (salted, cured tuna, as good as any air-dried ham) and the coelho ao escabeche (marinated rabbit). You have also developed quite an intimate relationship with the late bottled vintage portwine they have for sale and the recent addition of some artisanal high-quality Portuguese beers made you waggle outside quite happily whenever you had it together with your lunch.
7. You gladly pay a little bit more for such great quality. It could also be due to the fact that you always have just a bit too much to drink there. In any case, you’re glad. Shut up and
enjoy it.
8. Whenever you look out of the window with the decorated fish nets, you always feel a little bit at home.
9. Writing your blog article, all of a sudden you remember how a bacalhau com natas (gratinated codfish with cream) was the best you have ever had. The sauce was so creamy. And it really tasted of cod. Oh my cod!
10. You invite the owner to your place for a glass of wine and some homemade food because you want to give something back to them. Fernanda loved (or at least pretended to love) Nora’s South Korean soft tofu stew and its many delicious accompaniments. (You know we can do this for you too, right?)
11. You have to fall back on such a cheap blogger’s trick as making lists because you cannot find any other way to write about one of your favorite restaurants without sounding disingenuous or too uncritical.
(Nout Van Den Neste)
Opening hours: 9:00 to 20:00 (weekdays), 9:00 to 21:00 (Saturday), Sunday closed
Address: Rua de O Século 145
Phone Number: 966 921 280
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Jamie Barton, mezzo-soprano, sings at the Metropolitan Opera show at Central Park SummerStage. (Photographs by Aaron Adler/For New York Observer)
On Monday night, opera, which lately has seemed plain and puny, turned grand again.
The occasion wasn’t a lavishly staged opus of Wagner or Verdi with full orchestra and chorus; rather, this installment of the Metropolitan Opera Summer Recital Series, at Central Park SummerStage, featured just three singers in evening dress, accompanied by piano. But the richness of their voices and the sweep of their interpretations added up to sheer grandeur.
Jamie Barton, mezzo-soprano, sings the “Witch’s Aria” from Hansel & Gretel at the Metropolitan Opera show at Central Park SummerStage.
Now, “grand” can mean “opulent” or “magnificent,” which this concert’s singers certainly were. But it can also mean “large,” which these artists—soprano Amber Wagner, mezzo Jamie Barton and tenor Russell Thomas—are as well. If their voluptuous voices evoked the golden age of the late 19th century, so did their physiques, which recalled engravings in a century-old edition of The Victor Book of the Opera.
The uncomfortable chicken-egg riddle, “Are fat singers great or are great singers fat?” is as old as opera and as new as this spring’s headlines. In May, a zaftig mezzo named Tara Erraught garnered insulting reviews when she sang the part of Octavian—a young nobleman—at the Glyndebourne Festival. As the backlash to this criticism crested, the ancient question was revived: is it fair—or even plausible—to ask opera singers to look the part?
As with so many others of life’s great mysteries, the answer—at least based on Monday’s concert—is, “it depends.” Ms. Barton, for example, is nowhere near a size two, but that hardly matters, because she is a dynamo of a singer, wielding a large, deeply colored voice that never sounds heavy or ponderous. With a quick, energetic vibrato it sails all the way up to a resplendent high C, a note few other mezzos care to dwell upon.
There’s a hungry, slightly aggressive quality to her singing, as if she can’t wait to grab the audience by the lapels and shake them up. She and Ms. Wagner began the program with a duet from Verdi’s Aïda, a vocal catfight between princesses. Ms. Barton almost immediately plunged her voice into a biting, almost guttural chest register, and returned to this feral sound over and over again during the scene. The effect was electric, like a thoroughbred straining against its rider’s control.
Later, she sang a duet from Il Trovatore in which the gypsy Azucena has a vision of being burned at the stake, expressed in a wild phrase plunging from high G to low B. As this last note echoed through Central Park, applause broke out, right in the middle of the scene.
Ms. Barton is not primarily a vocal savage, though, as demonstrated in the elegance of her legato in the fervent aria “Acerba voluttà” from Cilea’s Adriana Lecouvreur. The slow lyrical section of this piece lies low in the voice, but she kept the tone round and controlled, the sound of an aristocrat’s passion. Two scenes from Bellini’s Norma served as a reminder of her smashing Met debut in the role of Adalgisa last season and whetted the appetite for more mezzo showcase parts such as Donizetti’s La Favorite.
Tenor Russell Thomas sings Rigoletto, “La Donna e Mobile” at Central Park SummerStage accompanied by pianist Dan Saunders.
If this had been The Jamie Barton Show, it would have been worth two hours of a summer night, but Russell Thomas revealed a world-class spinto tenor every bit as resplendent as Ms. Barton’s mezzo. Mr. Thomas has money notes to spare, tossing off high A-naturals nonchalantly in the tricky “Kleinzach” aria from Offenbach’s Les Contes d’Hoffmann and finishing the concert on a ringing top B in “La donna è mobile” from Rigoletto.
In between, he proved himself an adept Verdi stylist in scenes from Don Carlo, Macbeth and Il Trovatore. True, he’s stocky, like a fullback in the off-season, but he knows how to move onstage, and, more to the point, how to stand still. Like Ms. Barton, he will likely have a major future at the Met.
The issue of body shape became more problematic when Ms. Wagner took the stage. Her stately dramatic soprano took a couple of numbers to warm up, which meant the audience’s attention was free to wander to her avoirdupois, trussed up unflatteringly in an ill-fitting frock of slate-colored jersey.
When the voice finally broke through, it was undeniably a major instrument, a waterfall of sound perfect for the soaring climactic phrases of the Don Carlo duet and Elsa’s aria from Wagner’s Lohengrin. Other pieces, such as “Pace, pace mio Dio” from Verdi’s Forza del Destino were marred by the soprano’s sluggish rhythm. “My Man’s Gone Now” from Porgy and Bess, offbeat repertoire for the blonde Ms. Wagner, was her best number of the concert, performed with a dramatic gusto that otherwise eluded her.
So here’s a new attempt at a “fat singers” rule: it doesn’t matter how “grand” the body is, so long as the singing is grander. Ms. Barton and Mr. Thomas are evidence of that.
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\begin{document}
\maketitle\
\begin{abstract}
We study a model of random graph where vertices are $n$ i.i.d. uniform random points on the unit sphere $S^d$ in $\mathbb{R}^{d+1}$, and a pair of vertices is connected if the Euclidean distance between them is at least $2- \epsilon$. We are interested in the chromatic number of this graph as $n$ tends to infinity.
It is not too hard to see that if $\epsilon > 0$ is small and fixed, then the chromatic number is $d+2$ with high probability. We show that this holds even if $\epsilon \to 0$ slowly enough. We quantify the rate at which $\epsilon$ can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik--Schnirelman--Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor --- if $\epsilon \to 0$ faster than this, we show that the graph is $(d+1)$-colorable with high probability.
\end{abstract}
\section{Introduction}
Given $\epsilon>0$ and $d \ge 1$, the Borsuk Graph $\text{Bor}^d(\epsilon)$ is the graph with vertex set corresponding to points on the $d-$dimensional unit sphere $S^d\subset\R^{d+1}$ and edges $\{x,y\}$ if and only if $\d{x}{y}>2-\epsilon$, that is, if the two points are $\epsilon$-near to antipodal. Here distance is measured in the ambient Euclidean space $\R^{d+1}$. It is well known that when $\epsilon$ is sufficiently small, its chromatic number is $d+2$, in fact this is equivalent to the Borsuk--Ulam theorem.
The Borsuk graph was part of Lov\'asz's inspiration for his proof of the Kneser conjecture \cite{Lovasz1978}. Among other properties, this graph constitutes a nice example of a graph with large chromatic number and odd girth. See for example \cite{Gabor2011,Prosanov-Raigorodskii-Sagdeev2017,Erdos-Hajnal1966,Erdos-Hajnal1967}. It has also been studied because of its relation with Borsuk's conjecture and distance graphs \cite{Raigorodskii2012, Barg2014,Prosanov2018,Sagdeev18}.
We are interested in the chromatic number of random induced $n$-vertex subgraphs of the Borsuk graph. Our main point is that if $\epsilon \to 0$ slowly enough as $n \to \infty$, then topological lower bounds on chromatic number are tight. This contrasts with the situation studied by Kahle in \cite{Kahle2007}, where topological lower bounds are not efficient for the chromatic number of Erd\H{o}s--R\'enyi random graphs. Similar problems have also been studied for random Kneser graphs in \cite{Kupavskii2018} and \cite{Kiselev-Kupavskii2018}.
The rest of the paper is organized as follows. We finish this section with some definitions and notation. In section \ref{section_2} we prove Theorem \ref{thm_eps_constant} when $\epsilon$ is fixed.
\begin{theorem}\label{thm_eps_constant}
Let $d\geq 1$ and $0<\epsilon<2-\lambda_d$ be fixed. Then a.a.s. $$\chi\left(\Bord{\epsilon}{n}\right)=d+2.$$
\end{theorem}
In section \ref{section_3} we prove Theorem \ref{thm_lower_bound}, stating that the chromatic number is still the same when $\epsilon\to0$ slowly.
\begin{theorem}\label{thm_lower_bound}
Let $\epsilon(n)=C\left(\dfrac{\log{n}}{n}\right)^{2/d}$, where $$C\geq \dfrac{64}{3} \left( \dfrac{3\pi^2}{4} \right)^{1/d}.$$ Then a.a.s. $\chi\left(\Bord{\epsilon(n)}{n}\right)=d+2.$
\end{theorem}
Finally in section \ref{section_4} we prove Theorem \ref{thm_upper_bound}, showing that this rate is tight, up to a constant, in the sense that if $\epsilon\to 0$ faster, then the random Borsuk graph is $(d+1)$-colorable, a.a.s.
\begin{theorem}\label{thm_upper_bound}
Let $\epsilon(n) = C (\log n / n)^{2/d}$, where $$C< \frac{3(4-\lambda_d^2)}{64}\sqrt[d]{\frac{9}{4 d^2}}.$$
Then a.a.s. $\chi\left(\Bord{\epsilon(n)}{n}\right)\leq d+1$.
\end{theorem}
\begin{defi}[Random Borsuk graph]
Given $n \ge 1$, $d\geq 1$, and $\epsilon>0$, we define a \textbf{random Borsuk graph}, $\Bord{\epsilon}{n}$, as follows.
\begin{itemize}
\item Its vertices are $X_1, X_2,\cdots,X_n$, $n$ independent and identically distributed uniform random variables over the $d$-dimensional sphere $S^d\subset \R^{d+1}$ of radius 1.
\item $X_i$ and $X_j$ for $i\neq j$ are connected by an edge, if and only if $\d{X_i}{X_j}>2-\epsilon$, where $\lVert\cdot\rVert$ is the Euclidean distance.
\end{itemize}
\end{defi}
Throughout this paper we will think of random Borsuk graphs on $S^d$ for a fixed dimension $d$. However we will explicitly point out the constants that depend on $d$ in the statement of the results. We will denote the closed ball with center $x$ and radius $r$ by $B(x,r)=\left\{y\in\R^{d+1}: \norm{x-y}\leq r \right\}$. Similarly, we denote intersections of closed balls with the $d$-sphere by $\B{x}{r}$, and we call them spherical caps, so $$\B{x}{r}:=B(x,r)\cap S^d=\left\{y\in S^{d}: \norm{x-y}\leq r\right\}$$
Given a Borel set $F\subset \R^{d+1}$, we denote its volume, i.e. Lebesgue measure, as $\vol{F}$. Similarly, for a Borel set $F\subset S^d$, we denote its area on the surface of the sphere, by $\area{F}$. Also, we denote $\omega_d=\vol{S^d}$ and $\alpha_d=\area{S^d}$. Given a graph $G$, we denote its chromatic number by $\chi(G)$.
We say that an event happens \emph{asymptotically almost surely (a.a.s)} if the probability approaches $1$ as $n \to \infty$.
\section{Random Borsuk Graph with $\epsilon$ constant}\label{section_2}
We start by proving that when $\epsilon>0$ is constant and small, $\chi(\Bord{\epsilon}{n})=d+2$ a.a.s.
\begin{lemma}\label{lemma_min_distance}
For $x,y\in S^d$, $\norm{x-y}>2-\epsilon$ if and only if $\norm{x+y}<2\sqrt{\epsilon-\frac{\epsilon^2}{4}}$
\end{lemma}
\begin{proof}
Since $\overline{(-x)x}$ is a diameter, $\overline{(-x)y}\perp \overline{xy}$. Thus $\norm{x+y}^2=4-\norm{x-y}^2$, so the claim follows.
\end{proof}
Before getting into the analysis of the chromatic number, let us point out the fact that the odd girth of the Borsuk graph is $> 1/\sqrt{\epsilon}$. While this has been observed before (see \cite{Erdos-Hajnal1966,Erdos-Hajnal1967,Gabor2011}), we include a proof for completeness.
\begin{lemma}\label{lemma_odd_girth}
Let $\epsilon>0$ and $x_0\in S^d$. If $x_0y_1x_1y_2\cdots x_ny_{n+1}=x_0$ is an odd cycle in the Borsuk graph $\Bor{\epsilon}$, then $2n+1\geq 1/\sqrt{\epsilon}$. In other words, all odd cycles in $\Bor{\epsilon}$ have length greater than $1/\sqrt{\epsilon}$.
\end{lemma}
\begin{proof}
Since $\norm{x_i-y_i},\norm{y_i-x_{i-1}}>2-\epsilon$, for any $i$, by applying Lemma \ref{lemma_min_distance}, we get
\begin{align*}
\norm{x_i-x_{i-1}} &\leq \norm{x_i+y_i}+\norm{-y_i-x_{i-1}} \\
& =\norm{x_i+y_i} +\norm{y_i+x_{i-1}} \\
& \leq 4 \sqrt{\epsilon - {\epsilon^2}/{4}} \\
& <4\sqrt{\epsilon}.
\end{align*}
Thus $$\norm{x_n-x_0}\leq\norm{x_n-x_{n-1}}+\norm{x_{n-1}-x_{n-2}}+\cdots+
\norm{x_1-x_0}\leq 4n\sqrt{\epsilon}.$$
Finally, $$2=\norm{2x_0}=\norm{x_0+y_{n+1}}\leq \norm{x_0-x_n}+\norm{x_n+y_{n+1}}<2(2n+1)\sqrt{\epsilon}.$$
Therefore $$\frac{1}{\sqrt{\epsilon}}<2n+1.$$
\end{proof}
\begin{lemma}\label{lemma_color_d+2}
For each $d\geq 1$, there exist a constant $\lambda_d<2$ such that, whenever $0<r<2-\lambda_d$, the Borsuk graph $\text{Bor}^d(r)$ has a proper coloring with $d+2$ colors.
\end{lemma}
\begin{proof}
Let $\Delta$ be the regular $(d+1)$-simplex inscribed in the unit $d$-sphere $S^d$. Consider the map $\Phi:\partial\Delta\to S^d$ from the boundary of $\Delta$ to $S^d$ given by $\Phi(x)=x/\norm{x}$. Note then that $\Phi$ is a homeomorphism. Let $\tau\in\partial\Delta$ be a maximal face, and let $\lambda_d=\diam{\Phi(\tau)}$. Since $\Delta$ is regular, the value of $\lambda_d$ does not depend on the face $\tau$.
Note now that $\lambda_d<2$. To see this, suppose that $\lambda_d=2$. Since $\tau$ is closed, so is $\Phi(\tau)$, so there exist $x,y\in\tau$ such that $\norm{\Phi(x)-\Phi(y)}=2$. This means $\Phi(x)$ and $\Phi(y)$ are antipodal, and so $y=-\frac{\norm{y}}{\norm{x}}x$. Since $\tau$ is convex, $0=\sfrac{\norm{y}}{(\norm{x}+\norm{y})}x+\sfrac{\norm{y}}{(\norm{x}+\norm{y})}y$ must also be in $\tau$, but this is a contradiction, since $\tau\subset\partial\Delta$, proving the claim.
We now give a coloring for $S^d$ as follows. We start by coloring $\partial\Delta$: give a different color to each of the $d+2$ facets, and for the lower dimensional faces, assign an arbitrary color among the facets that contain them. Finally, color $\Phi(x)\in S^d$, with the color of $x$.
Note this is indeed a proper coloring for $\Bor{r}$, since all points in $S^d$ of the same color lie on the image of a facet $\Phi(\tau)$, of diameter $\lambda_d$; so if $x$ and $y$ have the same color, $\norm{x-y}\leq\lambda_d< 2-r$ so they are not connected by an edge in the Borsuk graph.
\end{proof}
The upper bound for the chromatic number follows immediately from Lemma \ref{lemma_color_d+2}. The proof we give below for the lower bound, is a direct application of the Lyusternik--Shnirelman--Borsuk Theorem \cite{LS30,Borsuk33}. We state this well-known theorem without proof; for more details and a self-contained proof see, for example, Chapter 2 of Matousek's book \cite{Matousek2008}.
\begin{theorem*}[Lyusternik--Shnirelman--Borsuk] For any cover $U_1, \dots , U_{d+1}$ of the sphere $S^d$ by $d+1$ open (or closed) sets, there is at least one set containing a pair of antipodal points.
\end{theorem*}
We note that B\'ar\'any gave a short proof of Kneser's conjecture using this theorem \cite{Barany78}. See also Greene's proof \cite{Greene02}. For the rest of the paper, we refer to this theorem as the LSB Theorem.
\begin{proof}[Proof of Theorem \ref{thm_eps_constant}]
By Lemma \ref{lemma_color_d+2}, since $\Bord{\epsilon}{n}\subset \text{Bor}^d(\epsilon)$, $d+2$ is an upper bound for the chromatic number of the random Borsuk graph.\\
Let $F_1,F_2,\dots, F_N$ be a cover of $S^d$ by Borel sets, such that $\text{diam}(F_i)\leq\frac{\sqrt{\epsilon}}{2}$ and $\area{F_i}>0$ for all $i$. Note we can construct such a family of sets in many ways, for instance, as we do in the next section, we can consider a $\delta$-net of $S^d$ and let the sets $F_i$ to be spherical caps centered on the $\delta$-net of radius $\sqrt{\epsilon}/4$ where $\delta\leq\sqrt{\epsilon}/4$. Note here that the sets $F_i$ and $N$ depend only on $\epsilon$, which is fixed. \\
Let $c=\min_{i}\frac{\area{F_i}}{\area{S^d}}$ and $G=\Bord{\epsilon}{n}$. The following computation shows that, a.a.s., $G$ contains at least one vertex in each of the sets $F_i$.
\begin{align*}\label{eqn-prob-Fi-has-vertices}
\P{\bigwedge_{i=1}^N \left(V(G)\cap F_i \neq \emptyset\right)} & =
1-\P{\bigvee_{i=1}^N\left(V(G)\cap F_i=\emptyset\right)}\\
&\geq 1 - \sum_{i=1}^N\P{V(G)\cap F_i=\emptyset}\nonumber\\
& = 1 - \sum_{i=1}^N \left( 1-\frac{\area{F_i}}{\area{S^d}}\right)^n \geq 1 -N(1-c)^n
\end{align*}
since $N$ and $c$ are constant, $1-N(1-c)^n\to 1$ as $n\to\infty$, proving the claim. \\
We may assume then, $G$ has a vertex $y_i\in F_i$ for $i=1,\dots, N$. Proceeding by way of contradiction, suppose there exists a proper coloring of $G$ with $d+1$ colors. For each $j=1,\dots, d+1$ define
$$U_j=\bigcup \B{y_k}{\frac{\sqrt{\epsilon}}{2}}$$
where the union is taken over all the $y_k$'s of color $j$.\\
Since $F_i\subset \B{y_i}{\frac{\sqrt{\epsilon}}{2}}$, the sets $U_1,\dots, U_{d+1}$ are a closed cover of $S^d$. Thus, by the LSB Theorem, there exists an antipodal pair in one of the closed sets. Without lost of generality, say $x,(-x)\in U_1$, so $x\in \B{y_1}{\frac{\sqrt{\epsilon}}{2}}$ and $(-x)\in \B{y_2}{\frac{\sqrt{\epsilon}}{2}}$, with both $y_1$ and $y_2$ having color 1. Then,
$$\norm{y_1+y_2}\leq \norm{y_1-x}+\norm{x+y_2}\leq \sqrt{\epsilon}\leq 2\sqrt{\epsilon-\frac{\epsilon^2}{4}},$$
where the last inequality holds because $\epsilon<2$. Lemma \ref{lemma_min_distance} then implies $\norm{y_1-y_2}>2-\epsilon$, so $y_1$ and $y_2$ are connected by an edge in $G$, giving the desired contradiction.
\end{proof}
\section{Random Borsuk Graph with $\epsilon\to 0$}\label{section_3}
\subsection{Lower Bound}
The proof we gave for the lower bound in Theorem \ref{thm_eps_constant}, suggests that we should be able to let $\epsilon\to 0$ and still a.a.s.\ get the same chromatic number. Indeed, this will be the case. We just need to control the number of sets $N$ we use to cover the sphere and their area, in such a way that
$$\lim_{n \to \infty} 1 - N (1-c)^n \to 1.$$
In this section we discuss how to do this using $\delta$-nets on $S^d$, and then adapt the proof of Theorem \ref{thm_eps_constant} to get Theorem \ref{thm_lower_bound}.
We start with a technical lemma on spherical caps.
\begin{lemma}\label{lemma_area_spherical_cap}
Given $x\in S^d$ and $0<r<1$, the following hold for the spherical cap $B=\B{x}{r}$.
\begin{enumerate}
\item The boundary $\partial B$, is a $(d-1)$-dimensional sphere with radius $$r'=r\sqrt{1-r^2/4}.$$
\item $B$ is indeed a cap, i.e. there exist a $d$-hyperplane in $\R^{d+1}$, such that $B$ is the portion of $S^d$ contained in one of the semi-spaces defined by the hyperplane.
\item The area $\area{B}$ satisfies the inequalities
$$\frac{1}{\pi}\left(\frac{\sqrt{3}}{2}\right)^{d-1}r^d\leq \frac{\area{\B{x}{r}}}{\area{S^d}}\leq\frac{d}{3} r^d. $$
\end{enumerate}
\end{lemma}
\medskip
Spherical caps are well studied in the literature. See, for example, Lemmas 2.2. and 2.3 in \cite{Ball97}. We note that Lemma 2.2 in \cite{Ball97} is better than our Lemma \ref{lemma_area_spherical_cap} in the case that $r$ is fixed and $d \to \infty$, but we are interested in the case that $d$ is fixed.
\begin{proof}[Proof of Lemma \ref{lemma_area_spherical_cap}]
Without loss of generality, we may choose $x=N=(0,\dots,0,1)$ to be the north pole by rotating the sphere. Thus $x\in\partial B$ if and only if $\norm{x}=1$ and $\norm{x-N}=r$. Then
\begin{align*}
r^2 &=\norm{x-N}^2 \\
&=x_0^2+\cdots +x_{d-1}^2+(x_d-1)^2 \\
&=x_0^2+\cdots+x_{d-1}^2+x_d^2-2x_d+1 \\
&=2-2x_d. \\
\end{align*}
So $x_d=1-\frac{r^2}{2}$. Therefore, the hyperplane $\{x_d=1-\frac{r^2}{2}\}$ determines the cap $B$, proving (2). To get (1), note $1=\norm{x}^2=x_0^2+\cdots+x_d^2$, so we get $x_0^2+\cdots+x_{d-1}^2=r^2-r^4/4=(r')^2$, for all $x\in\partial B$, proving it is indeed a $(d-1)$-dimensional sphere with the desired radius.\\
For (3), recall we get the area of $B=\B{x}{r}$ by integrating the length $\ell$ of the arc from $x$ to the boundary $\partial B$, over all the possible unit vectors. Thus
$$\area{\B{x}{r}}=\int_{\hat{u}\in S^{d-1}} \ell (r')^{d-1}d\!\hat{u}=\ell(r')^{d-1}\int_{\hat{u}\in S^{d-1}}d\!\hat{u}=\ell(r')^{d-1}\alpha_{d-1}.$$
Some planar geometry gives the bounds $$r\leq \ell=2\arcsin(r/2)\leq\frac{\pi}{3}r,$$ and $$\frac{\sqrt{3}}{2}r\leq r'\leq r,$$ since $0<r<1$. This gives
$$\left(\frac{\sqrt{3}}{2}\right)^{d-1}r^d\alpha_{d-1}\leq \area{B}\leq \frac{\pi}{3}r^d\alpha_{d-1}$$
Recall the formula for the surface area of the unit $d$-dimensional sphere $$\alpha_d=\dfrac{2\pi^{(d+1)/2}}{\Gamma\left((d+1)/2\right)}.$$
By using the fact that the Gamma function is increasing on $[2,\infty)$ and treating the first few cases separately, we have that for $d\geq 1$,
$$\frac{1}{\pi}\leq\frac{\alpha_{d-1}}{\alpha_d}=\frac{\Gamma((d+1)/2)}{\sqrt{\pi}\ \Gamma(d/2)}\leq\frac{d}{\pi},$$
from which the desired result follows.
\end{proof}
For the sake of completeness, we include the following discussion on $\delta$-nets. See, for example, \cite[Chapter 13]{Matousek2002} for more details.
\begin{defi}[$\delta$-Nets]
Given a metric space $X$ with metric $d$, a \textbf{$\boldsymbol{\delta}$-net} is a subset $\mathcal{B}\subset X$ such that for every $x\in X$, there exists $y\in \mathcal{B}$ with $d(x,y)<\delta$.
\end{defi}
We can then construct a $\delta$-net for any compact metric space $M$ inductively. Indeed, choose any point $y_1\in M$.
For each $m\geq 2$, if $B_m=\cup_{i=1}^m \B{y_i}{\delta}\subsetneq M$, choose any $y_{m+1} \in M \setminus B_m$. Otherwise, stop and let $\mathcal{B}=\{y_1,y_2,\dots\,y_m \}.$
Compactness ensures that the process stops. It's clear that $\mathcal{B}$ is a $\delta$-net and moreover it is also a maximal $\delta$-apart set. That is, $d(y_i,y_j)>\delta$ whenever $i\neq j$, and we can not add any other point to $\mathcal{B}$ without destroying this property. This implies that the balls of radius $\delta$ and center on the points $y_i$'s cover $M$, while the open balls of radius $\delta/2$ with center on the $y_i$'s are all disjoint. We now show that we can control the size of the $\delta$-net in the case that $M=S^d$.
\begin{lemma}\label{lemma-nets}
For every $d \ge 1$ and $0<\delta<1$ there exists a $\delta$-net $\mathcal{B}\subset S^d$ , such that (1) for every two points $y_i, y_j\in\mathcal{B}$, $\norm{y_i-y_j}>\delta$, and (2) its cardinality $\mathcal{N}=|\mathcal{B}|$ satisfies:
$$\frac{3}{d\delta^d}\leq \mathcal{N}\leq \frac{2(3^d)(d+1)}{\delta^d}.$$
\end{lemma}
\medskip
In the literature, it seems more common to find an upper bound such as
$$\mathcal{N}\leq(4 / \delta)^{d+1},$$ which is a better bound when $\delta$ is constant and $d\to\infty$ \cite[Lemma 13.1.1]{Matousek2002}. However, the bound we give is more useful for us since we are dealing with $d$ constant and $\delta\to0$.
\begin{proof}[Proof of Lemma \ref{lemma-nets}]
By letting $\mathcal{B}$ be the $\delta$-net defined above, we already have a $\delta$-net for the sphere $S^d$ that is also a $\delta$-apart maximal set. To prove the inequalities on its cardinality we give a volume and an area arguments. \\
Since the points $y_i\in\mathcal{B}$ are $\delta$ apart, the open balls $\text{int}\left(B\left(y_i,\frac{\delta}{2}\right)\right)$ are disjoint. So
$$ \vol{\bigcup_{i=1}^N B\left(y_i,\frac{\delta}{2}\right)}= \mathcal{N}\omega_{d}\left(\frac{\delta}{2}\right)^{d+1},$$
where $\omega_{d}$ denotes the volume of the $d+1$-dimensional unit ball in $\R^{d+1}$. Morever, all such balls are contained in the set $B\left(0,1+\frac{\delta}{2}\right) \setminus B\left(0,1-\frac{\delta}{2}\right)$. Thus
\begin{align*}
\vol{\bigcup_{i=1}^N B\left(y_i,\frac{\delta}{2}\right)} &\leq
\vol{B\left(0,1+\frac{\delta}{2}\right)} - \vol{B\left(0,1-\frac{\delta}{2}\right)}\\
& = \omega_{d}\left(\left(1+\frac{\delta}{2}\right)^{d+1} - \left(1-\frac{\delta}{2}\right)^{d+1}\right)\\
&= \omega_{d}\delta \sum_{r=0}^{d}\left(1+\frac{\delta}{2}\right)^{d-r}\left(1-\frac{\delta}{2}\right)^r \\
& \leq \omega_{d}\delta \sum_{r=0}^{d}\left(1+\frac{\delta}{2}\right)^{d}\\
&=\omega_{d}\delta (d+1)\left(1+\frac{\delta}{2}\right)^d \\
& \leq \omega_d\delta(d+1)\left(\frac{3}{2}\right)^d,
\end{align*}
and the upper bound for $\mathcal{N}$ follows.
For the lower bound, we consider the area of the spherical caps. Since all points in $S^d$ are within distance $\delta$ of the points $y_i\in\mathcal{B}$ we must have
$$\area{S^d}=\area{\bigcup_{i=1}^{\mathcal{N}}\B{y_i}{\delta}}\leq\sum_{i=1}^{\mathcal{N}}\area{\B{y_i}{\delta}}=\mathcal{N}\area{\B{y_1}{\delta}}$$
Therefore, Lemma \ref{lemma_area_spherical_cap} yields $\displaystyle\mathcal{N}\geq \frac{\area{S^d}}{\area{\B{y_0}{\delta}}}\geq\frac{3}{d\delta^d}$.
\end{proof}
We now proceed to prove Theorem \ref{thm_lower_bound}.
\begin{proof}[Proof of Theorem \ref{thm_lower_bound}]
Let $G=\Bord{\epsilon(n)}{n}$. Since $\epsilon\to 0$, eventually $\epsilon<2-\lambda_d$, so by Lemma \ref{lemma_color_d+2}, $\chi(G)\leq d+2$. We now proceed to prove the lower bound by a modification of the proof of theorem \ref{thm_eps_constant}. \\
Let $\delta = \sqrt{\epsilon} / 4$. Let $\mathcal{B}$ be the $\delta$-net given by Lemma \ref{lemma-nets}. Say $\mathcal{B}=\{y_1,y_2,\cdots,y_N\}$, where $$N\leq \frac{2(3^d)(d+1)}{\delta^d}=\frac{A_d}{\epsilon^{d/2}},$$ and $A_d$ is a constant which only depends on $d$. For each $i=1,\cdots, N$, define $F_i=\B{y_i}{\delta}$. Note then the $F_i's$ cover the sphere and $\diam{F_i}\leq2\delta=\frac{\sqrt{\epsilon}}{2}$. \\
Applying Lemma \ref{lemma_area_spherical_cap}, we have
$$\frac{\area{F_i}}{\area{S^d}}\geq
\frac{1}{4\pi}\left(\frac{\sqrt{3}}{8}\right)^{d-1}\epsilon^{d/2}=
B_d \epsilon^{d/2}=:c,$$
where $B_d$ is constant.\\
Finally, all that remains to prove is that $1-N(1-c)^n\to 1$ as $n\to\infty$, even when $N$ and $c$ depend on $n$. This is as follows
\begin{align*}
N(1-c)^n &\leq \frac{A_d}{\epsilon^{d/2}}\left(1-c\right)^n \\
&=
\frac{A_d}{\epsilon^{d/2}}\left(1-B_d\epsilon^{d/2}\right)^n \\
&=
\frac{A_dn}{C^{d/2}\log{n}}\left(1-\frac{B_dC^{d/2}\log{n}}{n}\right)^n\\
&\leq \frac{A_dn}{C^{d/2}\log{n}}\exp\left(-B_dC^{d/2}\log{n}\right)\\
&= \frac{A_d}{C^{d/2}\log{ n}} n^{1-B_dC^{d/2}}
\end{align*}
The last expression goes to zero as $n\to\infty$, since $$C\geq \frac{64}{3}\sqrt[d]{\frac{3\pi^2}{4}},$$ so $B_d C^{d/2}\geq 1$ and this completes the proof.
\end{proof}
\begin{coro}
If $$\frac{64}{3}\sqrt[d]{\frac{3\pi^2}{4}}\left(\dfrac{\log{n}}{n}\right)^{2/d}\leq \epsilon(n)\leq 2-\lambda_d$$ for all sufficently large $n$, then $\chi\left(\Bord{\epsilon(n)}{n}\right)=d+2$ a.a.s.
\end{coro}
\begin{proof}
The chromatic number is monotone with respect to $\epsilon$, so this follows directly.
\end{proof}
\subsection{Upper Bound}\label{section_4}
Theorem \ref{thm_lower_bound} and its Corollary shows that if $\epsilon \to 0$ sufficiently slowly then the chromatic number of the random Borsuk graph is a.a.s. $d+2$. In this section we show that the rate obtained is tight, up to a constant factor. That is, we show an upper bound for $\epsilon$ for which the random Borsuk graph is $(d+1)$-colorable.
We start our analysis by constructing a proper coloring of $\Bor{\epsilon}\setminus\B{x}{\delta}$ with exactly $d+1$ colors, for a suitable $\delta$ that depends on $\epsilon$. Lemma \ref{lemma_area_spherical_cap} establishes that the boundary of an spherical cap on $S^d$ is a $S^{d-1}$ with radius $\delta'$, and Lemma \ref{lemma_color_d+2} allows to color it with $d+1$ colors. We will provide the technical details to translate this coloring into a proper coloring of the desired graph.\\
For the following analysis consider the spherical cap $A=\B{N}{r}$, where $N$ is the north pole. For each $x\in S^d\setminus\{N,-N\}$, let $\gamma_x:[0,\pi]\to S^d$ be the great semi-circle going from $N$ to $-N$ and passing through $x$. Define $f: S^d\to\partial A$ by letting $f(x)$ be the intersection of $\gamma_x$ with $\partial A$. Note this is a well defined function, since if $x=(x_0,\dots, x_d)$, we can parametrize $$\gamma_x(t)=\left(\frac{\sin{t}}{\sqrt{1-x_d^2}}x_0,\dots,\frac{\sin{t}}{\sqrt{1-x_d^2}}x_{d-1},\cos{t}\right)$$
so its last coordinate takes all values in $[-1,1]$ exactly once for $0\leq t\leq \pi$, and from Lemma \ref{lemma_area_spherical_cap} we know $\partial A$ consists of all points with last coordinate $a:=1-\frac{r^2}{2}$.\\
The following lemmas construct the desired coloring.
\begin{lemma}\label{lemma_distance_geodesics}
Let $x, y\in S^d\setminus\{N,-N\}$ such that $\norm{x-y}\leq\delta$. Define $y'=(y'_0,\dots,y'_d)$ to be the point in the geodesic $\gamma_y$, such that $y'_d=x_d$. Then $\norm{x-y'}\leq 2\delta$.
\end{lemma}
\begin{proof}
Without lost of generality, we may assume $y=(0,0,\dots,0,\sqrt{1-y_d^2},y_d)$, since we can get this by a rotation of the sphere that leaves the last coordinate fixed. This rotation fixes the north and south poles, so it also transforms the geodesic through $y$ into another geodesic through $y$. Thus, the formula for the geodesic simplifies to
$$\gamma_y=(0,\dots,0,\sin{t},\cos{t}),\text{ for } 0\leq t\leq\pi.$$
So, $y'=(0,\dots,0,\sqrt{1-x_d^2},x_d)$. Then
\begin{align*}
\norm{x-y}^2
&=x_0^2+\cdots+x_{d-2}^2+\left(x_{d-1}-\sqrt{1-y_d^2}\right)^2+(x_d-y_d)^2\\
&= (1-x_d^2)+(1-y_d^2)-2x_{d-1}\sqrt{1-y_d^2}+(x_d-y_d)^2\\
\end{align*}
and
\begin{align*}
\norm{y-y'}^2
&=\left(\sqrt{1-y_d^2}-\sqrt{1-x_d^2}\right)^2+(x_d-y_d)^2\\
&= (1-x_d^2)+(1-y_d^2)-2\sqrt{1-x_d^2}\sqrt{1-y_d^2}+(x_d-y_d)^2
\end{align*}
Since $x_{d-1}\leq |x_{d-1}|\leq \sqrt{x_0^2+\cdots+x_{d-1}^2}=\sqrt{1-x_d^2}$, we get $\norm{y-y'}\leq\norm{x-y}$, and so $$\norm{x-y'}\leq\norm{x-y}+\norm{y-y'}\leq 2\norm{x-y}\leq 2\delta.$$
\end{proof}
\begin{lemma}\label{lemma_distance_F}
Let $x, y\in S^d\setminus\{N,-N\}$ such that $x\not\in A\cup (-A)$ and $\norm{x-y}\leq\delta$. Then $$\norm{f(x)-f(y)}\leq 2\delta.$$
\end{lemma}
\begin{proof}
Let $y'\in S^d$ such that its last coordinate is $y'_d=x_d$. From the parametrization for $\gamma_x$, we see $f(x)=\gamma_x(t_1)$, where $\cos{t_1}=a$ and $\sin{t_1}=\sqrt{1-a^2}=\delta'$, the radius of $\delta A$, hence
$$f(x)=\left(\frac{\delta'}{\sqrt{1-x_d^2}}x_0,\dots,\frac{\delta'}{\sqrt{1-x_d^2}}x_{d-1},a\right).$$
A similar expression holds for $f(y')$, with $y'_d=x_d$, so we get
\begin{align*}
\norm{f(x)-f(y')}&=\sqrt{\sum_{i=0}^{d-1}\frac{\delta'^2}{1-x_d^2}(x_i-y_i')^2}\\
&=\frac{\delta'}{\sqrt{1-x_d^2}}\sqrt{\sum_{i=0}^{d-1}(x_i-y'_i)^2}\\
&\leq \frac{\delta'}{\sqrt{1-x_d^2}} \norm{x-y'}
\end{align*}
Moreover, since $x\not\in A\cup (-A)$, $|x_d|<a$, so $\frac{\delta'}{\sqrt{1-x_d^2}}<1$, so $\norm{f(x)-f(y')}\leq \norm{x-y'}$. Finally, if we let $y'$ be the one defined in Lemma \ref{lemma_distance_geodesics}, $f(y')=f(y)$, and therefore $\norm{f(x)-f(y)}=\norm{f(x)-f(y')}\leq \norm{x-y'}\leq 2\delta$.
\end{proof}
\begin{lemma}\label{lemma_color_d+1}
Let $0<\epsilon<1$, such that $$ r=\frac{8\sqrt{\epsilon}}{\sqrt{3(4-\lambda_{d-1}^2)}}<1,$$ $x\in S^d$, and $A=\B{x}{r}$. Let $H$ be the induced subgraph of $\Bor{\epsilon}$ by the vertex set $S^d\setminus A$. Then $\chi(H)\leq d+1$.
\end{lemma}
\begin{proof}
Without loss of generality let $x=N$ the north pole, so $A=\B{N}{r}$. Lemma \ref{lemma_area_spherical_cap} says $\partial A$ is a $S^{d-1}$ sphere of radius $r'=r\sqrt{1-\frac{r^2}{4}}\geq\frac{\sqrt{3}}{2}r$. Thus adapting Lemma \ref{lemma_color_d+2}, we can color it in such a way that every two points with the same color are at a distance of at most $\lambda_{d-1}r'$. We then color $H$ by giving each point $y\in S^d\setminus A\setminus\{-N\}$ the color of $f(y)$, and giving the south pole $-N$ any color. We proceed to prove this is a proper coloring of $H$. \\
From Lemma \ref{lemma_min_distance}, the neighbors of the south pole lie in $\B{N}{\sqrt{\epsilon-\epsilon^2/4}}\subset A$, so $-N$ is isolated in $H$. Let $y,z\in S^d\setminus A\setminus\{-N\}$ such that $\norm{y-z}>2-\epsilon$. Lemma \ref{lemma_min_distance} implies $\norm{y+z}<\delta:=2\sqrt{\epsilon-\frac{\epsilon^2}{4}}$. If we had $(-y),(-z)\in A$, that would mean $y,z\in -A$, but then $\norm{y-z}\leq r\leq 2-\epsilon$ for small $\epsilon$. So we may assume $(-y)\not\in A$, and since $y\not\in A$, $(-y)\not\in -A$. Thus $(-y)\not\in A\cup (-A)$ and $\norm{-y-z}\leq\delta$, thus Lemma \ref{lemma_distance_F} implies
$$\norm{f(-y)-f(z)}\leq 2\delta=4\sqrt{\epsilon-\frac{\epsilon^2}{4}}<4\sqrt{\epsilon}=\sqrt{4-\lambda_{d-1}^2}\frac{\sqrt{3}}{2}r\leq \sqrt{4-\lambda_{d-1}^2}r'$$
From the definition of $f$, it is clear that $f(-y)=-f(y)$, thus Lemma \ref{lemma_min_distance} implies $\norm{f(y)-f(z)}>\lambda_{d-1}r'$, and so $f(y)$ and $f(z)$ have different colors, meaning $y$ and $z$ have different colors as well. Therefore $\chi(H)\leq d+1$.
\end{proof}
As an immediate application, if a random Borsuk graph leaves some spherical cap in $S^d$ of radius bigger than $r$ with no vertices, then it can be colored with $d+1$ colors. We will show that this is indeed the case when $\epsilon\to 0$ at the said rate. We now include some theorems about Poisson Point Processes and Poisson distributions. For their proofs and a complete discussion refer to \cite{Penrose2003} or \cite{Kingman1993}.
\begin{theorem}[Poissonization]\label{thm_poissonization}
Let $X_1, X_2, \dots$, be uniform random variables on $S^d$. Let $M\sim \text{Pois}(\lambda)$ and let $\eta$ be the random counting measure associated to the point process $P_\lambda$= $\{X_1,X_2,\dots,X_M\}$. Then $P_\lambda$ is a Poisson Point Process and for a Borel $A\subset S^d$, $\eta(A)\sim \pois{\lambda\frac{\area{A}}{\area{S^d}}}$.
\end{theorem}
\begin{lemma}\label{lemma_poisson}
For $n\geq 0$, $\P{\pois{2n}<n}\leq e^{-0.306n}.$
\end{lemma}
We are now ready to prove the Theorem \ref{thm_upper_bound}.
\begin{proof}[Proof of Theorem \ref{thm_upper_bound}]
Let $X_1, X_2, \dots$, be uniform random variables on $S^d$. Let $M\sim\pois{2n}$. Let $\eta$ be the random counting measure of the Poisson Point Process $\left\{X_1,\dots,X_M\right\}$. Similarly, let $\eta_1^n$ be the counting measure of the Random points $\left\{X_1,\dots, X_n\right\}$.\\
Let $$\delta=\frac{16\sqrt{\epsilon}}{\sqrt{3(4-\lambda_{d-1}^2)}}=A_d\sqrt{\epsilon},$$ where $A_d$ is a constant which only depends on $d$. Let $\mathcal{B}=\left\{y_1,\dots,y_N\right\}$ be the $\delta$-net given by Lemma \ref{lemma-nets}, so $$N \geq \frac{3}{d\delta^d}=\frac{B_d}{\epsilon^{d/2}},$$ and $B_d$ is constant. Let $F_i=\B{y_i}{\delta/2}$ for $i=1,\dots,N$ be spherical caps centered at the $\delta$-net. Thus, as in the proof of \ref{lemma-nets}, the $F_i$'s are disjoint.
Lemma \ref{lemma_area_spherical_cap} gives $$\frac{\area{F_i}}{\area{S^d}}\leq\frac{d}{3}\left(\frac{\delta}{2}\right)^d=D_d\epsilon^{d/2},$$
where $D_d$ is constant.
Note that these spherical caps have the same radius required by Lemma \ref{lemma_color_d+1}, so if we prove that a.a.s.\ one of these $F_i$'s doesn't contain any vertices of the random Borsuk graph, then it must be contained in $S^d\setminus F_i$, and the Lemma \ref{lemma_color_d+1} gives a proper $(d+1)$ coloring. This is what we do.\\
Note that
\begin{equation}
\P{\min_{1\leq i\leq N}\eta(F_i)=0} \leq \P{\min_{1\leq i\leq N}\eta_1^n(F_i)=0}+\P{M<n}.
\label{eqn_prob_unif_vs_poisson}
\end{equation}
We have
\begin{align*}
\P{\min_{1\leq i\leq N}\eta(F_i)=0}&=1-\P{\bigwedge_{i=1}^N\eta(F_i)>0}=1-\prod_{i=1}^N\P{\eta(F_i)>0}\\&=1-\left(1-\P{\pois{2n\frac{\area{F_1}}{\alpha_d}}=0}\right)^N\\
&\geq 1-\exp\left(-\exp\left(-2n\frac{\area{F_1}}{\alpha_d}\right)N\right)\\
&\geq 1-\exp\left(-\exp(-2nD_d\epsilon^{d/2})\frac{B_d}{\epsilon^{d/2}}\right)\\
&=1-\exp\left(-\frac{B_d}{C^{d/2}\log{n}}n^{1-2D_dC^{d/2}}\right).\\
\end{align*}
This last expression tends to 1 as $n\to\infty$, since $C$ is such that $1-2D_dC^{d/2}>0$.\\
Lemma \ref{lemma_poisson} assures that $$\P{M<n}=\P{\pois{2n}<n}\to 0$$ as $n\to \infty,$ and therefore (\ref{eqn_prob_unif_vs_poisson}) gives $\displaystyle\P{\min_{1\leq i\leq N}\eta_1^n(F_i)=0}\to 1$, as desired.\\
\end{proof}
\section{Further Questions}
\begin{enumerate}
\item It might be possible to find sharper constants in Theorems \ref{thm_lower_bound} and \ref{thm_upper_bound}. For $d=1$, it is certainly possible. The following can be achieved with similar methods to the ones used throughout this paper, so we include the statement without proof.
\begin{theorem}
Let $\epsilon=C \left(\log n / n \right)^2$.
\begin{enumerate}
\item If $C\geq 9\pi^2 /4$, then a.a.s. $\chi\left(\Borone{\epsilon}{n}\right)=3$.
\item If $C<\pi^2/4$, then a.a.s. $\chi \left( \Borone{\epsilon}{n} \right) \le 2$.
\end{enumerate}
\end{theorem}
\item We wonder whether there exist functions $\epsilon=\epsilon(n)$ such that the chromatic number of the random Borsuk graph $\Bord{\epsilon}{n}$ a.a.s.\ equals $i$, for $1 \le i \le d+1$.
\item We only studied here the case that $d$ is fixed and $\epsilon$ is either fixed or tends to zero at some rate. It also seems interesting to let $d \to \infty$ at some rate, or to let $d$ be fixed and $\epsilon \to 2$. See, for example, Raigorodskii's work on coloring high-dimensional spheres \cite{Raigorodskii2012}.
\end{enumerate}
\noindent We thank our anonymous referees for careful reading and helpful comments.
\bibliographystyle{plain}
\bibliography{References}
\end{document}
| 154,002
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Fisher is an actress, writer and lecturer best known for her portrayal of Princess Leia in the orignal Star Wars trilogy.
She won the Kim Peek Award for Disability in Media in 2012, for sharing the details of her battles with bipolar disorder.
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| 81,729
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Earn more with the same resources. In general, this is what we mean by improving the sales force’s productivity, one of the main goals of any modern company. To companies with sales force, this is defined as the relation between the number of sales achieved by sales force and the resources invested to obtain the aforesaid goals. Experience shows that the most quick and efficient way to improve sales force is to change the way in which it is managed.
Did you know that an unfocused and without leader sales team sells a 40% less? The figure of the Sales Manager is –due to its privileged position to influence, persuade and enhance sellers towards the highest levels of success– key for any company with sales force. However, this usually goes unnoticed due to its big amount of work, complicating this way the workers’ productivity at its expense. How to solve this? The 6 tips below will help you achieve it.
1. Support your sales team
No matter how big your sales staff may be, your duty as the Sales Manager is to know your team very well in order to reach the planned goals and achieve above average results. For your sales team work properly, sellers should be lined up, motivated and do a good performance.
It’s not enough to tell your employees what to do and how to do it focusing only on indicators and terms to try to boost their performance, because this will turn into a hostile environment and they will feel overwhelmed. You must know each of your workers and adapt to their needs: enhance their strong points and help them with their weaknesses. An effective Sales Manager is the one who finds the best way to reach his team with motivation and rewards that make the best of them. This is what will stimulate their productivity!
Confidence is gained in the “battlefield”, going to the sales visits and working side by side with them. The sentence “a happy employee works better“ is quite right. Coaching is the less cared responsibility by the Sales Managers because it requires more time. Sales Managers have realized that supporting their workers is the basis to obtain their teams’ confidence and unity.
2. Offer the best resources
Providing the best tools, a lot of training and a good working environment are essential things to develop and feed back your team. It is indispensable that, at least once a month, the sales force has the chance to renew the documents they use to reinforce their arguments and speeches when selling. Offer them the latest technologies so that they could sell your products more and better. Make their work easier with specific apps for sales force that will boost sales.
Furthermore, at least once a year, they should receive additional professional information to progress in their job function. Bear in mind that, in the most of the cases, sales force fails due to a lack of training rather than their attitude. To finish, don’t forget to create a proper working environment so that your sales team could feel motivated and therefore, they will increase their performance, fighting for an increment in the company’s sales.
3. Boost the internal communication
It is proved that those Sales Managers who share a lot of information and knowledge with their sales team become more effective. Hence the importance of does a meeting weekly and communicating every day with their workers.
To know exactly what to do, to have Sales Manager’s support via mobile, to share knowledge and transmit experience make the sales force more capable. Is because of this that we recommend to create a WhatsApp group including the members of sales team so that they could use it when a seller has difficulties and help him instantly. Workers appreciate having support, especially in the most complicated moments.
4. Manage sales force
Sales Manager’s essence has not changed in the last years: an efficient leader must be skilled in planning, organizing, managing, evaluating and controlling his area and his workers to boost their productivity.
A good Sales Manager knows his workers and how to manage them according their nature. The “star sellers” must be cared, motivated and rewarded to maximize their performance. On the contrary, you must work proactively with those workers who present small negative behavior patterns to prevent bad habits or low results that will lead to affect the annual productivity.
The best way to manage and unite the areas and the sales workers is to promote integration and cooperation activities. Make funny out-of-work activities in which everybody must work together to reach a goal. This way, you’ll find out what are the problems existing among the company’s staff. Furthermore, you’ll make that they trust each other in their jobs, because working together outside the working space unites the team.
The last point, and the most important one, is the tracking of all that influences the sales. Measure all the activities to see their efficiency and track the results accurately.
5. Recognize their merits
Win is funny and it must be celebrated. So, recognize the individual and team merits whenever you have the chance. The best way to relax is to reward their triumphs –even the most insignificant ones– as often as you can and use it as an opportunity to boost your team. A joint celebration creates motivation and unity to keep on selling and reaching their goals.
6. Commit to failures
To be a good Sales Manager you must commit to your workers’ failures too. To fail is not as bad as it seems: it means not to achieve a goal or an objective but also to learn how to avoid this mistake. It is very important to know: Why do we fail? How can we solve it? What do we need? What are we going to change the next time? How can we help our sales force?
To blame the responsible before analyzing the situation causes a bad working environment, suspicion and deception for that person. At this point, the sales team can both become unified or divide completely. If you are comprehensive with your worker, support him and help him to overcome, he’ll trust you and will fight more the next time. As popular sayings says:
“You must keep together in good times and in bad ones”
According to recent studies, the 70% of the staff gives up the boss, not the work. That’s why we recommend having a tracking of your workers, to be close to them, to support them, to help them when they need it the most and to build ties to make them feel comfortable inside the community and motivate them to sell. Do you think your company follows these advices?
| 263,957
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TITLE: FEM: Steady-State heat diffusion and convection
QUESTION [0 upvotes]: So the strong form of the heat diffusion and convection PDE is given as
$\rho c_m \mathbf{v}\cdot\nabla T - \nabla \cdot \nabla T = \dot{q}\\
T(\mathbf{x},t) = T_e(\mathbf{x},t)~~~ on ~~~\Gamma_e ~~~~(\text{Dirichlet-BC})\\
k \frac{\partial T}{\partial \mathbf{n}} = q_n~~~ on ~~~\Gamma_n ~~~~(\text{Neumann-BC})\\
\mathbf{v}....\text{velocity, given}$
Then I introduced a test function $w$ and derived the weak form of the PDE ($\Omega$ - volume, $\Gamma$ - surface):
$
\int_\Omega w \rho c_m \mathbf{v} \cdot\nabla T~ d\Omega + \int_\Omega (\nabla w)\cdot (k \nabla T) ~d\Omega = \int_\Omega w \dot{q} ~d\Omega + \int_{\Gamma_n} w q_n~d\Gamma
$
From that I want to derive Galerkin's finite element formulation by approximating the domain $\Omega$ and the function spaces $T(\mathbf{x},t)$ and $w(\mathbf{x})$ by known shape functions $N_i(\mathbf{x})$.
$
\int_\Omega (\nabla w)\cdot (k \nabla T)~ d\Omega \approx \sum_{m=1}^M \int_{\Omega^{(m)}} (\nabla w)\cdot (k \nabla T)~ d\Omega \\
w(\mathbf{x}) \approx \sum_{i=1}^{N^{(m)}} w_i N_i(\mathbf{x})\\
~~~~~~~~~~~~~~N_i(\mathbf{x_j}) = \delta_{ij}
$
When inserting this approximations in the weak form of the PDE, I can obtain a single equation for node i=k by assuming that $w_k = 1$ at the node and $w_k=0$ otherwise. I get the final result as:
$
\sum_{m=1}^{M_e} \sum_{m=1}^{N^{(m)}} [T_j \int_{\Omega^{(m)}} (N_i(\mathbf{x})\rho c_m \mathbf{v} \cdot \nabla N_j(\mathbf{x})+k(\nabla N_i(\mathbf{x})\cdot \nabla N_j(\mathbf{x}))) d\Omega]= \sum_{m=1}^{M_e} \int_{\Omega^{(m)}} N_i(\mathbf{x}) \dot{q} d\Omega + \sum_{m=1}^{M_n} \int_{\Gamma_n^{(m)}} N_i(\mathbf{x}) q_n d\Gamma
$
which I can write in Matrix-Vector form as the following:
$\sum_{j=1}^{N^{(m)}} K_{ij}^{(m)}T_j^{(m)} = f_i^{(m)}, ~~~~~~j = 1,...,N^{(m)} $
with the system matrix
$K_{ij}^{(m)} = \int_{\Omega^{(m)}} (N_i(\mathbf{x})\rho c_m \mathbf{v} \cdot \nabla N_j(\mathbf{x})+k(\nabla N_i(\mathbf{x})\cdot \nabla N_j(\mathbf{x}))) d\Omega$
But now my question: can this be right? Because I read that the system matrix normally should be symmetric and this is not fullfilled here?
Thanks for any help!
REPLY [0 votes]: The calculation seems fine to me. I have not gone in details but from an overlook, it looks fine. Now to answer your main question, why isn't the matrix symmetric? The answer is the matrix can be non-symmetric! The Symmetricity of the matrix depends on the symmetric nature of the bilinear form. In fact, the bilinear form is symmetric if and only if the matrix is symmetric.
Mostly standard books on FEM considers the Poisson problem, there we have the symmetric nature of the bilinear form and hence we get a symmetric matrix. The problem you have doesn't have a symmetric bilinear form
| 134,588
|
Description
Directions for use:
Step 1 – Using the included scoop, add powder to paint. Two scoops per 8 oz. of paint will provide proper consistency. Increase or decrease amount to your preference. Mixture should be a consistency between cake batter and a thick frosting.
Step 2 – Apply mixture with brush, using a pouncing motion, creating peaks. Try applying in a variety of thicknesses. This will create more of an authentic crusted look. Allow to dry slightly, but not completely. Once most top areas are dry, use a rubber spatula or spackle knife to knock down and flatten the peaks, creating a textured look.
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| 2,511
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\section{Conclusions and outlook}
In this paper the basic concepts of deriving full-moment Kershaw closures were derived. These models provide a huge gain in efficiency compared to the state-of-the-art minimum-entropy models, since they can be closed (in principle) analytically using the available realizability theory. Benchmark tests confirm that Kershaw closures can indeed compete with minimum-entropy models. In some situations, they are even better.
Although the gain in efficiency is very big, recent results for minimum-entropy models showed that using high-order numerics is still advantageous \cite{Schneider2015a,Schneider2015b}. The big problem in high-order schemes for moment models is that the property of realizability has to be preserved during the simulation, since otherwise the fluxes cannot be evaluated. Future work will have to investigate how to adapt the scheme in \cite{Schneider2015a} to Kershaw closures. Furthermore, different scattering operators should be taken into account, like the slightly more complicated (in terms of realizability preservation) Laplace-Beltrami operator.
Finally, the concepts have to be lifted to higher dimensions. While fully three-dimensional first-order variants of Kershaw closures exist \cite{Ker76,Schneider2015c}, no higher-order models or a completely closed theory is available.
| 148,776
|
TITLE: Convergence in probability implies equality with probability approaching 1?
QUESTION [1 upvotes]: Suppose $X_n\overset{p}{\to} X$. Does this imply $\Pr(X_n=X)\to 1$ as $n\to\infty$?
My approach would be to prove the equivalent statement $\Pr(X_n\neq X)\to 0$ as $n\to\infty$. Since $\big\{ X_n\neq X \big\}=\bigcup\limits_{k=1}^{\infty} \Big\{\lvert X_n-X \lvert > \frac{1}{k} \Big\} $, we have $\Pr(X_n\neq X)\leq \sum\limits_{k=1}^{\infty}\Pr(\lvert X_n-X \lvert > \frac{1}{k})$ by countable subadditivity. Because we assumed $X_n\overset{p}{\to} X$, each term inside the sum can be made arbitrarily small if $n$ is chosen large enough. Am having trouble with filling the details. Any help is greatly appreciated.
REPLY [4 votes]: Let $X_n = 1/n$ and $X=0$. Then for any $\epsilon > 0$, the probability $P(|X_n - X| > \epsilon) = \mathbf{1}_{\{1/n > \epsilon\}}$ is zero for all sufficiently large $n$, so $X_n \overset{p}{\to} X$. However, $P(X_n = X) = 0$ for all $n$.
Regarding your proof attempt: it is true that each term in the sum can be made arbitrarily small if $n$ is large enough, but it may not be true that you can find an $n$ that simultaneously makes all terms small enough for the entire sum to be to be small.
| 60,070
|
TITLE: Why does 17 change signs when solving this equation?
QUESTION [0 upvotes]: Currently working through some problem sets on straight line equations but I'm just not grasping the last part. Any further explanation or intuition to help the concept sink in would be great.
$$y - y_1 = m(x - x_1) \\
y - 7 = \frac 23(x - 2) \\
3y - 21 = 2x - 4$$
Then putting it in the form $0 = ax + by + c$
$$0 = 2x - 3y + 17$$
Setting $x = 0$
$$0 = -3y + 17$$
Setting $y = 0$
$$0 = 2x + 17$$
The answer I've given says $x = \frac{-17}2$ <---- Why does the $17$ turn into $-17$? I believe that both $2x$ and $17$ is divided by $2$ to solve for $x$ but I can't seem to get my head around why it turns to $-17$ and I know when someone tells me, the answer is going to be obvious!
Thanks
REPLY [2 votes]: $$\require{cancel}\begin{align}0 &=2x+17 \\ (0) \color{red}{-17} &= (2x+\cancel{17})\color{red}{-\cancel{17}} & (\text{subtract $17$ from both sides}) \\ -17 &= 2x & (\text{simplify}) \\ \color{red}{\frac{\color{black}{-17}}{2}} &=\color{red}{\frac{\color{black}{\cancel{2}x}}{\cancel2}} & (\text{divide both sides by $2$}) \\ \frac{-17}{2} &=x & (\text{simplify})\end{align}$$
In words: our goal was to get the $x$ by itself (on the right) and everything else over to the other side (on the left). So first we got rid of the $17$ on the right by subtracting it from both sides (remember, anything you do to one side you must do to the other). Then there was still that $2$ left over on the right so we divided both sides by it to get rid of it.
Does that help?
| 161,014
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\begin{document}
\def\dis{\displaystyle}
\def\dotminus{\mathbin{\ooalign{\hss\raise1ex\hbox{.}\hss\cr
\mathsurround=0pt$-$}}}
\begin{center}
{\LARGE{\sf $\aleph_0$-categorical spaces contain $\ell_p$ or
$c_0$ }} \vspace{10mm}
{
{\bf \sf Karim Khanaki}} \vspace{3mm}
{\footnotesize
Faculty of Fundamental Sciences,\\
Arak University of Technology,\\
P.O. Box 38135-1177, Arak, Iran; \\
e-mail: khanaki@arakut.ac.ir, and
\medskip
School of Mathematics, \\
Institute for Research in Fundamental Sciences (IPM),\\
P.O. Box 19395-5746, Tehran, Iran; \\
e-mail: khanaki@ipm.ir} \vspace{5mm}
\end{center}
{\sc Abstract.}
{\small This paper has three parts. First, we establish some of
the basic model theoretic facts about $\mathbb{T}$, the Tsirelson
space of Figiel and Johnson \cite{FJ}. Second, using the results
of the first part, we give some facts about general Banach spaces.
Third, we study model-theoretic dividing lines in some Banach
spaces and their theories. In~particular, we show: (1) every
\emph{strongly separable} space, such as $\mathbb{T}$, has the
\emph{non independence property} (NIP); (2) every
\emph{$\aleph_0$-categorical space} is stable if and only if it
is strongly separable; consequently $\mathbb{T}$ is not
$\aleph_0$-categorical; equivalently, its first-order theory (in
any countable language) does not characterize $\mathbb{T}$, up to
isometric isomorphism; (3) every \emph{explicitly definable}
space contains isometric copies of $c_0$ or $\ell_p$ ($1\leqslant
p<\infty$).
}
\medskip
{\small{\sc Keywords}: Tsirelson's space, continuous logic,
strongly separable, stable space, Rosenthal space, non
independence property, $\aleph_0$-categorical, explicitly
definable }
AMS subject classification: 46B04, 46B25, 03C45, 03C98.
{\small \tableofcontents
\bigskip
{\bf References} {\hspace{\stretch{0.01}} \bf \pageref{ref}} }
\noindent\hrulefill
\section{Introduction}
A famous conjecture in Banach space theory had predicted that
every Banach space contains at least one of the classical spaces
$c_0$ or $\ell_p$ for some $1\leqslant p<\infty$. All the Banach spaces
known until 1974 were {\em normal} and this conjecture was
confirmed. Tsirelson's example \cite{Ts} was the first space not
containing isomorphic copies any of the classical sequence
spaces, and the first space whose norm was defined implicitly
rather than explicitly.
This phenomenon, i.e. implicit definability, later played an important role in
Banach space theory when new spaces with implicitly defined norms
yielded solutions to many of the most long standing problems in
the theory (see \cite{Gowers}). Given than all new spaces whose
norms were implicitly defined do not contain any of the classical
sequences, some Banach space theorists, such as Gowers
\cite{Gowers, Gowers-weblog} and Odell \cite{Od-2}, asked the
following question:
\medskip
\textbf{(Q1)}~``Must an `explicitly defined' Banach space contain
$\ell_p$ or $c_0$?''
\medskip
To provide a positive answer to this question one must first
provide a precise definition of `explicitly defined space'.
Second, he/she must show that explicitly defined spaces contain
$\ell_p$ for some $1\leqslant p<\infty$, or $c_0$. Third, all
{\em normal} Banach spaces such as $L_p$ spaces, Lorentz spaces,
Orlicz spaces, Schreier spaces, etc., are explicitly defined but
the Tsirelson example and similar spaces are not (see
\cite{Gowers-weblog} for more discussions).
On the other hand, the notion of definability plays a basic and
key role in model theory and its applications in algebraic
structures. Recall that a subset $A$ of a first-order structure
$M$ is definable if $A$ is the set of solutions in $M$ of some
formula $\phi(x)$. However, the problem there is that the usual
first-order logic does not work very well for structures in
Analysis.
Chang and Keisler \cite{CK} and Henson \cite{Hen} produced the
{\em continuous logics} and the {\em logic of positive bounded
formulas}, respectively, to study structures in analysis using
model theoretic techniques. Since then, numerous attempts to
provide a suitable logic has been going on (e.g. see \cite{HenM,
HI, Ben-cat}), and eventually led to the creation of the {\em
first order continuous logic} (see \cite{BU, BBHU} for more
details). In many ways the latter logic is the best and the
ultimate.
The main purpose of this article is to give a definition of
`definable norm' (or `explicitly defined space') using the
continuous logic presented in \cite{BU, BBHU} and to prove that
Banach spaces whose norms are definable contain $\ell_p$ for some
$1\leqslant p<\infty$ or $c_0$. Since the dichotomy
stable/unstable (in both logics, classic and continuous) has been
widely studied in literature, and some model theoretic
properties, such as IP and SOP, in continuous logic (or Banach
space theory) have received less attention, the second goal of
this paper is to provide examples of these concepts. The third
and final purpose is to get a better understanding of
communication between both fields (model theory and Banach space
theory) so that techniques from one field might become useful in
the other.
To summarize the results of this paper, in the first part
(Sections \ref{strong separability} and \ref{aleph_0
categorical}), we study the example of Tsirelson as a
guidance. Since, by the Krivine-Maurey theorem which asserts that
every stable space contains some $\ell_p$ almost isometrically,
Tsirelson's space is not stable, so we consider the following
question: Does Tsirelson's space have the non independence
property (NIP)? The NIP is a model theoretic property weaker than
stability.\footnote{See section~\ref{section-type} for a
definition of NIP.} By a result due to E. Odell \cite{Od} which
asserts that the type space of Tsirelson's space is {\em strongly
separable}, we show that the answer to the latter question is
positive.\footnote{See section~\ref{strong separability} for a
definition of strongly separable space.} On the other hand, a
recent result due to T. Ibarluc\'{i}a \cite{Iba} implies that an
$\aleph_0$-categorical space is stable if and only if its type
space is strongly separable. Roughly speaking a separable Banach
space is $\aleph_0$-categorical if it can be exactly
characterized by its first order theory (in a countable
language), up to isometric isomorphism. Using these observations,
we conclude that the theory of Tsirelson's space (in any
countable language) is not $\aleph_0$-categorical, equivalently,
its norm can not be {\em defined} by first order axioms in
continuous logic. This fact leads us to the following question:
\medskip
\textbf{(Q2)}~Does an $\aleph_0$-categorical Banach space contain
$\ell_p$ or $c_0$?
\medskip
In the second part of paper (Section \ref{explicit definable} and
\ref{perturb}), we answer to this question. In fact, the answer
is yes, and this is a positive answer to the question
\textbf{(Q1)} above; furthermore we claim that
$\aleph_0$-categoricity is the `true and correct' notion of
`explicitly defined space'. We will discuss this subject in
detail and offer other observations that could be interesting in
themselves. Then we study perturbed $\aleph_0$-categorical
structures and prove that separable structures which are
$\aleph_0$-categorical up to small perturbations
contain $\ell_p$ or $c_0$.
In the third part of the paper (Sections \ref{correspondence} and
\ref{examples}), we first remark some results in \cite{K4} about
correspondences between dividing lines in model theory and
Banach space theory which will be used in Section 9. Then, we
study model theoretic dividing lines in some Banach spaces and
their theories.
In Section \ref{appendix} (Appendix), we remark an observation
about Rosenthal's dichotomy communicated to us by Michael
Megrelishvili which asserts that Fact~\ref{fact3} below is not a
dichotomy in noncompact Polish case. To our knowledge this
observation itself does not appear somewhere in literature in a
clear form.
It is worth recalling another line of research. In \cite{Iov},
Jose Iovino studied the connection between the definability of
types and the existence of $\ell_p$ and $c_0$ in the framework of
Henson's logic, i.e. the logic of positive bounded formulas. He
showed that the existence of enough definable types guarantees
the existence of $\ell_p$ or $c_0$ as a subspace. Despite this,
his result is `local' and does not give any `global' information
about the norm. Unlike him, in the present paper, we study some
global properties of a norm which yield the existence of $\ell_p$
and $c_0$ subspaces (see Section \ref{explicit definable}).
After preparing a number of editions of the present paper we
came to know that, independently from us, Alexander Usvyatsov
\cite{Usv} has observed that $\aleph_0$-categorical spaces contain
some classical sequences. We guess that his approach is `local'
too.
This paper is organized as follows: In the second section, we give
some prerequisites. In the third section, we briefly review
continuous logic and the notions of types in model theory and
Banach space theory. In the fourth section, we recall the
definition of Tsirelson's space which is used later. In the fifth
section, we show that every Banach space with strongly separable
types, as well as Tsirelson's space, has NIP. In the sixth
section, we study $\aleph_0$-categorical spaces and point out
that every $\aleph_0$-categorical space is stable if and only if
it is strongly separable, and consequently Tsirelson's space is
not $\aleph_0$-categorical. In the seventh section, we prove the
main theorem that every explicitly definable space contains $c_0$
or $\ell_p$. In the eighth section,
we study perturbed
$\aleph_0$-categorical structures and give an extension of the
main result of the seventh section.
In the ninth section, we present a result similar
to the Krivine-Maurey theorem. In the tenth section, we briefly
review some model theoretic dividing lines and their connections
to some functional analysis notions presented in \cite{K4}. In
the eleventh section, we study some examples and investigate
their model theoretic dividing lines. In the appendix, we give
some remarks on Rosenthal's dichotomy.
\bigskip\noindent
{\bf Acknowledgements.} I want to thank Alexander Berenstein,
Gilles Godefroy, Richard Haydon, Ward Henson, Tom\'{a}s
Ibarluc\'{i}a, Michael Megrelishvili and Alexander Usvyatsov for
their valuable comments and observations.
I am grateful to Thomas Schlumprecht for sending a copy of
\cite{Od} to me. I would like to thank the Institute for Basic
Sciences (IPM), Tehran, Iran. Research partially supported by IPM
grants no. 92030032 and 93030032.
\noindent\hrulefill
\section{Preliminaries}
\subsection{Independence family of functions}
\begin{dfn} \label{indep} \begin{itemize}
\item [(i)] A sequence $\{f_n\}$ of real valued functions on a set $X$ is said to be {\em independent } if there exist real numbers $s<r$ such that
$$\bigcap_{n\in P} f_n^{-1}(-\infty,s)\cap\bigcap_{n\in M} f_n^{-1}(r,\infty)\neq\emptyset \ \ \ \ \ \ \ \ \boxtimes$$
for all finite disjoint subsets $P,M$ of $\mathbb
N$. A family $F$ of real valued functions on $X$ is called {\em independent} if it contains an independent sequence; otherwise it is called
{\em strongly} (or {\em completely}) {\em dependent}.
\item [(ii)] We say that the sequence $\{f_n\}$ is {\em strongly} (or {\em completely})
{\em independent} if $\boxtimes$ holds for all {\em infinite}
disjoint subsets $P,M$ of $\Bbb N$. A family $F$ of real valued functions is called {\em strongly} (or {\em completely})
{\em independent} if it contains a strongly independent sequence; otherwise it is called {\em dependent}.
\end{itemize}
\end{dfn}
It is an easy exercise to check that when $X$ is a compact space
and functions are continuous the above two notions are the same,
but this does not hold in general (see Example~\ref{c_0} below).
\begin{lem} \label{dependent=strong dependent}
Let $X$ be a {\em compact} space and $F\subseteq C(X)$ a bounded
subset. Then the following conditions are equivalent:
\begin{itemize}
\item [(i)] $F$ is dependent.
\item [(ii)] $F$ is strongly dependent.
\end{itemize}
\end{lem}
\begin{lem}[Rosenthal's lemma] \label{lemma-1} Let $X$ be a compact space and $F\subseteq C(X)$ a bounded subset. Then the following conditions are equivalent:
\begin{itemize}
\item [(i)] $F$ does not contain an independent subsequence.
\item [(ii)] Each sequence in $F$ has a convergent subsequence
in $\mathbb R^X$.
\end{itemize}
\end{lem}
Rosenthal \cite{Ros} used the above lemma for proving his famous
$\ell^1$ theorem: a sequence in a Banach space is either `good'
(it has a subsequence which is weakly Cauchy) or `bad' (it
contains an isomorphic copy of $\ell^1$). We will shortly discuss
this topic (see Section~\ref{appendix}).
\noindent\hrulefill
\section{Type space} \label{section-type}
We assume that the reader is familiar with continuous logic (see
\cite{BBHU} or \cite{BU}). We will study the notion of type in
continuous logic and its connection with the notion of type in
Banach space theory. In this paper, local types are more
important for us.
\subsection{Types in model theory}
A difference between our frame work and the continuous logic
presented in \cite{BBHU} is that we study real-valued formulas
instead of $[0,1]$-valued formulas. One can assign bounds to
formulas and retain compactness theorem in a local way again.
\begin{con}
In this paper, we do not need to study unbounded metric
structures. In fact, the unit ball $B=\{x\in V:\|x\|\leqslant
1\}$ of a Banach space $V$ is sufficient. Likewise, although each
formula can have an arbitrary bound, but our focus is on the
formula $\phi(x,y)=\|x+y\|$. So, we can assume that the atomic
formulas are $[0,2]$-valued.
\end{con}
Suppose that $L$ is an arbitrary language. Let $M$ be an
$L$-structure, $A\subseteq M$ and $T_A=Th({M}, a)_{a\in A}$. Let
$p(x)$ be a set of $L(A)$-statements in free variable $x$. We
shall say that $p(x)$ is a {\em type over} $A$ if $p(x)\cup T_A$
is satisfiable. A {\em complete type over} $A$ is a maximal type
over $A$. The collection of all such types over $A$ is denoted by
$S^{M}(A)$, or simply by $S(A)$ if the context makes the theory
$T_A$ clear. The {\em type of $a$ in $M$ over $A$}, denoted by
$\text{tp}^{M}(a/A)$, is the set of all $L(A)$-statements
satisfied in $M$ by $a$. If $\phi(x,y)$ is a formula, a {\em
$\phi$-type} over $A$ is a maximal consistent set of formulas of
the form $\phi(x,a)\geqslant r$, for $a\in A$ and
$r\in\mathbb{R}$. The set of $\phi$-types over $A$ is denoted by
$S_\phi(A)$.
\subsubsection{The logic topology and $d$-metric}
We now give a characterization of complete types in terms of
functional analysis. Let $\mathcal{L}_A$ be the family of all
interpretations $\phi^{M}$ in $M$ where $\phi$ is an
$L(A)$-formula with a free variable $x$. Then $\mathcal{L}_A$ is
an Archimedean Riesz space of measurable functions on $M$ (see
\cite{Fremlin3}). Let $\sigma_A({M})$ be the set of Riesz
homomorphisms $I: {\mathcal L}_A\to \mathbb{R}$ such that
$I(\textbf{1}) = 1$. The set $\sigma_A({M})$ is called the {\em
spectrum} of $T_A$. Note that $\sigma_A({M})$ is a weak* compact
subset of $\mathcal{L}_A^*$. The next proposition shows that a
complete type can be coded by a Riesz homomorphism and gives a
characterization of complete types. In fact, by compactness
theorem, the map $S^{M}(A)\to\sigma_A({M})$, defined by $p\mapsto
I_p$ where $I_p(\phi^M)=r$ if $\phi(x) = r$ is in $p$, is a
bijection.
\begin{rmk} \label{rem2} For an Archimedean Riesz space $U$ with order unit
$\textbf{1}$, write $X$ for the set of all normalized Riesz
homomorphisms from $U$ to $\Bbb R$, or equivalently,
the positive extreme points of unit
ball of the dual space $U^*$, with its weak* topology. Then $X$ is
compact by Alaoglu's theorem and the
natural map $u\mapsto \hat{u}:U\to \mathbb{R}^X$
defined by setting $\hat{u}(x)=x(u)$ for $x\in X$ and $u\in U$,
is an embedding from $U$ to an order-dense and norm-dense
embedding subspace of $C(X)$.
\end{rmk}
Now, by the above remark, $\mathcal L_A$ is dense in $C(S^M(A))$.
\begin{fct} \label{key}
Suppose that $M$, $A$ and $T_A$ are as above.
\begin{itemize}
\item [(i)] The map $S^{M}(A)\to\sigma_A({M})$ defined by $p\mapsto I_p$ is bijective.
\item [(ii)] $p\in S^{M}(A)$ if and only if there is an elementary
extension $N$ of $M$ and $a\in N$ such that
$p=\text{tp}^{N}(a/A)$.
\end{itemize}
\end{fct}
We equip $S^{M}(A)=\sigma_A({M})$ with the related topology
induced from $\mathcal{L}_A^*$. Therefore, $S^{M}(A)$ is a compact
and Hausdorff space. For any complete type $p$ and formula
$\phi$, we let $\phi(p)=I_p(\phi^{M})$. It is easy to verify that
the topology on $S^{M}(A)$ is the weakest topology in which all
the functions $p\mapsto \phi(p)$ are continuous. This topology
sometimes called the {\em logic topology}. The same things are
true for $S_\phi(A)$.
\begin{dfn}[The $d$-metric] The space $S_\phi(M)$ has another
topology. Indeed, define $d(p,q)=\sup_{a\in
M}|\phi(p,a)-\phi(q,a)|$. Clearly, $d$ is a metric on $S_\phi(M)$
and it is called the {\em $d$-metric } on the type space
$S_\phi(M)$.
\end{dfn}
\begin{rmk} \label{uniform top} The $d$-metric sometimes is called the {\em uniform topology}.
For unbounded logics, there is a third topology on the type space,
namely the {\em strong topology}. Indeed, define
$d_s(p,q)=\sum_1^k\frac{1}{2^k}\sup_{\|a\|\leqslant k}
|\phi(p,a)-\phi(q,a)|$. R. Haydon has informed us that he has
constructed a Banach space $M$ for which its type space is
strongly separable but not uniformly separable. Of course, for
bounded continuous logic, and hence in this paper, the strong and
uniform topologies are the same.
\end{rmk}
\begin{dfn} A (compact) \emph{topometric} space is a triplet $\langle X, \tau,\frak{d}\rangle$,
where $\tau$ is a
(compact) Hausdorff topology and $\frak d$ a metric on $X$, satisfying:
(i) The metric topology refines the topology.
(ii) For every closed $F \subseteq X$ and $\epsilon > 0$, the closed
$\epsilon$-neighbourhood of $F$ is closed in $X$ as well.
\end{dfn}
\begin{fct} Let $M$ be a structure and $\phi(x,y)$ a formula. The triplet $\langle S_\phi(M), \frak{T},d\rangle$ is
a compact topometric space where $\frak{T}$ and $d$ are the logic
topology and the $d$-metric on the type space $S_\phi(M)$,
respectively.
\end{fct}
Note that $(S_\phi(M),\frak{T})$ is the weakest topology in which
all functions $p\mapsto\phi(p,a)$, $a\in M$, are continuous, and
$(S_\phi(M),d)$ is the weakest topology in which all functions
$p\mapsto\phi(p,a)$, $a\in M$, are uniformly continuous with a
{\em modulus of uniform continuity} $\Delta_\phi$ (see
\cite{BBHU}).
\subsection{Types in Banach space theory}
Let $M$ be a Banach space and $a$ be in the unit ball $B_M=\{x\in
M:\|x\|\leqslant 1\}$. In Banach space theory, a function
$f_a:B_M\to[0,2]$ defined by $x\mapsto\|x+a\|$ is called a {\em
trivial type} on the Banach space $M$. A {\em type} on $M$ is a
pointwise limit of a family of trivial types. The set
$\frak{T}(M)$ of all types on $M$ is called the {\em space of
types}. In fact $\frak{T}(M)\subseteq[0,2]^M$ is a compact
topological space with respect to the product topology. This
topology is called the {\em weak topology} on types, and denoted
by $\frak{T}'$.
$\frak{T}(M)$ has another topology. Define $d'(f,g)=\sup_{a\in
M}|f(a)-g(a)|$ for $f,g\in \frak{T}(M)$. Clearly, $d'$ is a metric
on $\frak{T}(M)$.
\begin{fct}
The triplet $\langle \frak{T}(M),\frak{T}',d' \rangle$ is a
compact topometric space.
\end{fct}
Let $\phi(x,y)=\|x+y\|$. Then there is a correspondence between
$S_\phi(M)$ and $\frak{T}(M)$. Indeed, let $a\in M$, and consider
the quantifier-free type $a$ over $M$, denoted by
$\mathrm{tp}_{\mathrm{qf}}(a/M)$. It is easy to verify that
$\mathrm{tp}_{\mathrm{qf}}(a/M)$ corresponds to the function
$f_a:M\to[0,2]$ defined by $x\mapsto\|x+a\|$. Also, there is a
correspondence between $\mathrm{tp}_{\mathrm{qf}}(a/M)$ and the
Riesz homomorphism $I_a:{\mathcal{L}_M}\to[0,2]$ defined by
$I_a(\phi(x,y))=\phi^{\M}(x,a)$ for all $x\in M$. To summarize,
the following maps are bijective:
$$\mathrm{tp}_{\mathrm{qf}}(a/M)\rightsquigarrow f_a$$
$$\mathrm{tp}_{\mathrm{qf}}(a/M)\rightsquigarrow I_a$$
Now, it is easy to check that their closures are the same:
\begin{fct}
Assume that $M$ is a Banach space and $\phi(x,y)=\|x+y\|$. The
compact topometric spaces $\langle S_\phi(M),\frak{T},d \rangle$
and $\langle \frak{T}(M),\frak{T}',d' \rangle$ are the same. More
exactly, $(S_\phi(M),\frak{T})$ and $(\frak{T}(M),\frak{T}')$ are
homeomorphic, and $(S_\phi(M),d)$ and $(\frak{T}(M),d')$ are
isometric.
\end{fct}
\subsection{Local stability and NIP}
\begin{dfn} \label{stable-formula}
Let $\M$ be a structure, and $\phi(x,y)$ a formula. The following
are equivalent and in any of the cases we say that $\phi(x,y)$ is
stable on $M\times M$ (or on $M$).
\begin{itemize}
\item [(i)] Whenever $a_n,b_m\in M$ form two sequences we have $$\lim_n\lim_m\phi(a_n,b_m)=\lim_m\lim_n\phi(a_n,b_m),$$
provided both limits exist.
\item [(ii)] The set $A=\{\phi(x,b):S_x(M)\to{\Bbb R}~|b\in M\}$ is relatively weakly compact in $C(S_x(M))$.
\end{itemize}
\end{dfn}
The equivalence (i)~$\Leftrightarrow$~(i) is called Gothendieck's
criterion (see~\cite{K3}).
\begin{dfn} A Banach space (or Banach structure) $\M$ is {\em stable} if
the formula $\phi(x,y)=\|x+y\|$ is stable on $\M$.
\end{dfn}
\begin{dfn} \label{NIP-formula}
Let $\M$ be a structure, and $\phi(x,y)$ a formula.
The following are equivalent and in any of the cases we say that
$\phi(x,y)$ is NIP on $M\times M$ (or on $M$).
\begin{itemize}
\item [(i)] For each sequence $(a_n)\subseteq M$, and
$r>s$, there are some {\em finite} disjoint subsets $P,M$ of ${\Bbb N}$ such that $$\Big\{b\in M:\big( \bigwedge_{n\in P}\phi^{\M}(a_n,b)\leqslant
s\big)\wedge\big(\bigwedge_{n\in M}\phi^{\M}(a_n,b)\geqslant r\big)\Big\}=\emptyset.$$
\item [(ii)] For each sequence $(a_n)\subseteq M$, each elementary extension ${\N}\succeq{\M}$, and
$r>s$, there are some {\em arbitrary} disjoint subsets $P,M$ of ${\Bbb N}$ such that $$\Big\{b\in N:\big( \bigwedge_{n\in P}\phi^{\N}(a_n,b)\leqslant
s\big)\wedge\big(\bigwedge_{n\in M}\phi^{\N}(a_n,b)\geqslant r\big)\Big\}=\emptyset.$$
\item [(iii)] For each sequence $\phi(a_n,y)$ in the set
$A=\{\phi(a,y):S_y(M)\to{\Bbb R}~|~a\in M\}$, where $S_y(M)$ is
the space of all complete types on $M$ in the variable $y$, and $r>s$
there are some {\em finite} disjoint subsets $P,M$ of ${\Bbb N}$ such that $$\Big\{y\in S_y(M):\big( \bigwedge_{n\in P}\phi(a_n,y)\leqslant
s\big)\wedge\big(\bigwedge_{n\in M}\phi(a_n,y)\geqslant r\big)\Big\}=\emptyset.$$
\item [(iv)] The condition (iii) holds for
{\em arbitrary} disjoint subsets $P,M$ of ${\Bbb N}$.
\item [(v)] Every sequence $\phi(a_n,y)$ in $A$ has a convergent subsequence.
\end{itemize}
\end{dfn}
By the compactness theorem, (i)~$\Leftrightarrow$~(ii), and by
the compactness of the type space, (iii)~$\Leftrightarrow$~(iv).
(iii)--(iv) by Rosenthal's lemma and Fact~\ref{dependent=strong
dependent}. (ii)~$\Leftrightarrow$~(iv), by the definition of
elementary extension in model theory.
\begin{rmk} Note that for each separable structure $\M$, every sequence
$\phi^{\M}(a_n,y):M\to\Bbb R$ has a convergent subsequence (see
\ref{diagonal-lemma}), but this does not imply that every sequence
$\phi(a_n,y):S_\phi(M)\to\Bbb R$ in $A$ has a convergent
subsequence (see Example~\ref{c_0}).
\end{rmk}
\begin{dfn} A Banach space (or Banach structure) $\M$ has {\em NIP} if
the formula $\phi(x,y)=\|x+y\|$ is NIP on $\M$.
\end{dfn}
\begin{fct}
Let $\M$ be a structure, and $\phi(x,y)$ a formula. If $\phi$ is
stable on $\M$ then $\phi$ has NIP on $\M$. In particular, every
stable Banach space has NIP.
\end{fct}
\noindent\hrulefill
\section{Tsirelson's space} \label{Tsirelson's space}
Although we only use some known properties of
Tsirelson's example \cite{Ts} or actually the dual space of the
original example as described by Figiel and Johnson \cite{FJ},
for the sake of completeness, we present Tsirelson's space and
remind some its properties which are used in this paper. One of
its properties will be presented in the next section (see
Fact~\ref{Odell-fact} below).
A collection $(E_i)_1^n$ of finite subsets of natural numbers is
called {\em admissible} if $n\leqslant E_1<E_2<\cdots<E_n$. By
$E<F$ we mean $\max E<\min F$ and $n\leqslant F$ means
$n\leqslant\min F$. For $x\in c_{00}$ and $E\subseteq {\Bbb N}$
by $Ex$ we mean the restriction of $x$ to $E$, i.e. $Ex(n)=x(n)$
if $n\in E$ and $0$ otherwise.
Now we recall the definition $\Bbb T$ (the Tsirelson of
\cite{FJ}). For all $x\in c_{00}$ set
$$\|x\|=\max\Big(\|x\|_\infty,~\sup\Big\{\frac{1}{2}\sum_{i=1}^n\|E_ix\|: (E_i)_1^n \text{ is admissible} \Big\}\Big).$$
$\Bbb T$ is then defined to be the completion of
$(c_{00},\|\cdot\|)$. Note that this norm is given implicitly
rather than explicitly.
One can define this norm on $c_{00}$ by induction. Indeed, for
$x\in c_{00}$, set $\|x\|_0=\|x\|_\infty$ and inductively
$$\|x\|_{n+1}=\max\Big(\|x\|_n,~\sup\Big\{\frac{1}{2}\sum_{i=1}^n\|E_ix\|_n: (E_i)_1^n \text{ is admissible} \Big\}\Big).$$
Then $\|x\|=\lim_n\|x\|_n$ is the desired norm.
\begin{fct} \label{properties}
Tsirelson's space has the following properties.
\begin{itemize}
\item [(i)] $\Bbb T$ is reflexive.
\item [(ii)] $\Bbb T$ does not contain a subspace
isomorphic to $c_0$ or $\ell_p$ ($1\leqslant
p<\infty$).
\end{itemize}
\end{fct}
In addition, it is easy to verify that $(e_n)$ is a normalized
{\em unconditional basis} for $\Bbb T$ and any {\em spreading
model} of $\Bbb T$ is isomorphic to $\ell_1$. See \cite{AK} for
definitions and proofs.
The following result, due to Krivine and Maurey, gives a partially
answer to the main question of the present paper.
\begin{thm}[Krivine--Maurey, \cite{KM}] \label{Krivine--Maurey} Every (separable) stable Banach space contains
almost isometric copies of $\ell_p$ for some $1\leqslant
p<\infty$.
\end{thm}
\begin{cor} \label{unstable}
$\Bbb T$ is not stable.
\end{cor}
\begin{proof}
Immediate by Fact~\ref{properties} and
Theorem~\ref{Krivine--Maurey}.
\end{proof}
A Banach space is {\em weakly stable} if the condition~(i) in
Definition~\ref{stable-formula} holds whenever $(a_n)$ and
$(b_m)$ are both weakly convergent. It is proved that every weakly
stable space contains almost isometric copies of $\ell_p$ for
some $1\leqslant p<\infty$ or $c_0$ (see \cite{ANZ}). So, $\Bbb
T$ is not too weakly stable.
\noindent\hrulefill
\section{Strong separability and NIP} \label{strong separability}
We will show that Tsirelson's space is strongly independent,
equivalently it has NIP. Indeed , we show that every (separable)
Banach structure with strongly separable space of types has NIP.
Then the desired result will be achieved from
Fact~\ref{Odell-fact} below. For this, First we consider the
stable case.
\medskip
Recall that a function $f$
form a metric space $X$ to a metric space $Y$ is $k$-Lipschitz
($k\geqslant 0$) if for all $x,y$ in $X$, $d(f(x),f(y))\leqslant
k\cdot d(x,y)$. Now we give the following easy lemma.
\begin{lem} \label{1-Lip} Assume that $(X,d)$ is a metric spaces, and $(f_\alpha)$ is a
pointwise convergent net of $k$-Lipschitz functions from $X$ to
$\Bbb R$. Then the pointwise limit of $(f_\alpha)$ is
$k$-Lipschitz.
\end{lem}
\begin{proof}
Let $f=\lim_n f_\alpha$. If $d(x,y)<\epsilon$, then
$|f_\alpha(x)-f_\alpha(y)|\leqslant k\epsilon$ for all $\alpha$.
Now $|f(x)-f(y)|\leqslant
|f(x)-f_\alpha(x)|+|f_\alpha(x)-f_\alpha(y)|+|f_\alpha(y)-f(y)|$
for all $\alpha$. So, $|f(x)-f(y)|\leqslant k\epsilon$.
\end{proof}
\begin{dfn} We say a (separable) Banach space is {\em strongly separable},
or equivalently {\em its type space is strongly separable}, if
$(S_\phi(M),d)$ is separable where $\phi(x,y)=\|x+y\|$ and $d$ is
the $d$-metric on $\phi$-types.
\end{dfn}
Krivine and Maurey \cite{KM} showed that separable stable Banach
spaces are strongly separable. For a separable and stable space
$M$, this is actually a consequence of the separability of $M$ and
the definability of types in stable models (see \cite{Ben-Gro} or
\cite{K3, K4}). We thank Gilles Godefroy for communicating to us
the following argument (see \cite{Od}, and compare with
Proposition~1 therein).
\begin{fct}[Definability of types] \label{stable->strong} Assume that $M$ is a separable Banach structure and
$\phi(x,y)=\|x+y\|$ is stable on $M$. Then the type space
$S_\phi(M)$ is strongly separable.
\end{fct}
\begin{proof}
Since $\phi$ is stable, by Grothendieck criterion, the set
$F=\{\phi(x,a):a\in M\}$ is a relatively compact subset of
$C(S_\phi(M))$ for the topology of pointwise convergence. Recall
that $(S_\phi(M),\frak{T},d)$ is a compact topometric space,
$\frak{T}$ is a compact Polish space, and $d$ is complete. Also,
$\frak{T}$ is the weakest topology such that all functions in
$\overline{F}$, i.e. the closure of $F$ with respect the topology
pointwise convergence, are $\frak{T}$-continuous, and $d$ is the
weakest topology such that all functions in $\overline{F}$ are
$1$-Lipschitz for $d$-metric. (Indeed, note that by
Lemma~\ref{1-Lip}, every $f\in\overline F$ is $1$-Lipschitz.)
$\overline{F}\subseteq C(S_\phi(M))$ is compact for the topology
of pointwise convergence, and since $(S_\phi(M),\frak{T})$ is
separable, the topology $\tau$ of pointwise convergence is
metrizable on $\overline F$. Now $S_\phi(M)$ is a subset of
$C(\overline F)$ when $\overline F$ is equipped with $\tau$. And
since $d$ is the weakest topology such that all functions in
$\overline F$ are 1-Lipschitz, the distance $d$ is the
restriction of the norm topology of $C(\overline F)$. But since
$(\overline{F}, \tau)$ is a metrizable compact space, the
separability of $C(\overline F)$ and thus of $S_\phi(M)$ follows.
\end{proof}
The converse does not hold. In fact, Edward Odell proved that
\begin{fct}[Odell, \cite{Od}] \label{Odell-fact}
Tsirelson's space is strongly separable. Furthermore, its type
space in unbounded continuous logic is uniformly separable (see
Remark~\ref{uniform top} above).\footnote{Does the type space of
a Tsirelson-like space such as Schlumprecht's space ${\mathcal
S}_f$ is strongly separable? Repeat the argument in the proof in
\cite{Od}. See \cite[Corollary 13.31]{BL}.}
\end{fct}
By the Krivine--Maurey theorem, this space is not stable (see Corollary~\ref{unstable} above).
We will show that this space has NIP. For this, we need the
following lemma.
\begin{lem} \label{diagonal-lemma}
Assume that $(X,d)$ is a metric spaces, and $F$ a bounded family
of $1$-Lipschitz functions from $X$ to $\Bbb R$. If $X$ is
separable then every sequence in $F$ has a (pointwise) convergent
subsequence.
\end{lem}
\begin{proof} The proof is an easy diagonal argument. Indeed, assume that $(g_m)$ is a sequence in $F$ and let
$A=\{a_1,a_2,\ldots\}$ be a countable dense subset of $X$. By a
diagonal argument one can find a subsequence $(f_m)\subseteq(g_m)$
such that for each $n$ the sequence $f_m(a_n)$ converges. Let $b$
be an arbitrary element of $X$, and $(c_n)\subseteq A$ such that
$d(c_n,b)<\frac{1}{n}$. Since all functions are $1$-Lipschitz,
for each $m$, $|f_m(c_n)-f_m(b)|\leqslant\frac{1}{n}$. Therefore,
$f_m(c_n)-\frac{1}{n}\leqslant f_m(b)\leqslant
f_m(c_n)+\frac{1}{n}$ for all $m,n$. So,
\begin{align*}
& r_n-\frac{1}{n}\leqslant \liminf_mf_m(b)\leqslant\limsup_mf_m(b)\leqslant r_n+\frac{1}{n} \mbox{ where } r_n=\lim_mf_m(c_n), \\
& \mbox{and for all } n, \ \ \limsup_m
f_m(b)-\liminf_mf_m(b)\leqslant \frac{1}{n}.
\end{align*}
Therefore, $\limsup_mf_m(b)=\liminf_mf_m(b)=\lim_mf_m(b)$.
\end{proof}
In the above lemma, instead
of $1$-Lipschitz functions, one can deal with an uniformly
equicontinuous family of functions.
\begin{thm} \label{strong->NIP}
Assume that $M$ is a separable Banach structure and
$\phi(x,y)=\|x+y\|$. If the type space $S_\phi(M)$ is strongly
separable, then $\phi$ has NIP on $M$.
\end{thm}
\begin{proof}
Recall that $(S_\phi(M),\frak{T},d)$ is a
compact topometric space where $(S_\phi(M),\frak{T})$ is compact and
Polish, and $(S_\phi(M),d)$ is a complete metric space. (Here, $\frak{T}$ is the logic topology and $d$
is the $d$-metric.) Note that $d$ is separable since the type space is strongly separable. Also, the functions of the form
$\phi(\cdot,a),a\in M$, are $1$-Lipschitz with respect to $d$.
Now, use Rosenthal's lemma (Lemma~\ref{lemma-1}) and
Lemma~\ref{diagonal-lemma}.
\end{proof}
\begin{cor} (i) Tsirelson's space has NIP.
(ii) The space of definable predicates on Tsirelson's space is not
weakly sequentially complete.
\end{cor}
\begin{proof}
(i) Immediate, since the type space of Tsirelson's space is
strongly separable (see Fact~\ref{Odell-fact} above).
(ii) Use the Eberlein--\v{S}mulian theorem.
\end{proof}
Unfortunately, life is not {\em simple}; in
Theorem~\ref{strong->NIP}, the reverse implication fails. The
following example is communicated to us by Gilles Godefroy. Let
$E$ be a separable Banach space not containing $l^1$, whose dual
space is not separable (e.g. the James tree space). We take $F$
to be the unit ball of $E$, and $X$ to be the unit ball of $E^*$.
The distance $d$ is the norm of $E^*$, and the topology $\frak T$
is any distance inducing the weak* topology on $X$. Then every
sequence in $F$ has a pointwise convergent subsequence on $X$ by
Rosenthal's theorem, but $(X, d)$ is not separable. So, the
predual of the James tree space has NIP.
With the above observations, we divide separable Banach spaces to
some classes such as stable, containing some $\ell_p$
($1\leqslant p<\infty$), with strongly separable type space, with
NIP, not containing $c_0$, and these classes form a hierarchy
$$\textrm{stable}\varsubsetneq\textrm{strongly separable}\subset_{?}\textrm{NIP}\subset_{?}\textrm{ not containing $c_0$}$$
Still we don't know the answers to
the following:
\begin{que}
Must a space not containing $c_0$ have NIP?
\end{que}
\begin{que}
Must a space with NIP have strongly separable types?
\end{que}
However we strongly believe that the answers to the above
questions are negative. In the 90's Gowers \cite{Gowers-94} has
constructed a space $X_G$ not containing $\ell_p$ and no reflexive
subspace. By the result of Haydon and Maurey \cite{HM} which
asserts that the spaces with strongly separable types contain
$\ell_1$ or have a reflexive subspace, the type space of $X_G$ is
not strongly separable and this space does not contain $c_0$, so
\{strongly separable\} $\varsubsetneq$ \{not containing $c_0$\}.
Therefore, the answer of one of the above two questions is
certainly negative.
In classical model theory, the stable models and the models with
the independence property (IP) are considered `simple' and
`complex', respectively. Since, the Bancah spaces containing
$c_0$ have IP (see the appendix), and we strongly suspect that
the spaces with IP may not be containing
$c_0$, such spaces seem to be very complex.
With the above observations, one can divide the
spaces into three types: the ``very simple" spaces (i.e. the
stable spaces), the ``very complex" spaces (i.e. the spaces
containing $c_0$), and the ``moderate" spaces (i.e. the
non-stable and not containing $c_0$). So, the $\ell_p$ spaces
($1\leqslant p<\infty$) are very simple, $c_0$ is very complex,
and Tsirelson's space is moderate. In other word, the space not
containing any $\ell_p$ or $c_0$ are moderate.
\noindent\hrulefill
\section{$\aleph_0$-categorical structures} \label{aleph_0
categorical} In this section we translate a result due to
Ibarluc\'{i}a \cite{Iba}, that is, weakly almost periodic
functions and Asplund functions on every Roelcke precompact
Polish group are the same, and conclude that the theory of
Tsirelson's space can not be $\aleph_0$-categorical in any
countable language.
A complete theory $T$ in a countable language is {\em
$\aleph_0$-categorical} (or {\em $\omega$-categorical}) if it has
an infinite model and any two models of size $\aleph_0$ are
isomorphic. An {\em $\aleph_0$-categorical} (or {\em
$\omega$-categorical}) {\em structure} is a separable structure
$M$ whose theory is $\aleph_0$-categorical.
Let $L$ be a language and $T$ a complete $L$-theory. Suppose that
$M$ is a model of $T$ and $A\subseteq M$. Denote the
$L(A)$-structure $(M,a)_{a\in A}$ by $M_A$, and set $T_A$ to be
the $L(A)$-theory of $M_A$.
\medskip
For begin with, we give an easy but interesting consequence of an
important model theoretic result, namely the Ryll-Nardzewski
theorem for continuous logic.
\begin{fct} \label{omega-categorical}
Suppose $M$ is separable, $A\subseteq M$ countable and dense. If
$T_A$ is $\aleph_0$-categorical, then $M$ is stable.
\end{fct}
\begin{proof} First note that $S(A)=S(M)$, because $A$ is dense in $M$ (see \cite{BBHU}, Proposition~8.11(4)). On the other hand, by
the Ryll-Nardzewski theorem (see Theorem~12.10 in \cite{BBHU}),
the logic topology and the metric topology on $S(A)$ are the same.
Equivalently, the metric topology is compact. So, every formula
$\phi$ on $S_\phi(M)$ is stable. Indeed, since $S(M)$ is
strongly separable, every sequence $\phi(x,a_n):S_\phi(M)\to\Bbb
R$ has a pointwise convergent subsequence and its limit is
Lipschitz (see Lemmas~\ref{1-Lip} and \ref{diagonal-lemma}). So
its limit is continuous since the logic topology coincides with
the metric topology. Thus, by the Eberlein-\v{S}mulian theorem,
the set $\{\phi(x,a):a\in M\}$ is relatively weakly compact in
$C(S_\phi(M))$, equivalently, $\phi$ is stable on $M$.
\end{proof}
\begin{cor} \label{not-categorical} Let $M$ be Tsirelson's space $\Bbb T$ and
$A=c_{00}$. Then $T_A$ is not $\aleph_0$-categorical.
\end{cor}
\begin{proof}
Immediate, since the formula $\phi(x,y)=\|x+y\|$ is unstable.
\end{proof}
It is easy to check that for a separable model $M$, and countable
subsets $A\subseteq B$ of $M$, if $T_B$ is $\aleph_0$-categorical
then $T_A$ is also $\aleph_0$-categorical (see \cite{BBHU},
Corollary~12.13); however, the converse does not hold in a strong
form (see Remark~12.14 in \cite{BBHU}). Now we get the result more
powerful than Corollary~\ref{not-categorical}; for this we need a
definition.
\begin{dfn}[\cite{Iba}, 1.10] A structure $M$ is called
$\emptyset$-saturated if every type in any countable variable is
realized in $M$.
\end{dfn}
By the Ryll-Nardzewski theorem for continuous logic (see
\cite{BU-Ryll}, Fact~1.14), every $\aleph_0$-categorical structure
has a stronger saturation property, namely {\em approximate
$\omega$-saturation}.
\medskip
The key fact to prove of the main result of this section is the
following.
\begin{fct}[\cite{Iba},~2.5] \label{key fact} Let $M$ be $\emptyset$-saturated. If a
formula $\phi(x,y)$ has the order property on $M$, then there are
distinct real numbers $r,s$ and $(a_i)_{i\in{\Bbb Q}}\subseteq M$,
$(b_j)_{j\in{\Bbb R}}\subseteq S_\phi(M)$ such that
$\phi(a_i,b_j)=r$ for $i<j$ and $\phi(a_i,b_j)=s$ for $j\leqslant
i$.
\end{fct}
The following fact is a translation of Proposition~2.10 in
\cite{Iba}. (Tom\'{a}s Ibarluc\'{i}a pointed out to us that
$\emptyset$-saturation has a key role and Proposition~2.10 is a
slight generalization of Theorem~2.9 in his paper \cite{Iba}.)
\begin{pro} \label{stable=Asplund} Let $M$ be $\aleph_0$-categorical and $\phi(x,y)$ a
formula. If $S_\phi(M)$ is strongly separable, then $\phi(x,y)$ is
stable on $M$.
\end{pro}
\begin{proof} By $\aleph_0$-categoricity, $M$ is
$\emptyset$-saturated. Suppose that $\phi$ has order property on
$M$, and let $(a_i)_{i\in{\Bbb Q}}$, $(b_i)_{i\in{\Bbb R}}$ and
$r,s\in{\Bbb R}$ be as given by Fact~\ref{key fact}. Then, for
real numbers $j<k$, $d(b_k,b_j)=\sup_{a\in
M}|\phi(a,b_i)-\phi(a,b_j)|\geqslant|\phi(a_i,b_k)-\phi(a_i,b_j)|\geqslant|r-s|$
where $k<i<j$. Since $(b_j)_{j\in{\Bbb R}}$ is uncountable,
$S_\phi(M)$ is not $d$-separable.
\end{proof}
\begin{cor} \label{strong-not-categorical} Suppose that $T$ is the
complete theory of Tsirelson's space $\Bbb T$ in a language with
only a countable number of nonlogical symbols. Then $T$ is not
$\aleph_0$-categorical. In~particular, $T_{c_{00}}=Th(M,a)_{a\in
c_{00}}$ is not $\aleph_0$-categorical.
\end{cor}
\begin{proof} Suppose, if possible, that $T$ is
$\aleph_0$-categorical. Then, since $S_\phi(M)$ is $d$-separable
where $\phi(x,y)=\|x+y\|$, so $\phi(x,y)$ is stable on $\Bbb T$.
This is a contradiction, since $\phi$ is not stable on $\Bbb T$.
\end{proof}
\begin{note} We will say that a formula $\phi(x,y)$ is {\em fragmened on a subset
$A$ of a model $M$}, if the family
$F=\{\phi(a,y):S_\phi(M)\to{\Bbb R}\}_{a\in A}$ is fragmented in
the sense of Definition~2.1 in \cite{GM}, that is, for every
$\epsilon>0$ and nonempty subset $B$ of $S_\phi(M)$, there exists
a non-void open set $O$ in $S_\phi(M)$ such that
$\textrm{diam}(f(O\cap B))<\epsilon$ for every $f\in F$. The
latter definition is equivalent to say that $\phi(x,y)$ is {\em
Asplund on a subset $A$}, in the sense of \cite{Iba}. Also, this
notion coincide with the $d$-separability of $S_\phi(M)$ (see
also \cite{GM}, Theorem~2.14).
\end{note}
The following fact is a generalization of
Proposition~\ref{stable=Asplund}. Let $T$ be a theory and
$\phi(x,y)$ a formula. Then we say that {\em $\phi$ is stable in
$T$ } if for every model $M$ of $T$, $\phi$ is stable on $M$,
i.e., $\phi$ has not order property on $M$.
\begin{pro} \label{stable<->strong} Let $T$ be a countable theory and $\phi(x,y)$ a formula.
Then the following are equivalent.
\begin{itemize}
\item [(i)] For every separable model $M$ of $T$, $S_\phi(M)$ is strongly separable.
\item [(ii)] $\phi$ is stable on every separable model of $T$.
\item [(iii)] $\phi$ is stable in $T$.
\item [(iv)] For every model $M$ of $T$, $\|S_\phi(M)\|\leqslant\|M\|$.
\end{itemize}
\end{pro}
\begin{proof}
(i) $\Rightarrow$ (ii): Suppose that $M$ is a separable model of
$T$ and $(a_m'),(b_n')$ are sequences in $M$ such that
$$\lim_m\lim_n\phi(a_m',b_n')=r<s=\lim_n\lim_m\phi(a_m',b_n').$$
Then by an argument similar to the
proof of Fact~2.5 in \cite{Iba}, one
can show that there exist a separable model $K$ of $T$ and an
elementary extension $N$ of $K$, and elements $(a_i)_{i\in{\Bbb
Q}}\subseteq K,(b_j)_{j\in{\Bbb R}}\subseteq N$ such that
$\phi(a_i,b_j)=r$ for $i<j$ and $\phi(a_i,b_j)=s$ for $j\leqslant
i$. Therefore, $S_\phi(K)$ is not strongly separable.
(ii) $\Rightarrow$ (iii): Use the downward L\"{o}wenheim-Skolem
theorem (Proposition~7.2 in \cite{BBHU}).
(iii) $\Leftrightarrow$ (iv) is standard (see \cite{Ben-Gro}).
(iv) $\Rightarrow$ (i): Clear.
\end{proof}
The above observation explains why {\em $\aleph_0$-stability } is
not a `local' property in the sense of model theory, i.e.,
formula by formula.
\begin{cor}
Suppose that $T$ is the theory of Tsirelson's space (in a
countable language). Then there exists a separable model $M$ of
$T$ which its type space is not strongly separable.
\end{cor}
\begin{proof}
Immediate from Proposition~\ref{stable<->strong} and
Corollary~\ref{unstable}. (Clearly, this model is different from
Tsirelson's space by Fact~\ref{Odell-fact}.)
\end{proof}
\noindent\hrulefill
\section{$c_0$- and $\ell_p$-subspaces of $\aleph_0$-categorical
spaces} \label{explicit definable} As we saw earlier the norm of
Tsirelson's space is given implicitly rather than explicitly. The
above statements show that the norm of Tsirelson's space can not
be characterized by axioms in continuous logic. This observation
and the fact that Tsirelson's space does not contain $c_0$ or
$\ell_p$ lead us to a natural question as Gowers and Odell asked:
Must an ``explicitly defined" Banach space contain $c_0$ or
$\ell_p$? (See \cite{Gowers, Gowers-weblog, Od-2}.) Now, we give
an explicit definition of an ``explicitly defined" Banach space,
and give a positive answer to the above question.
\begin{dfn} We say that a separable Banach space is {\em explicitly
definable} if its complete theory in some countable language is
$\aleph_0$-categorical.
\end{dfn}
We will shortly discuss why the above definition is the `true and
correct' notion for our purpose.
\subsection{$c_0$- and $\ell_p$-types}
\begin{dfn} If $p\in[0,\infty)$ and $\epsilon>0$, the following set of statements
with countable variables $\x=(x_0,x_1,\ldots)$ will be called the
{\em $\epsilon-\ell_p$-type}: for every natural number $n$, and
scalars $r_0,\ldots,r_n$
$$(1+\epsilon)^{-1}\Big\|\sum_0^n r_ix_i\Big\|\leqslant\Big\|\big(\sum_0^n|r_i|^p\big)^{\frac{1}{p}}x_0\Big\|\leqslant (1+\epsilon)\Big\|\sum_0^n r_ix_i\Big\|.$$
The following set of statements with countable variables
$\x=(x_0,x_1,\ldots)$ will be called the {\em
$\epsilon-c_0$-type}: for every natural number $n$, real number
$\epsilon>0$ and scalars $r_0,\ldots,r_n$
$$(1+\epsilon)^{-1}\Big\|\sum_0^n r_ix_i\Big\|\leqslant\Big\|\big(\max_{0\leqslant i\leqslant n}|r_i|\big)x_0\Big\|\leqslant (1+\epsilon)\Big\|\sum_0^n r_ix_i\Big\|.$$
(Warning: the notion of $\epsilon-\ell_p$-type
($\epsilon-c_0$-type) is a formal definition, and is different
from the notion of types in model theory or Banach space theory.
Of course, if $T$ is a complete theory and there exists a model
$M$ of $T$ such that the $\epsilon-\ell_p$-type
($\epsilon-c_0$-type) is realized in $M$, then it is a type in
the sense of model theory, i.e. it belongs to
$S_{\x}(\emptyset)$.)
\end{dfn}
\subsection{Krivine's theorem}
\begin{dfn}
Let $p\in[1,\infty]$ and $(x_i)$ a sequence of elements of some
Banach space. We say that $\ell_p$ (resp. $c_0$) is block finitely
represented in $(x_i)$ if $p\in[1,\infty)$ (resp. $p=\infty$) and
for every $\epsilon>0$ and positive integer $n$, there are $n+1$
finite subsets $F_0,\ldots,F_n$ of the positive integers $N$ with
$\max F_j<\min F_{j+1}$, for all $0\leqslant j\leqslant n-1$ and
elements $y_0,\ldots,y_n$ with $y_j$ in the linear span of $\{x_i
: i\in F_j\}$ for all i so that for all scalars $r_1,\ldots,r_n$,
$$(1-\epsilon)\Big\|\big(\sum_0^n|r_i|^p\big)^{\frac{1}{p}}y_0\Big\|\leqslant\Big\|\sum_0^n r_iy_i\Big\|\leqslant(1+\epsilon)\Big\|\big(\sum_0^n|r_i|^p\big)^{\frac{1}{p}}y_0\Big\|$$
(where $(\sum_0^n|r_i|^p)^\frac{1}{p}=\sup|r_i|$, if $p=\infty$).
\end{dfn}
In the year in which Tsirelson's example appeared in print
\cite{Ts}, J.-L. Krivine \cite{Kr} publised his celebrated
theorem.
\begin{thm}[Krivine's theorem] Let $(x_n)$ be a sequence in a
Banach pace with infinite-dimensional linear span. Then either
there exists a $1\leqslant p<\infty$ so that $\ell_p$ is block
finitely represented in $(x_n)$ or $c_0$ is block finitely
represented in $(x_n)$.
\end{thm}
\subsection{The main theorem}
The main theorem is a consequence of Krivine's theorem and a
powerful tool in model theory, namely the Ryll-Nardzewski theorem.
We note that at the heart of the latter theorem, although hidden,
is the omitting types theorem for continuous logic (see
\cite{BBHU}, Theorem~12.6 or \cite{BU-Ryll}, Theorem~1.11).
\begin{thm} \label{main theorem} Every explicitly definable Banach space $M$ contains
isometric copies of $c_0$ or $\ell_p$ ($1\leqslant p<\infty$).
\end{thm}
\begin{proof} Let $(x_n)$ be a sequence in a
Banach space with infinite-dimensional linear span. By Krivine's
theorem, with out loss generality, we can assume that there
exists a $p\in[1,\infty)$ so that $\ell_p$ is block finitely
represented in $(x_n)$. For an arbitrary $\epsilon>0$, let
$t_{\ell_p,\epsilon}=tp_{\ell_p,\epsilon}(\x)$ be the
$\epsilon-\ell_p$-type with countable variables
$\x=(x_1,x_2,\ldots)$, defined in above.
Now, clearly $t_{\ell_p,\epsilon}$ is finitely realized in $M$, by Krivine's
theorem. So $t_{\ell_p,\epsilon}$ is a (partially) type in the
sense of model theory.
Set ${\bf
t}=t_{\ell_p,0}=\bigcup_{\epsilon>0}t_{\ell_p,\epsilon}$. Since
$t_{\ell_p,\epsilon}$ is a partial type (for each $\epsilon>0$),
so ${\bf t}$ is a type too. It is easy to check that
${\bf t}$ is a complete type, i.e. ${\bf t}\in S_{\x}(\emptyset)$.
Since, $M$ is $\aleph_0$-categorical, by the Ryll-Nardzewski theorem, it is
$\emptyset$-saturated, and so $\bf t$ is realized in $M$.
Therefore, there exists a sequence $(a_n)$ in $M$ which is
equivalent to the unit basis of $\ell_p$. Similarly for
$p=\infty$.
\end{proof}
After preparing the final versions of this article, Ward Henson
informed us that one can prove that every $\aleph_0$-categorical
space contains $\ell_2$. He presented a sketch of proof using
Dvoretzky's Theorem. Of course, his statement is a consequence of
Theorem~\ref{main theorem}. Indeed, since $c_0$ is universal for
finite dimensional spaces (up to $(1+\epsilon)$-isomorphism) and
$\ell_2$ is finitely representable in $\ell_p$, so $t_{\ell_2,0}$
is a type for each $\aleph_0$-categorical space. Therefore:
\begin{cor}
Every explicitly definable Banach space contains isometric copy
of $\ell_2$.
\end{cor}
\subsection{Definability of classical sequences}
Recall that a type ${\bf t}(\x)\in S(T)$ is {\em definable} (or
{\em isolated}) if the predicate $\text{dist}(\x,{\bf t})$ from
the monster model to $\Bbb R$, defined by $\a\mapsto d(\a,{\bf
t})$, is $\emptyset$-definable (see \cite{BU-Ryll},
Definition~1.7). Roughly speaking, this predicate is the uniform
limit of a set of formulas, equivalently, $\text{dist}(\x,{\bf
t})$ is a {\em formula} (see \cite{BBHU}, Definitions~9.1 and
9.16).
\begin{dfn} Let $p\in[1,\infty]$, $L$ a countable first order language, $M$ a separable Banach space,
and $T$ the complete theory of $M$ in the language $L$. We say that
$\ell_p$ (resp. $c_0$) is definable in the theory $T$ if for every
$\epsilon>0$, there exists a definable (complete) type $\bf t$ in
$S(T)$ (with countable variables) which contains the
$\epsilon-\ell_p$-type (resp. $\epsilon-c_0$-type).
\end{dfn}
Recall that a class $\mathcal C$ of $L$-structures is called {\em
axiomatizable} if there is a set $T$ of closed $L$-conditions
such that $\mathcal C$ is all models of $T$. (See \cite{BBHU},
Defintion~5.13.)
The following fact is (again) a consequence of the omitting types
theorem.
\begin{pro}[Explicit Definability] \label{definability}
Let $M$ be a separable Banach space. Then the
following are equivalent.
\begin{itemize}
\item [(i)] $M$ contains almost isometric copies of $c_0$ or $\ell_p$
($1\leqslant p<\infty$).
\item [(ii)] There exist some $p\in[1,\infty)$ (resp. $p=\infty$) and a countable complete theory $T$ of $M$ such that $\ell_p$ (resp. $c_0$) is
definable in $T$.
\item [(iii)] There exist some $p\in[1,\infty)$ (resp. $p=\infty$) and an axiomatizable class
$\mathcal C$ (in a countable language) of Banach spaces such that
$M\in{\mathcal C}$, and
all structures in $\mathcal C$ contain $\ell_p$ (resp. $c_0$).
\item [(iv)] There exist some $p\in[1,\infty)$ (resp. $p=\infty$) and a class
$\mathcal C$ of Banach space structures (in a countable language) such that
$M\in{\mathcal C}$, all structures in $\mathcal C$ contain $\ell_p$ (resp.
$c_0$), $\mathcal C$ is closed under ultraproducts
and isomorphisms, and the complement of $\mathcal C$
is closed under ultrapowers.
\end{itemize}
\end{pro}
\begin{proof}
(i)~$\Rightarrow$~(ii): Let $L$ be a countable language which
contains a constant symbols $c_n$ for each natural number $n$. For
a formula $\phi(x_1,\ldots,x_n)$ (with $n$ variables) in the
$\epsilon-\ell_p$-type, let $\phi(\c)$ be the sentence
$\phi(c_1,\ldots,c_n)$. Let $T$ be the complete theory of $M$,
where $c_n$ is interpreted by $e_n$ (the $n$th element of the
standard unit basis of $\ell_p$). Then $\phi(\c)$ belongs to $T$,
and so the interpretation of $c_n$ in every model of $T$ is $e_n$.
Let ${\bf t}(\x)=tp^{M}(e_1,e_2,\ldots)\in S(T)$. Since $\bf t$ is
realized in every model of $T$, by the omitting types theorem,
$\bf t$ is definable.
(ii)~$\Rightarrow$~(i): Immediate, since the definable types are
realized in every model of $T$.
(ii)~$\Leftrightarrow$~(iii): Immediate by the omitting types
theorem.
(iii)~$\Leftrightarrow$~(iv): Immediate by Proposition~5.14 in
\cite{BBHU}.
\end{proof}
\begin{rmk}
Note that ``containing $\ell_p$" is not axiomatizable in any
countable language. Indeed, let $L$ be a countable language and
$\mathcal C$ the class of all Banach space $L$-structures
containing $\ell_p$. Then, consider a Banach space $L$-structure
$N$ in the complement of $\mathcal C$ such that $\ell_p$ is block
finitely representable in $M$. Let $\mathcal U$ be a non-principle
ultrafilter on $\Bbb N$. It is easy to verify that $(N)_{\mathcal
U}$, i.e. the ultrapower of $N$, contains $\ell_p$, almost
isometrically. Therefore, by Proposition~5.14 in \cite{BBHU},
$\mathcal C$ is not axiomatizable in $L$. Note that there is no
conflict between this observation and
Proposition~\ref{definability}.
\end{rmk}
A natural question is the following: Is a countable complete
theory which its models contain classical sequences,
$\aleph_0$-categorical? The answers is negative. There are some
stable and non $\aleph_0$-categorical spaces such as Nakano spaces
and Orilcz spaces. In the next section, we give a stronger result
such that these spaces are included in our account.
\subsection{Concluding remarks}
(1) As previously mentioned, Krivine and Maurey \cite{KM} proved
that every {\em stable} Banach space contains a copy of $\ell^p$
for some $1\leqslant p<\infty$. Also, it was noticed that
the type space of every separable stable Banach
space is {\em strongly separable}. Later, Haydon and Maurey
\cite{HM} showed that a Banach space with strongly separable
types contains either a reflexive subspace or a subspace
isomorphic to $\ell^1$.
\medskip\noindent
(2) It is known that there is a unique separable space such that
it has the {\em universal and ultra-homogeneous properties} in
the class of all separable Banach spaces, namely the {\em Gurarij
space}. Note that the norm of the Gurarij space is not defined by
a `formula', and all seperable Banach spaces, such as $\ell_p$
($1\leqslant p<\infty$) and $c_0$, live inside the Gurarij space.
Moreover, the Gurarij space is $\aleph_0$-categorical (see
\cite{BH}). This fact confirms that definability by a `formula'
is inadequate for our purpose, and $\aleph_0$-categoricity is
more suitable for ``explicit definability". Of course, the notion
of definability by a {\em formula} in the sense of model theory,
as we already saw in Fact~\ref{definability}, is necessary and
sufficient. Note that the Gurarij space is not stable, because
$c_0$ lives inside it, and so its type space is not strongly
separable.
\medskip\noindent
(3) One might expect that $\aleph_0$-categoricity implies that
a large number of $\ell_p$, $p\in[1,\infty)$, or $c_0$, are involved in the
space. This is not true; for example, the theory $\ell_2\cong L^2=L^2([0,1],\lambda)$
is $\aleph_0$-categorical (see \cite{BBHU}), but $\ell_2$
does not have any subspace isomorphic to $c_0$ or $\ell_q$ for all
$q\neq 2$ (see \cite{AK}, Corollary~2.1.6).
In other words, a direct proof of Theorem~\ref{main theorem}, without using Krivine's Theorem,
does not imply a stronger result.
\medskip\noindent
(4) In \cite{Iov-2005}, J.~Iovino showed that
the existence of enough definable types guarantees the existence
of $\ell_p$ subspaces, but he does not give a clear connection
between the notion of type definability and the notion of
``explicit definability'' for norms. In fact, his result is
`local' and does not give any `global' information about the norm.
\medskip\noindent
(5) Although $\aleph_0$-categoricity and
$\aleph_0$-stability do not have any connection in general, but to
our knowledge the most examples of $\aleph_0$-stable theories are
studied in continuous model theory are $\aleph_0$-categorical
(see \cite{BBHU}).
\noindent\hrulefill
\section{Perturbed $\aleph_0$-categorical structures} \label{perturb}
It is known that some classes of Nakano spaces (and Orilcz spaces)
are axiomatizable, but their theories are not
$\aleph_0$-categorical (see \cite{Poitevin}). In
\cite{Ben-Nakano}, Ben Yaacov proved that assuming small
perturbation, these theories are $\aleph_0$-categorical. In this
section, we relax some conditions of the main theorem
(Theorem~\ref{main theorem} above)
assume small perturbation; any (suitable) perturbed $\aleph_0$-categorical space contains some
classical sequences $c_0$ or $\ell_p$.
Let $M,N$ be two Banach spaces and $a\in M$ and $b\in N$. For
$\epsilon>0$, a $(1+\epsilon)$-isomorphic between $(M,a)$ and
$(N,b)$ is a linear map $f:M\to N$ such that $f(a)=b$ and
$\|f\|,\|f^{-1}\|\leqslant 1+\epsilon$. Let us Fix a complete
theory $T$ (in a countable language of Banach spaces) and a
monster model $\mathcal{U}$ of it.
\begin{dfn}[Banach--Mazur perurbation] The Banach--mazur
perturbation system $\mathfrak{p}_{_{BM}}$ (for theory $T$) is the
perturbations system defined by the Banach--mazur distance
$d_{BM}$ in the sense of Definition~1.23 in \cite{Ben-perturb},
that is, for each $\epsilon>0$,
$\mathfrak{p}_{_{BM}}(\epsilon)=\{(p,q)\in
S_n(T)^2:d_{BM}(p,q)<\epsilon\}$ where
$$d_{BM}(p,q)=\inf\left\{\log(1+\epsilon):
\begin{aligned}[c]
& \text{there is a $(1+\epsilon)$-isomorphism between $(\mathcal{U},a)$ } \\
& \text{ and $(\mathcal{U},b)$, and $\mathcal{U}\vDash p(a),q(b)$}
\end{aligned}\right\}$$
\end{dfn}
See also Jose Iovino, \cite{Iovino-stability-I}.
\begin{dfn}[$\mathfrak{p}_{_{BM}}$-isomorphism]
\begin{itemize}
\item [(i)] Two separable models $M$, $N$ are called $\mathfrak{p}_{_{BM}}$-isomorphic if for each $\epsilon>0$ there
is a bijective map $f:M\to N$ such that $(\text{tp}^M(a),\text{tp}^N(f(a)))\in\mathfrak{p}_{_{BM}}(\epsilon)$ for all $a\in M$.
\item [(ii)] A theory $T$ is called $\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical if every two separable models $M, N\vDash T$
are $\mathfrak{p}_{_{BM}}$-isomorphic. A separable Banach space
is called $\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical if
its continuous first order theory (in a countable
language) is $\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical.
\end{itemize}
\end{dfn}
Clearly, every two models of a
$\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical theory are {\em
almost isometric} in the sense of Banach space theory.
\begin{thm} \label{perturbed main theorem} Every $\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical space
contains almost isometric copies of $c_0$ or $\ell_p$ ($1\leqslant
p<\infty$).
\end{thm}
\begin{proof} Let $M$ be a separable model of $T$. By Krivine's
theorem, some $\ell_p$ (or $c_0$) is block finitely representable
in $M$. So, for every $\epsilon>0$, the $\epsilon-\ell_p$-type is
a type in $S_\omega(T)$. By the perturbed Ryll-Nardzewski theorem,
$M$ is $\mathfrak{p}_{_{BM}}$-approximately $\aleph_0$-saturated in the sense of Definition~2.2 in \cite{Ben-perturb}, and since $\epsilon$ is arbitrary,
the $\epsilon-\ell_p$-type is (exactly) realized in $M$.
Therefore, for every $\epsilon>0$, there is a sequence in $M$ which is
$(1+\epsilon)$-isomorphic to $\ell_p$ (or $c_0$). Again, since
$\ell_2$ is finitely representable in $c_0$ and $\ell_p$, so $\epsilon-\ell_2$-type
is a partial type in the sense of model theory, and so $M$ contains almost isometric copies of $\ell_2$.
\end{proof}
Recall that for two perturbation systems
$\mathfrak{p},\mathfrak{p}'$, the perturbation system
$\mathfrak{p}$ is stricter than $\mathfrak{p}'$, denoted by
$\mathfrak{p}<\mathfrak{p}'$, if for each $\epsilon>0$ there is
$\delta>0$ such that
$\mathfrak{p}(\delta)\subset\mathfrak{p}'(\epsilon)$. Clearly, if
$\mathfrak{p}<\mathfrak{p}'$ and $N, N$ are
$\mathfrak{p}$-isomorphic then they are
$\mathfrak{p}'$-isomorphic too. Therefore, if $\mathfrak{p}$ be
any perturbation system stricter than $\mathfrak{p}_{_{BM}}$, then
the above theorem holds for $\mathfrak{p}$. So the main theorem
(Theorem~\ref{main theorem}) is a consequence of
Theorem~\ref{perturbed main theorem} assuming the identity
perturbation system $\mathfrak{p}_{\text{id}}$. The following is
a nontrivial example.
\begin{exa} (i) Let $T=Th(\mathcal{AN}_{\subseteq[1,r]})$ be the theory of Nakano spaces presented
in \cite{Ben-Nakano}. The theory $T$ is not
$\aleph_0$-categorical, but it is $\lambda$-stable for
$\lambda\geqslant \mathfrak{c}$ (see Theorem~3.10.9 in
\cite{Poitevin}). Ben~Yaacov proved that $T$ is
$\aleph_0$-categorical and $\aleph_0$-stable up to small
perturbations of the exponent function. This perturbation system
is stricter than $\mathfrak{p}_{_{BM}}$ in the language of Banach
spaces (see Proposition~4.6 in \cite{Ben-Nakano}), and so the
theory of Nakano spaces is
$\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical.
(ii) (non-example) The theory of the 2-convexification of
Tsirelson space, is denoted by $\mathbb{T}_2$, as presented by B.
Johnson (in the language of Banach spaces) is not
$\mathfrak{p}_{_{BM}}$-$\aleph_0$-categorical because this space
does not contain any $\ell_p$ or $c_0$. Of course, it is known
that every ultrapower of this space is linearly homeomorphic to a
canonical direct sum of the space and a Hilbert space of suitable
dimension. So, $\mathbb{T}_2$ and $\mathbb{T}_2\oplus\ell_2$ are
two non $\mathfrak{p}_{_{BM}}$-isomorphic separable models of
this theory.
\end{exa}
\noindent\hrulefill
\section{A Krivine-Maurey type theorem}
We show that a NIP space has a spreading model containing $c_0$
or $\ell_p$ for some $1\leqslant p<\infty$. Compare the following
statement with the Krivine-Maurey theorem \cite{KM}.
\begin{thm} Let $X$ be a separable space with NIP. Then there
exists a spreading model of $X$ containing $c_0$ or $\ell_p$ for
some $1\leqslant p<\infty$.
\end{thm}
\begin{proof}[Skech of proof]
By Krivine's Theorem, there is some $p\in[1,\infty]$ such that
$\ell_p$ (or $c_0$) is a type. By a theorem of Bourgain, Fremlin,
and Talagrand \cite{BFT}, every type is Baire 1 definable (see
\cite{K3}, Theorem 3.20). This implies that some $\ell_p$ or
$c_0$ are spreading models of $X$.
\end{proof}
Full proof will be presented elsewhere.
\noindent\hrulefill
\section{Dividing lines in Banach spaces and model theory} \label{correspondence}
In this section, for the sake of completeness, we recall some
facts which were observed by \cite{K4}, and in the next section
we will use them.
This section must be rewritten . . .
Recall that a Banach space $X$ is reflexive if a
certain natural isometry of $X$ into $X^{**}$ is onto. This
mapping is $\widehat{\ }:X\to X^{**}$ given by
$\hat{x}(x^*)=x^*(x)$.
Now, we analyze the weak sequential compactness. Obviously, a
Banach space $X$ is weakly sequentially compact if the following
conditions hold:
\begin{itemize}
\item [(a)] every bounded sequence $(x_n)$ of $X$ has a weak Cauchy subsequence,
(i.e. there is $(y_n)\subseteq (x_n)$ such that for all $x^*\in X^*$ the sequence $(x^*(y_n))_{n=1}^\infty$ is a convergent sequence of reals, so
$\hat{y}_n\to x^{**}$ weak* in $X^{**}$ for some $x^{**}\in X^{**}$), and
\item [(b)] every weak Cauchy sequence $(x_n)$ of $X$ has a weak limit (i.e. if $\hat{x}_n\to x^{**}$ weak* in $X^{**}$ then $x^{**}\in \widehat{X}$).
\end{itemize}
It is easy to check that the condition (a) corresponds to NIP and
the condition (b) corresponds to NSOP in model theory (see
below). In functional analysis, the condition (a) is called {\em
Rosenthal property}, and the condition (b) is called {\em weak
sequential completeness} (short WS-completetness). Clearly, a
weakly sequentially compact set is weakly* sequentially compact,
but the converse fails. Indeed, the sequence $y_n=(
\underbrace{1,\ldots,1}_{n-times},0,\ldots)$ form a weakly Cauchy
sequence in $c_0$ without weak limit.
In the next subsection we study the connections between model
theory and Banach space theory more exactly.
\subsection{Banach space for a formula} Let $\M$ be an
$L$-structure, $\phi(x,y):M\times M\to \Bbb R$ a formula (we
identify formulas with real-valued functions defined on models).
Let $S_\phi(M)$ be the space of complete $\phi$-types over $M$ and
set $A=\{\phi(x,a),-\phi(x,a)\in C(S_\phi(M)): a\in M\}$. The
(closed) convex hull of $A$, denoted by ($\overline{conv}(A)$)
$conv(A)$, is the intersection of all (closed) convex sets that
contain $A$. $\overline{conv}(A)$ is convex and closed, and
$\|f\|\leqslant \|\phi\|$ for all $f\in\overline{conv}(A)$. So,
by normalizing we can assume that $\|f\|\leqslant 1$ for all
$f\in\overline{conv}(A)$. We claim that $B=\overline{conv}(A)$ is
the unit ball of a Banach space. Set
$V=\bigcup_{\lambda>0}\lambda B$. It is easy to verify that $V$
is a Banach space with the normalized norm and $B$ is its unit
ball. This space will be called the {\em space of linear
$\phi$-definable relations}. One can give an explicit description
of it: $$V=\Big(~\overline{\big\{\sum_{i=1}^n
r_i\phi(x,a_i):a_i\in M, r_i\in{\Bbb R},n\in{\Bbb
N}\big\}}~;~~\||\cdot\|| ~ \Big)$$ where $ \||\cdot\||$ is the
normalized norm.
\medskip
Note that $V$ is a subspace of $C(S_\phi(M))$. Recall that for an
infinite compact Hausdorff $X$, the space $C(X)$ is neither
reflexive nor weakly complete. So, if $V$ is a lattice (or
algebra), then it is neither reflexive nor weakly complete
(since, in this case, $V$ is isomorphic to $C(X)$ for some
compact Hausdorff space $X$).
\subsection{Stability and reflexivity}
A formula $\phi:M\times M\to {\Bbb R}$ has the order property if there
exist sequences $(a_m)$ and $(b_n)$ in $M$ such that
$$\lim_m\lim_n\phi(a_m,b_n)\neq \lim_n\lim_m\phi(a_m,b_n)$$
We say that $\phi$ has the double limit property (DLP) if it has
not has the order property.
\begin{dfn} We say that $\phi(x,y)$ is {\em unstable } if either $\phi$
or $-\phi$ has the order property. We call $\phi$ {\em stable }
if $\phi$ is not unstable.
\end{dfn}
\begin{fct}[\cite{K4}]
Assume that $\phi(x,y)$, $\M$, $B$ and $V$ are as above. Then the
following are equivalent:
\begin{itemize}
\item [(i)] $\phi$ is stable on $\M$.
\item [(ii)] $B$ is weakly compact.
\item [(iii)] The Banach space $V$ is reflexive.
\end{itemize}
\end{fct}
Recall that for an infinite compact Hausdorff space $X$, the
space $C(X)$ is not reflexive.
\medskip
In \cite{Iov} Jose Iovino pointed out the correspondence between
stability and reflexivity. He showed that a formula $\phi(x,y)$
is stable iff $\phi$ is the pairing map on the unit ball of
$E\times E^*$, where $E$ is a reflexive Banach space. In this
paper, we gave a `concrete and explicit' description of the
Banach space $V$, such that it is reflexive iff $\phi$ is stable.
This space is uniquely determined by $\phi$ and the formula
$\phi$ is completely coded by $V$. The value of $\phi$ is exactly
determined by the evaluation map $\langle
\cdot,\cdot\rangle:V\times V^*\to\Bbb R$ defined by $\langle
f,I\rangle=I(f)$.
\subsection{NIP and Rosenthal spaces} We say that a formula
$\phi(x,y)$ has {\em NIP } on a model $\M$ if the set
$A=\{\phi(x,a),-\phi(x,a)\in C(S_\phi(M)): a\in M\}$ does not
contain an independent sequence, in the sense of Definition
\ref{indep}.
\begin{dfn}[\cite{GM}, 2.10] A Banach space $X$ is said to be {\em Rosenthal} if it
does not contain an isomorphic copy of $\ell^1$.
\end{dfn}
\begin{fct}[\cite{K4}]
Assume that $\phi(x,y)$, $\M$, and $V$ are as above. Then the
following are equivalent:
\begin{itemize}
\item [(i)] $\phi$ is NIP on $\M$.
\item [(ii)] $V$ is Rosenthal Banach space.
\item [(iii)] Every bounded sequence of $V$ has a weak Cauchy subsequence.
\end{itemize}
\end{fct}
\subsection{NSOP and weak sequential completeness} Let
${\M}(={\mathcal U})$ be a monster model (of theory $T$) and
$\phi(x,y)$ a formula. The following is a consequence of
Rosenthal $\ell^1$-theorem and the Eberlein--\v{S}mulian theorem.
\begin{fct}[\cite{K4}]
Assume that $\phi(x,y)$, $\M$, and $V$ are as above. Then the
following are equivalent:
\begin{itemize}
\item [(i)] $\phi$ is NSOP (on $\M$).
\item [(ii)] $V$ is weakly sequentially complete, i.e. every weak Cauchy sequence has a weak limit (in $V$).
\end{itemize}
\end{fct}
Note that for a compact Hausdorff space $X$, the space $C(X)$
contains an isomorphic copy of $c_0$, and so $C(X)$ is not weakly
sequentially complete (see \cite{AK}, Proposition 4.3.11).
\bigskip
To summarize:
\bigskip\bigskip
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {\scriptsize Shelah}
~~~~~~ Stable ~~~~~~~~~~~~ $\Longleftrightarrow$ ~~~~~~~~~ NIP
~~~~~~~~ $+$ ~~~~~ NSOP
\bigskip
~~~~~~~~ $\Updownarrow$ {\scriptsize Grothendieck}
~~~~~~~~~~~~~~~~~~~~~ $\Updownarrow$
{\scriptsize
Rosenthal}
~~~~~~~~~~~~ $\Updownarrow$
\medskip
~~~~~~~~~~~~~~~~~~~~~~~~ {\scriptsize Eberlein--\v{S}mulian}
~~~~~~ Reflexive ~~~~~~~~ $\Longleftrightarrow$ ~~~~~~~ Rosenthal
~~~ $+$ ~~~ WS-complete
\bigskip
\noindent\hrulefill
\section{Dividing lines in some examples} \label{examples}
In this section, we list some classical Banach spaces and some
theories and point out their model theoretic dividing lines. In
fact the model theoretic properties depend on two factors: (a)
the space that one want to study it, (b) the language that one
use to study the space. In the following we study model theoretic
dividing lines for theories and special models.
\begin{exa} \label{exam1}
\begin{itemize}
\item [(i)] $L^p$ Banach lattices ($1\leqslant p<\infty$) are
reflexive and so Rosenthal and weakly sequentially
complete. By Kakutani representation theorem of $L$-spaces (\cite[4.27]{AB}),
it was proved that the classes of $L^p$ Banach lattices are
axiomatizable (in a suitable language) and their theories, denoted
by $ALpL$, are stable (see \cite{BBHU}, Section~17).
\item [(ii)] $L^1$ Banach lattices are neither reflexive nor Rosenthal in
general (e.g. $\ell^1$). Although, they are weakly sequentially complete.
Nevertheless, their theory, $AL1L$, is stable (see
(i) above). However, it is not established in some extensions of the language. For
example, Alexander Berenstein showed that $L^1({\Bbb R})$ with
convolution is unstable (see \cite{Be}), later did it appear that
$\ell^1(\mathbb{Z},+)$ with convolution is
unstable (see \cite{FHS1}, Proposition~6.2). In fact it has SOP (see \ref{exam2} below). On the other hand, since the theory atomless
probability measure algebra, denoted by $APA$, is interpretable in the
theory $L^1$ Banach lattices, the prior also is stable.
\item [(iii)] Hilbert spaces are reflexive. The theory infinite dimensional Hilbert spaces,
as described in \cite{BBHU}, is stable.
\item [(iv)] \label{c0} $c_0$ is neither reflexive nor weakly
sequentially complete; but it is Rosenthal. Let $(e_n)_1^\infty$ be
standard basis for $c_0$ and $s_n=e_1+\cdots+e_n$.
$(s_n)_1^\infty$ is called summing basis for $c_0$.
Then $\|e_m+s_n\|=2$ if $m\leqslant n$ and $=1$ if
$m>n$. So, the formula $\phi(x,y)=\|x+y\|$ has order
property. Now let $x=(a_n)_1^\infty\in\ell^1=(c_0)^*$
then $x^*(s_n)\to \sum_1^\infty a_k$. So $(s_n)_1^\infty$ is a weak Cauchy sequence with no weak
limit. It is trivial that $c_0$ is Rosenthal, because $(c_0)^*=\ell^1$ is separable but $(\ell^1)^*=\ell^\infty$ is nonseparable. Let $\psi(x,s_n)=\max(\|x+s_n\|,\|x-s_n\|)$.
Then $\psi(x,s_m)\leqslant\psi(x,s_n)$ and
$\psi(e_n,s_m)<\psi(e_m,s_n)$ for all $m<n$. So
$\psi(x,y)$ has SOP on $c_0$ (see \cite{K3}).
\medskip
$\phi(x,y)$ has IP on $c_0$ but the family $\{\phi(x,a):B_{c_0}\to[0,1]~| a\in B_{c_0}\}$ is not strongly independent. (a) Michael
Megrelishvili pointed out to us that the sequence
$f_n:B_{c_0}\to[0,2]$, $x\mapsto\|x+e_n\|$, is
independent in the sense of Definition~\ref{indep}. Indeed, for every infinite disjoint subsets $I$ and
$J$ of $\Bbb N$ (the naturals) define a binary vector $x\in c_0$ as the
characteristic function of $J$: $x_j=1$ for every $j\in J$ and $x_k=0$ for every $k\notin J$ (in particular, for every $i\in I$).
Then $x\in c_0$, $\|x_j\| = 1$ and $f_i(x)=1$ for every $i\in I$, $f_j(x)=2$
for every $j\in J$. So the formula
$\phi(x,y)=\|x+y\|$ has IP on $c_0$. (Indeed, we note
that if $S_\phi(c_0)$ is the space of complete
$\phi$-types on $c_0$, then by compactness of $S_\phi(c_0)$, the family
$\{\widehat{\phi}(x,e_n):S_\phi(c_0)\to[0,2]:n\in{\Bbb N}\}$ is
strongly independent, i.e. $\phi$ has IP on $c_0$.)
Particularly, if $c_0$ is a model of a theory $T$, then
by the compactness theorem (or compactness of $S_\phi(c_0)$), $T$ has IP.
(b) The family $\{\phi(x,a):B_{c_0}\to[0,1]~| a\in B_{c_0}\}$ (where $\phi(x,a)=\|x+a\|$) is not strongly independent. Indeed, let
$A=\{a_1,a_2,\ldots\}$ be a countable dense subset of
$c_0$ and $\phi(x,y)$ a formula. By Lemma~\ref{diagonal-lemma}, every
sequence $\phi(x,b_n)$, $b_n\in c_0$ has a convergent
subsequence and so $c_0$ is not strongly independent (see \ref{lemma-1}
above).
\item [(v)] $\ell^\infty$ is neither stable nor
Rosenthal or weakly sequentially complete because $c_0$ and $\ell^1$ live
inside $\ell^\infty=C(\beta\Bbb N)$ (see below). The formula $\psi$ as described above has SOP
in $\ell^\infty$. This fact explains why the case $p=\infty$ is excluded from (i) above.
Since $c_0$ has IP, $\ell^\infty$ has IP.
\item [(vi)] $C(X,{\Bbb R})$ space for an (infinite) compact
Hausdorff space is neither reflexive nor
Rosenthal or weakly sequentially complete (see (vii) below). By
Kakutani representation theorem of $M$-spaces, the
class of $C(X)$-spaces is axiomatizable in the language of Banach lattices. Indeed,
replace the axioms of abstract $L^P$-spaces in \cite{BBHU} by the $M$-property
$\|x^++y^+\|=\max(\|x^+\|,\|y^+\|)$.
Now we add a constant symbol $\textbf{1}$ to our language
as an order unite, i.e. for all $x$, $\|x\|=1$ implies that $x^+\leqslant
\textbf{1}$ or equivalently,
$\sup_x((\|x\|-1)\dotminus\|x^+\vee{\bf 1}-{\bf
1}\|)$. Note that $\|\cdot\|$ is an order unite norm
if for all $x$ and $\alpha$, $x^+\leqslant \alpha{\bf
1}$ implies that $\|x\|\leqslant\alpha$ or
equivalently $\sup_x(\|x^+\vee\alpha{\bf 1}-{\bf
1}\|\dotminus(\|x\|-\alpha))$. Since $\ell^\infty$
has SOP, the theory of $C(X)$-spaces is not NSOP.
Since $\ell^{\infty}$ has IP, then the theory of $C(X)$-spaces has not
NIP.
\item [(vii)] $C_0(X,{\Bbb C})$ for an (infinite) locally compact space $X$ is neither reflexive nor
Rosenthal or weakly sequentially complete. Note that, by Gelfand-Naimark
representation theorem of Abelian $C^*$-algebra, the class
$C_0(X,{\Bbb C})$-spaces is axiomatizable in a suitable language (see \cite{FHS1} for the noncommutative theory).
First, it is known that all Banach spaces, in particular $\ell^1$ and
$c_0$, live inside $C(K)$-spaces where $K$'s are compact. So
$C(K)$ is not is neither reflexive nor
Rosenthal or weakly sequentially complete in general.
Second, by an example of \cite{FHS1}, we will show that the theory $C_0(X)$-spaces has SOP and IP. Indeed, in $X$ find a sequence of distinct $x_n$ that
converges to some $x$ (with $x$ possibly in the
compactification of $X$). For each $n$ find a positive $a_n\in
C_0(X)$ such that $\|a_n\|=1$ and $a_n(x_i)=1$ if
$i\leqslant n$ and $=0$ if $i>n$. By replacing $a_n$
with maximum of $a_j$ for $j\leqslant n$, we may
assume that this sequence is increasing. Also, we can
assume that the support $K_n$ of each $a_n$ is
compact and $a_m(K_n)=1$ for $n<m$. So $a_ma_n=a_n$
if $n\leqslant m$. Now let $\phi(x,y)=\|(x-1)y\|$.
Then $\phi(x,a_i)\leqslant\phi(x,a_j)$ and
$\phi(a_j,a_i)+1\leqslant\phi(a_i,a_j)$ for $i<j$. So
$\phi$ has SOP. Since $c_0$ has IP, this theory is
not NIP.
\item [(viii)] $C^*$-algebras are neither reflexive nor
Rosenthal or weakly sequentially complete because
some infinite-dimensional Abelian $^*$-subalgebra lives inside a $C^*$-algebra and so by
Gelfand-Naimark theorem it has a $C_0(X)$-subalgebra (see \cite{FHS1},
Lemma~5.3). Note that the Gelfand-Naimark-Segal
theorem ensure that this class is axiomatizable
(\cite{FHS1}). So, the theory $C^*$-algebra has
SOP. Since $c_0$ has IP, this theory is not NIP.
\item [(ix)] The class of (Abelian) tracial von
Neummman algebra is axiomatizable (see \cite{FHS2}).
A tracial von Neumman algebra is stable if and only
if it is of type I. The theory of Abelian tracial von Neumann algebras is stable
because it is interpretable in the theory of
probability measure algebras (see above and also
\cite{FHS1}, Lemma~4.5).
\end{itemize}
\end{exa}
\begin{exa}[$\ell^1({\Bbb Z},+)$ with convolution has SOP] \label{exam2}
In \cite{FHS2} it is shown that the formula
$\phi(x,y)=\inf_{\|z\|\leqslant 1}\|x*z-y\|$ has order property in
$\ell^1({\Bbb Z},+)$. We show that $\phi$ also has SOP. Indeed,
we need to show that: (i) $\phi(x_i,y)\leqslant \phi(x_j,y)$ for
$i\leqslant j$, and (ii) $\phi(x_i,x_i)<\phi(x_j,x_i)$ for $i<j$,
where $x_i$'s are elements of $\ell^1$ as described in
\cite[Proposition~6.2]{FHS2}. It is easy to verify that (ii)
holds. We check that (i) also is true. Indeed, for any $z$,
$\|z\|\leqslant 1$ and any $y$, we have
$|\|y\|-\|z\||\leqslant\sup_t|z(t)-y(t)|\leqslant\|y\|+1$ where
$\|\cdot\|$ is uniform norm on $C({\Bbb T})$. So, if we assume
that $\|y\|\geqslant 1$, then $\inf_{\|z\|\leqslant
1}\|z-y\|=|\|y\|-1|$. (Note that we can replace $y$ with
$\max(y,1)$ and so assume that $\|y\|\geqslant 1$. In fact we
define $\phi(x,y)=\inf_{\|z\|\leqslant 1}\|x*z-\max(y,1)\|$.)
Therefore if $i\leqslant j$, then $\inf_{\|z\|\leqslant 1}\|(\cos
t )^{2i}z-y\|\leqslant\inf_{\|z\|\leqslant 1}\|(\cos
t)^{2j}z-y\|$. Equivalently, $\phi(x_i,y)\leqslant \phi(x_j,y)$.
To summarize, $\phi$ has SOP. \begin{que} Does $\ell^1({\Bbb
Z},+)$ with convolution have NIP?
\end{que}
\end{exa}
The following amazing fact affirms some of our observations.
\begin{fct}[\cite{Guer}, Theorem III.5.1] Every stable (separable) Banach space is weakly sequentially
complete.
\end{fct}
So, every Banach space containing $c_0$ is unstable. This
directly implies that the theories $C(X)$-spaces, $C_0(X,{\Bbb
C})$-spaces, and $C^*$-algebras are not stable (see (vi)--(viii)
above). (There is even something stronger (see
Proposition~\ref{c0->IP}).) Of course, the converse does not hold,
e.g. Tsirelson's spaces $T$ and $T^*$ are unstable but they are
reflexive and so weakly sequentially complete.
\begin{que} In \cite{M} the author proved that for $1\leqslant
p<\infty$, $p\neq 2$, the non-comutative $L^p$ space
$L^p({\mathcal M})$ is stable iff $\mathcal M$ is of type I. (See
also \cite{Hand2}, p. 1479.) Later, in \cite{FHS1} the authors
showed that a tracial von Neumman algebra is stable if and only
if it is of type I. We ask a similar question: Which tracial von
Neumman algebra (or non-comutative $L^p$ spaces) are NIP (or
NSOP)?
\end{que}
\noindent\hrulefill
\section{Appendix: Remarks on Rosenthal's dichotomy} \label{appendix}
\begin{fct} \label{fact3} If $X$ is a compact space and
$\{f_n\}$ a pointwise bounded sequence in $C(X)$, then
\begin{itemize}
\item [(1)] either $\{f_n\}$ has a (pointwise) convergent subsequence, or
\item [(2)] $\{f_n\}$ has a $\ell^1$-subsequence, equivalently, $\{f_n\}$
has an independent subsequence.
\end{itemize}
\end{fct}
Note that for non-compact spaces, Fact~\ref{fact3} is not a
dichotomy (see below). Michael Megrelishvili
informed us with details about this
observation and also pointed out to an Example which is in
\cite{Dulst}. Of course, we are not sure that the observation
itself (that Fact~\ref{fact3} is not always dichotomy in
noncompact Polish case) appears somewhere in the literature in a
clear form. For compact space, independence and strong
independence are the same.
\begin{fct}[Rosenthal's $\ell^1$-theorem] \label{fact1} If $(x_n)$ is a bounded sequence in a Banach
space $V$ then
\begin{itemize}
\item [(1)$'$] either $(x_n)$ has a weakly Cauchy subsequence, or
\item [(2)$'$] $(x_n)$ has a $\ell^1$-subsequence.
\end{itemize}
\end{fct}
Fact~\ref{fact1} is a consequence of
Fact~\ref{fact3}. Indeed, let $B^*$ be the unit ball of $V^*$,
and define $f_n:B^*\to{\Bbb R}$ by $x^*\mapsto x^*(x_n)$ for all
$x^*\in B^*$.
\begin{fct}[Rosenthal's dichotomy] \label{fact2} If $X$ is a Polish space and
$\{f_n\}$ a pointwise bounded sequence in $C(X)$, then
\begin{itemize}
\item [(1)$''$] either $\{f_n\}$ has a (pointwise) convergent subsequence, or
\item [(2)$''$] $\{f_n\}$ has a subsequence whose its closure in ${\Bbb
R}^X$ is homeomorphic to $\beta\Bbb N$, equivalently, it has a
strong independent subsequence.
\end{itemize}
\end{fct}
\begin{rmk} Note that (1)$'$~$\Rightarrow$~(1)=(1)$''$, but
(1)$''$~$\nRightarrow$~(1)$'$ in general. Also,
(2)$''$~$\Rightarrow$~(2)$'$, but (2)$'$~$\nRightarrow$~(2)$''$
(see below). In fact (2)$'$ is equivalent to independence
property. For Polish space $X$, (2)$''$ holds iff there is a
compact subset $K\subseteq X$ such that $\{f_n|_K\}$ contains a
strong independent subsequence, equivalently, $\{f_n\}$ has a
strong independent subsequence (see \cite{BFT}, Lemma~2B,
Theorem~2F and Corollary~4G).
\end{rmk}
We thank Michael Megrelishvili for communicating to us the
following example.
\begin{exa} \label{c_0}
Let $c_0=\{x=(x_n)_n\in\ell^\infty:\lim_nx_n=0\}$ with
$\|x\|=\sup_{n\in{\Bbb N}}|x_n|$. Let $B_{c_0}$ be the unit ball
of $c_0$, i.e. $B_{c_0}=\{x\in c_0:\|x\|\leqslant 1\}$. For each
$a\in B_{c_0}$, define $f_a:B_{c_0}\to[0,2]$ by $x\mapsto
\|x+a\|$. Let ${\mathcal F}_{c_0}=\{f_a:a\in B_{c_0}\}$. (i) The
family ${\mathcal F}_{c_0}$ contains an independent sequence.
Indeed, let $(e_n)_n$ be the standard basis of $c_0$.
For each $n$, define $f_n(x)=\|x+e_n\|$ for all $x$. Then
$(f_n)_n$ is independent.
Indeed, for every infinite disjoint subsets $I$ and $J$ of $\Bbb
N$ (the naturals) define a binary vector $x\in c_0$ as the
characteristic function of $J$: $x_j=1$ for every $j\in J$ and
$x_k=0$ for every $k\notin J$ (in particular, for every $i\in
I$). Then $x\in c_0$, $\|x_j\| = 1$ and $f_i(x)=1$ for every
$i\in I$, $f_j(x)=2$ for every $j\in J$. So the family ${\mathcal
F}_{c_0}$ contains an independent sequence. (ii) The family
${\mathcal F}_{c_0}$ is weakly precompact in ${\Bbb
R}^{B_{c_0}}$. Indeed, since $c_0$ is separable and the functions
$f_a$ are $1$-Lipschitz, So by a diagonal argument, every
sequence has a point-wise convergent subsequence. This show that
Fact~\ref{fact3} is not a dichotomy for non-compact spaces. (iii)
By Fact~\ref{fact1}, the sequence $(f_n)$ has not a weakly Cauchy
subsequence. (iv) By Fact~\ref{fact2}, $(f_n)$ has not a strong
independent subsequence.
\end{exa}
\medskip
Let $V$ be Banach space, $B_V$ be the unit ball of it, i.e.
$B_V=\{x\in V:\|x\|\leqslant 1\}$. For each $a\in B_V$, define
$f_a:B_V\to[0,2]$ by $x\mapsto \|x+a\|$. Let ${\mathcal
F}_V=\{f_a:a\in B_V\}$.
\medskip
\textbf{(Q3)} Is there any Banach space $V$ such that every
sequence in ${\mathcal F}_V$ has a weak Cauchy subsequence but
(the Banach space generated by) ${\mathcal F}_V$ is not weakly
sequentially complete?
\medskip
Clearly, the response is positive. Tsirelson's space has
the above properties.
\begin{rmk} (i) Define $f_n:B_{c_0}\to[0,2]$ by
$f_n(x)=\|x+s_n\|$. Then $f_n$ converges to the continuous
function $f(x)=1+\|x\|$. Now, let $\widehat{c_0}$ be the
\v{C}ech-Stone compactification of $c_0$ and $\hat{f_n},\hat{f}$
be the extensions of $f_n,f$ respectively. Then
$\hat{f_n}\nrightarrow \hat{f}$. Indeed, note that
$\hat{f_n}|_{c_0}=f_n$ and $\hat{f}|_{c_0}=f$, and
$\hat{f}(e)=\lim_m\hat{f}(e_m)=\lim_mf(e_m)=\lim_m\lim_nf_n(e_m)=2\neq
1=\lim_n\lim_mf_n(e_m)=\lim_n\lim_m\hat{f_n}(e_m)=\lim_n\hat{f_n}(e)$
where $e$ is a cluster point of $(e_m)$ in $\widehat{c_0}$. Note
that $c_0$ is dense in $\widehat{c_0}$ and $f_n\to f$, but
${\hat{f_n}}\nrightarrow\hat{f}$. (ii) Define $g_n(x)=\|x+e_n\|$.
Then $g_n\to g$ where $g(x)=\max(\|x\|,1)$. But $(\hat{g_n})$
does not contain a convergent subsequence. Because, $(g_n)$ has
independent property (see Example~\ref{c_0} above). (iii) $(g_n)$
is a poinwise convergent sequence but it has not a weakly Cauchy
subsequence. Because, by Rosenthal $\ell^1$-theorem, either a
sequence has a weakly Cauchy subsequence, or it has a
$\ell^1$-subsequence (equivalently, it contains an independent
subsequence). Note that weakly Cauchy is stronger than pointwise
convergence, in general. But, for locally compact spaces, they
are equivalent.
\end{rmk}
\begin{pro} \label{c0->IP} Suppose that $X$ is a Banach space structure containnig
$c_0$.Then $X$ has the independence property (IP).
\end{pro}
\begin{proof}
The sequence $(f_n)_n$ in Example~\ref{c_0} works well.
\end{proof}
\begin{que}
Does a Banach space with IP contain $c_0$?
\end{que}
\noindent\hrulefill
| 177,249
|
TITLE: Find $\frac{\mathrm{d} }{\mathrm{d} x}\frac{1}{\sqrt[3]{x+2}}$ using only the definition of the derivative
QUESTION [1 upvotes]: I am trying to find
$$\frac{\mathrm{d} }{\mathrm{d} x}\frac{1}{\sqrt[3]{x+2}}$$
using only the definition of the derivative.
I have gotten to this point $$\lim_{h \to 0}\;\dfrac{\dfrac{1}{\sqrt[3]{x+2}}+\dfrac{1}{\sqrt[3]{x+h+2}}}{h(\sqrt[3]{x^2+xh+4x+2h+4})}$$ by finding a common denominator and then expanding. I'm not sure where to go from here, or if I should try using $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ instead.
I have looked at Finding derivative of $\frac{1}{\sqrt{x+2}}$ using only the definition of the derivative $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ and the link within that page, and am also wondering if the solution is similar to that one, being that the only difference is a cube root instead of a square root.
REPLY [2 votes]: Using the limit definition of derivative: $$f'(x) = \lim_{h\to 0} \frac{(x+2+h)^{-1/3} - (x+2)^{-1/3}}{h}$$
Now let $x+2+h = u^3$ and $x+2 = v^3$. So now we have as $h \to 0$, $u^{3}-v^3 \to 0$. So $u\to v$
$$\begin{align}
\lim_{u\to v} \frac{u^{-1}-v^{-1}}{u^{3}-v^{3}} &= \lim_{u\to v}\frac{-1}{uv(u^2+uv+v^2)}\\
&= \frac{-1}{v^2(v^2+v^2+v^2)}\\
&= \frac{-1}{3v^4} \\
&=\frac{-1}{3 } (x+2)^{-4/3}
\end{align}$$
| 193,264
|
Could landlords be offered tax relief for offering longer-term tenancies? Government set to announce new incentives to support renters
- Calls have been made for tax incentives for landlords offering longer-term tenancies
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Landlords are set to benefit from new incentives if they offer longer agreements to their tenants.
Perks could include clawing back tax relief that is currently being withdrawn from thousands of landlords, a leading property expert has suggested.
The government wants to offer greater support to those who rent, as the number of people who do not own their own home continues to grow. Policies introduced could benefit landlords who comply.
Secretary of State for Communities and Local Government Sajid Javid indicated in his Conservative party conference speech that details will be revealed in the Autumn Budget on November 22.
Landlords profits are being squeezed amid the tax clampdown, with experts calling for a repeal of the punitive tax regime for those offering longer-term tenancies
'All landlords should be offering tenancies of at least 12 months to those who want them,' he said. 'That is why, at the Autumn Budget, we will bring forward new incentives for landlords who are doing the right thing.'
David Cox, chief executive of ARLA Propertymark - the trade body for the lettings industry, suggested that one such incentive could be tax relief for landlords who comply.
However, he argued that such a move should be combined with the introduction of a new housing court.
This would speed up the time it takes for a landlord to evict a tenant - one of the main concerns landlords have about offering longer-term deals.
'Tax incentives and easier access to justice are the two key things which will give landlords the confidence to offer longer-term tenancies.
'Therefore, a specialist housing court with judges who are housing law experts will both speed up the process and provide much greater consistency in judgments.
'Combining the housing court with tax incentives such as repealing the punitive restrictions on mortgage interest relief should provide landlords with both confidence in the legal system and financial incentives to offer longer-term tenancies.'
A four-bedroom flat on the third floor of this block in Scotland's Edinburgh is available to rent via Coulters Lettings for £2,400 a month
This three-bedroom terrace in Essex's Colchester is available to rent via Michaels property consultants for £1,600 a month
Increased tax relief could be the difference between whether or not buy-to-let is viable for some landlords.
Tax relief on mortgage interest is being reduced over a four-year period since the government announced the move in 2015.
Tax relief will taper down and be replaced with a 20 per cent tax credit by 2020, hitting profits for many landlords hard.
This, combined with other recent moves to tip the scales away from landlords and in favour of first-time buyers - such as a three per cent stamp duty on second homes - mean that many landlords are questioning their future in the buy-to-let market.
This one-bedroom basement flat in Wiltshire's Salisbury is available to rent via Visum letting agents for £695 a month
Mr Cox highlighted two further tax incentives that the Government could consider to improve the viability of buy-to-let in the wake of these changes.
The first would be to reinstate LESA - the landlords energy savings allowance - which was up to £1,500 per property per year to offset against income tax for energy efficient improvements.
And second, Mr Cox called for landlords to be allowed to roll over capital gains tax if reinvesting in another rental property. This is 28 per cent of higher rate taxpayers and 18 per cent for basic rate taxpayers.
Landlords are facing further pressure on their budgets amid the ban on letting fees, which could see agents switch their focus from tenants to landlords to get their administrative and regulatory costs covered.
Latest figures suggest that landlords are already taking steps to protect their investments by increasing rents.
The average rent across Britain edged closer to £1,000 a month, rising by 2.1 per cent in September to £927, according to the HomeLet rental index.
Martin Totty, chief executive of HomeLet, said: 'Landlords are facing a deluge of higher costs from new regulation, taxation changes on buy-to-let mortgages and the prospect of a near-term rise in interest rates. There's also the added uncertainty over the fall-out from the Government's intention to ban letting agents from charging up-front fees to tenants.
'In a sector where demand for rental properties generally outstrips supply, most informed commentators suggest higher externally imposed costs on landlords will inevitably translate into higher rents to tenants. This may prove to be the start of that upward movement, especially if tenants are left competing for fewer rental properties because some landlords decide the returns from property investment are being eroded by factors beyond their control.'
Data from UK finance found that buy-to-let remortgaging in August was 5 per cent lower than in July, while borrowing for house purchase among property investors was up 11 per cent. However, borrowing for house purchases for investment purposes remains at a lower level than before the introduction of the higher stamp duty rate.
>.
| 263,231
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In this article I will be extoling the virtues of making oyster extract powder part of your heath programme. However there are still many more virtues to be discussed but this article will look at oyster extract from the bodies biochemistry perspective, in particular the concept of balancing trace mineral levels in the body to maintain peak hormonal and enzymatic functioning.
Have you ever had that feeling that you are missing something in your diet, that you are just not up to par, knowing there is some imbalance in there that you just can’t figure out? Well this is a frequent reason many of us attend our GP or therapist. However in some countries certain physicians take a very holistic approach to this and work by a process of elimination to try and figure the problem out, rather than going down the orthodox prescribed medication route straight away. Oyster extract is one of the tools they often try to eliminate other possibilities.
Many people will present with a biochemical imbalance. Doctors and therapists will try to identify this by various means. They may opt for blood tests, hair analysis and use various other diagnostic tools and methods to reach a conclusion. Some therapists however will use oyster powder..
Furthermore, where deficiencies are identified by the more traditional analytical methods and micro-nutrients are subsequently administered, certain imbalances may result which can lead to more symptoms. With oyster extract capsules this does not happen, as the total array of trace elements are presented in a balanced biochemical matrix. Each and every one of the trace elements we need are presented in the relative concentrations as nature intended.
The results speak for themselves. Improved energy levels, stronger immunity, smoother complexion and better sexual health. All these benefits come from ensuring the trace elements necessary for peak performance in all these biochemical systems are balanced, by nature-as nature intended. You only get this from oyster extract.
With winter fast approaching and many of use reaching for the zinc lozenges, consider taking a course of oyster extract capsules. Not only are they the highest natural source of zinc, they also contain each and every one of the 59 trace elements we need to get us through the winter months.
Make sure you discuss your oyster extract with your supplier and choose a potent product. Some cheap oyster extract products on the market can have up to 60% tapioca starch mixed in along with other bulking agents. Only choose pure, high zinc, oyster extract capsules. Think before you zinc!
| 393,486
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TITLE: Vector rotations and quantum mechanics
QUESTION [0 upvotes]: Quantum mechanics deals with wave function and complex numbers that can be seen as vectors in 2D plane. I am interested in vector rotations and there use in quantum mechanics.
What is the role that rotations play in quantum mechanics? Given a set of $N$ vectors in the complex plane, What is the quantum mechanical interpretation of rotating every vector by 45 degrees?
REPLY [1 votes]: There's quite a lot of physics behind this question. I will give a sloppy (but didactic, I hope) answer.
We know that we can use complex numbers to parametrize our states in a quantum mechanical system. For example, in position representation we have a wave function $\psi(x)$ that describes a system as the particle in a box or something like that. This wave function is complex-valued in general.
In order to extract physical information of the system we have, though, to compute expectation values of physical observables (hermitian operators) $H$. For example, the expected value of $H$ when the system is in the state $\psi$ is
$$\langle\psi|H|\psi\rangle = \int dx\ \psi(x)^*(H\psi(x)) $$
The thing is that I can change $\psi(x)\mapsto e^{i\theta}\psi(x)$ (with $\theta$ a constant) and the result will be the same. So, if I am able to parametrize all the states of my system with functions $\psi_k$ this means that an equivalent one-to-one parametrisation in terms of physical information is $e^{i\theta}\psi_k$. This operation is basically rotating the space of states (the rotation in the complex-plane that the OP was talking about). It is a symmetry of the quantum mechanics, called phase invariance and what it tells us is that our physical description of the reality is redundant.
When one tries to generalize the QM to describe quantum fields, you want to get sure that this phase invariance occurs independently at every point in the space-time, so you promote the constant $\theta$ to a function of the coordinates $\theta(x)$ and then you get what is called gauge invariance, which is the most fundamental aspect in our formulation of the theories describing the elemental particles (fields).
| 107,695
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Increase drinking age to 21, advises report.
The report, supported by the Labor senators on the cross-party committee, recommended that governments consider measures including:
■ uniform national laws with heavy penalties for people supplying alcohol to teenagers;
■ stricter limits on the number of liquor outlets, their opening hours and the volume of take-away sales allowed;
■ raising the drinking age to 21 in the light of evidence that at this age the brain is more resilient to the adverse effects of alcohol;
■ reducing the allowable alcohol content in ready-to-drink products to 3 per cent.
The Minister for Health, Nicola Roxon, said the tax increase was an important part of the Government's strategy for curbing binge drinking. "We are glad that the committee, having looked at the evidence, agrees that this is an important measure," she said.
Liberal senators wrote a report opposing the excise increase, saying there was no clear evidence it would deter alcohol abuse, but it would impose a $3.1 billion burden on consumers.
Jobs in the alcohol and hospitality industries would be lost, the report said.
send photos, videos & tip-offs to 0424 SMS SMH (+61 424 767 764), or us.
Did you know you could pay less than $1 a day for a subscription to the Herald? Subscribe today.
| 118,208
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\begin{document}
\maketitle
\begin{abstract}
We gain tight rigorous bounds on the renormalisation fixed point function for period doubling in families of unimodal maps with degree 2 critical point. By writing the relevant eigenproblems in a modified nonlinear form, we use these bounds, together with a contraction mapping argument, to gain tight bounds on the essential eigenvalues and eigenfunctions of the linearised renormalisation operator at the fixed point and also those of the operator encoding the universal scaling of added uncorrelated noise.
We gain bounds on the corresponding power series coefficients and universal constants accurate to over 400 significant figures, confirming and (in the case of noise) extending the accuracy of previous numerical estimates, by using multi-precision interval arithmetic with rigorous directed rounding to implement operations on a space of analytic functions.
\end{abstract}
\section{Introduction}
Existence of the fixed point of the doubling operator for maps with degree 2 critical point was first proved by computer-assisted means by Lanford~\cite{Lan82}, by analytic means by Campanino and Epstein~\cite{Cam81} and, most recently, in full generalisation by Lyubich~\cite{Lyu99}. It is some decades since the first computer-assisted proofs. We revisit the problem with improved processing power, high precision computations, and parallel processing, and extend its application to bound eigenfunctions of the linearised operator and also the operator controlling the scaling of uncorrelated noise.
In this paper we follow the method established in our paper~\cite{Bur20} where we studied the universality class corresponding to degree~4 critical points. In section~\ref{sec:renormfp}, we work with a modified renormalisation operator, on a suitable space of analytic functions, corresponding to the action of the usual doubling operator on even maps. We identify a ball of functions centred on an approximate fixed point, given by a polynomial of degree $1280$, with $\ell^1$-radius $\rho\simeq 10^{-409}$. We prove that a variant of Newton's method for the fixed-point problem (corresponding to our modified operator) is a contraction map on this ball, yielding rigorous bounds on the fixed-point function itself and thereby on the associated universal constant, $\alpha$, controlling scaling in the state variable for families of maps in the corresponding universality class.
In section~\ref{sec:DTG}, we consider the eigenproblem for the derivative of our renormalisation operator at the fixed point. We approach the eigenproblem in a novel way, by rewriting it in a modified nonlinear form, and apply a contraction mapping argument, using the ball of functions proven to contain the renormalisation fixed point, to bound the eigenfunctions corresponding to essential eigenvalues. In particular, we gain tight bounds on the eigenfunction-eigenvalue pair for eigenvalue $\delta$, the universal constant controlling scaling in the parameter for the relevant families of unimodal maps.
In section~\ref{sec:LW}, we further adapt the technique to the eigenproblem corresponding to the universal scaling of added uncorrelated noise, gaining rigorous bounds on the relevant eigenfunction and hence on the eigenvalue $\gamma$.
Numerical approximations to the Feigenbaum constants $\alpha$ and $\delta$ (for the universality class of maps with degree $2$ critical points) with over $1,000$ digits have been computed by Broadhurst in 1999~\cite{Bro99}, and $10,000$ digits of each by Molteni in 2016~\cite{Mol16} using Chebychev polynomials. An approximation to $\gamma$ with $15$ digits was provided by Kuznetsov and Osbaldestin in 2002~\cite{Kuz02}. The rigorous bounds that we compute provide over $400$ confirmed digits for each constant (and for each one of $640$ nonzero coefficients of the relevant power series, together with bounds on all higher-order terms).
\section{The renormalisation fixed point}\label{sec:renormfp}
We consider the operator $R$ defined by:
\begin{equation}
Rg(x):= a^{-1}g(g(ax)),
\end{equation}
where $a:= g(1)$ is chosen to preserve the normalisation $g(0)=1$.
We first seek a nontrivial fixed point, $g^{*}$, of $R$, with a critical point of degree 2 at the origin.
It suffices to restrict to even functions. To this end, we define $X=Q(x)=x^2$,
and write
\begin{equation}
g(x)=G(Q(x))=G(X),
\end{equation}
with $G$ in the Banach algebra $\mathcal{A}(\Omega)$ of functions analytic on an open disc $\Omega=D(c,r) :=\{z\subset \mathbb{C}:|z-c|<r\}$ and continuous on its closure, $\overline{\Omega}$, with finite $\ell^1$-norm. Specifically, we write $f\in\mathcal{A}(\Omega)$ as
\begin{equation}
f(z)=\sum_{k=0}^{\infty}a_k\left(\frac{z-c}{r}\right)^k,
\label{eqn:powerseries}
\end{equation}
i.e., we take the monomials $e_k:z\mapsto\left(\frac{z-c}{r}\right)^k$ as Schauder basis, with the corresponding $\ell^1$-norm,
\begin{equation}
\|f\|:=\sum_{k=0}^{\infty}|a_k|.\label{eqn:norm}
\end{equation}
We first define a modified operator $T$, corresponding to the action of $R$ on even functions, by
\begin{equation}
TG(X):= a^{-1}G(Q(G(Q(a)X))),
\label{eq:TGX}
\end{equation}
in which $a:= G(1)$.
The above formulation is in contrast to~\cite{Lan82} in which the ansatz $g(x)=1+x^2h(x^2)$ is taken, with $h$ varying in a suitable space of functions equipped with a modified norm.
We start by finding an accurate polynomial approximation, $G^0$, to the fixed point of $T$.
It is important, for what follows, to prove: (i) that the operator $T$ is well-defined on a certain $\ell^1$-ball, $B\subset\mathcal{A}(\Omega)$, of radius $\rho$ around $G^0$, (ii) that it is differentiable there, and (iii) that the derivative is compact.
We take domain $\Omega=D(1,2.5)$ to define the space $\mathcal{A}(\Omega)$, and establish that the `domain extension' or `analyticity-improving' property~\cite{Mac93} holds for our operator on the chosen ball in this space, i.e., that for all $G\in B$:
\begin{align}
\overline{a^2\Omega} &\subset \Omega,\label{eqn:de1}\\
\overline{Q(G(a^2\Omega))}&\subset\Omega,
\end{align}
in which the overline indicates topological closure. (Note that, since $a=G(1)$, the universal quantifier on $G$ is not vacuous for equation~(\ref{eqn:de1}).)
This proves~\cite{Mac93} that the operator is well-defined on the ball $B$, that it is differentiable there, and that the derivative $DT(G)$ is compact for all $G\in B$. Figure~\ref{fig:de} demonstrates domain extension via a rigorous covering of the relevant sets.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=7.5cm]{figure1}
\caption{Domain extension for the modified operator $T$ on $\mathcal{A}(\Omega)$ for the ball $B=B(G^{0},\rho)$, illustrated using a covering of the boundary $\partial\Omega\subset\mathbb{C}$ (dashed lines) of $\Omega=D(c,r)$ by $256$ rectangles, showing the corresponding coverings $\Gamma_1$ of $a^2\partial\Omega$ (red in colour copy) and $\Gamma_2$ of $Q(G(a^2\partial\Omega))$ (green in colour copy) valid for all $G\in B$.\label{fig:de}}
\end{center}
\end{figure}
We then use a contraction mapping argument on the ball $B$ to bound a fixed point of $T$. Note, however, that $T$ is not itself contractive at the fixed point we seek.
Following~\cite{Lan82} and~\cite{Eck84}, we therefore consider a Newton-like operator corresponding to our modified operator $T$,
\begin{equation}
\Phi : G\mapsto G-\Lambda F(G),\label{eqn:newtonlike}
\end{equation}
where $F(G):=T(G)-G$ and $\Lambda$ is a fixed linear operator approximating $[DF(G^0)]^{-1}$. We establish invertibility of $\Lambda$ and note that $\Phi$ therefore has the same fixed points as $T$.
To establish (uniform) contractivity of $\Phi$ on $B$, we bound a suitable norm of the derivative of $\Phi$:
\begin{equation}
\|D\Phi(G)\|\leq\kappa<1\quad
\text{for all } G\in B,\label{eqn:contractive}
\end{equation}
and appeal to the mean value theorem.
Further, to establish that $\Phi$ is indeed a contraction map on $B$, we bound the movement of the approximate fixed point under $\Phi$:
\begin{equation}
\|\Phi(G^0)-G^0\|\leq\varepsilon,
\end{equation}
and establish the cruical inequality
\begin{equation}
\varepsilon<\rho(1-\kappa),
\end{equation}
so that $\Phi(B)\subset B$.
The contraction mapping theorem then yields the existence of a (locally unique) fixed point $G^{*}\in B$, of $\Phi$, and therefore also of $T$.
In~(\ref{eqn:newtonlike}), above, we choose $\Lambda$ to be a fixed linear operator. The Fr\'echet derivative of $\Phi$ is thus given by
\begin{equation}
D\Phi(G): \delta G \mapsto \delta G - \Lambda (DT(G)\delta G - \delta G).
\label{eqn:DPhiG}
\end{equation}
Note that the Fr\'echet derivative, $DT(G)$, of our modified operator $T$ at $G$ is given by:
\begin{align}
DT(G):\delta G\mapsto &-a^{-2}\delta a G(Q(G(a^2X)))\label{eq:term1}\\
&+a^{-1}\delta G(Q(G(a^2X)))\label{eq:term2}\\
&+a^{-1}G'(Q(G(a^2X)))\cdot 2G(a^2X)\cdot \delta G(a^2X)\label{eq:term3}\\
&+a^{-1}G'(Q(G(a^2X)))\cdot 2G(a^2X)\cdot G'(a^2X)\cdot 2XG(1)\delta a\label{eq:term4},
\end{align}
where $\delta a=\delta G(1)$.
A simpler linear operator in which terms corresponding to (\ref{eq:term1}) and (\ref{eq:term4}), which involve variations in $a$, are absent is often used in the literature when estimating the spectrum of $DR(g)$ nonrigorously. (In numerical calculations for the spectrum the simpler operator suffices with minor alterations to the spectral characteristics, see for instance \cite{Var11}.)
Note also that, in~(\ref{eqn:contractive}), we bound $D\Phi(G)$ by considering the `maximum column-sum norm': we note that
\begin{equation}
\|D\Phi(G)\|
:=
\sup_{\|f\|=1}\|D\Phi(G)f\|
\leq \sup_{k\ge 0}\|D\Phi(G)e_k\|,
\label{eqn:mcsnorm}
\end{equation}
where the norm on the left hand side of the inequality is the standard operator norm, and recall that $e_k$ denotes the $k$-th basis element.
For the rigorous calculations we use interval arithmetic, with rigorous directed rounding modes to bound operations in the corresponding space of analytic functions $\mathcal{A}(\Omega)$. The first detailed exposition of such a framework applied to renormalisation operators was provided in \cite{Eck84}, where it was applied with standard precision arithmetic to operations on functions of 2 real variables in the study of area-preserving maps.
For our computations, we introduce multi-precision arithmetic with directed rounding, specialise our implementation to spaces $\mathcal{A}(\Omega)$ and the corresponding complex Banach algebra, and use parallel computation to establish the bounds on contractivity. (For complex values, a straightforward analogue of interval arithmetic, namely rectangle arithmetic with intervals bounding real and imaginary parts, is used.) Specifically, individual functions are written as the sum of a polynomial part (to some chosen truncation degree, $N$), and a high-order part, $f=f_P+f_H$. We maintain bounds on the power series coefficients such that those of $f_P$ lie in computer-representable intervals, and those of $f_H$ are bounded in norm, $\|f_H\|\le v_H$.
Following~\cite{Eck84}, in order to accommodate balls of functions and to absorb errors in the case where it would be undesirable to do so in the polynomial and high-order bounds, we write $f=f_P+f_H+f_E$, with $f_E$ a general `error' function bounded in norm, $\|f_E\|\le v_E$. We ensure that all computed operations deliver intervals bounding polynomial coefficients and upper bounds on the norms of the respective high-order and error parts that guarantee inclusion of the exact result.
The challenge of bounding the supremum in equation~(\ref{eqn:mcsnorm}) is reduced to a finite computation in two parts: Firstly, we bound $\|D\Phi(G)e_k\|$ (for all $G\in B$) for $k=0,1,\ldots,N$, by bounding the expressions in~(\ref{eqn:DPhiG})--(\ref{eq:term4}) evaluated at the polynomial basis elements. This first computation is well-suited to a parallel implementation, in which care is taken to ensure the safety of directed rounding modes across processes. Secondly, we bound the action of $D\Phi(G)$ (for all $G\in B$) on a single ball $B_H$ of high-order functions $f_H$, such that $\|f_H\|\le 1$, that therefore contains all of the high-order basis elements $e_k$ for $k>N$. The latter requires careful consideration of the action of $D\Phi(G)$ on high-order perturbations $\delta G$ in order to minimise dependencies on $\delta G$ when implementing equation~(\ref{eqn:DPhiG}) in order to gain a suitable bound $\kappa<1$~\cite{Bur20}. (We additionally make use of closures in order to avoid recomputation of bounds on those subexpressions in the Fr\'echet derivative $D\Phi(G)\delta G$ that do not depend on $\delta G$.)
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{figure2a}
& \includegraphics[width=7.5cm]{figure2b}\\
(a) & (b)\\
\includegraphics[width=7.5cm]{figure2c}
& \includegraphics[width=7.5cm]{figure2d}\\
(c) & (d)
\end{tabular}
\caption{Rigorous covering by rectangles of the graphs of the fixed point functions, (a,c) $G^{*}(X)$ of the operator $T$ and (b,d) $g^{*}(x)$ of the operator $R$. In (a) and (b) we use coverings of the interval $\Omega\cap\mathbb{R}$, respectively its preimage (for $X\ge 0$) under $Q$, by $10$ (dashed lines) and $1,000$ (solid lines) subintervals. The dotted lines indicate the universal constant $G^{*}(1)=g^{*}(1)=a\simeq -0.3995$. In (c) and (d) the functions $G^{*}$ and $g^{*}$ are extended to larger domains by making use of the fixed-point equations, $T(G^{*})=G^{*}$ and $R(g^{*})=g^{*}$, recursively.\label{fig:bigandlittleg}}
\end{center}
\end{figure}
It would be sufficient, in order to gain a proof of existence of the fixed point by this technique, to use truncation degree $N=20$ for the polynomial part of $G$ (thus degree $2N=40$ for $g$) with standard double-precision accuracy and careful use of directed rounding modes. However, we tighten the resulting bounds significantly by increasing the truncation degree and using multi-precision interval arithmetic.
Using truncation degree $N=640$ for the polynomial part of $G$ (thus degree $2N=1280$ for $g$) and precision corresponding to (approximately) $P=\lfloor 2N/3\rfloor=426$ digits in the significand, we are able to choose a function ball of radius $\rho=10^{-409}$ (actually a close dyadic rational approximation to $10^{-409}$, with $1415$ bits in the significand) that results in bounds $\varepsilon<7\times 10^{-410}$, and $\kappa<1.3\times 10^{-99}$ (the numbers listed here are safely upwards-rounded decimal conversions of the corresponding dyadic rational computer-representable bounds). We confirm rigorously that $\varepsilon<\rho(1-\kappa)$, establishing that the Newton-like operator $\Phi$ corresponding to our modified operator $T$ is a contraction map on $B(G^0,\rho)$. The fixed-point functions $G^{*}$ and $g^{*}$ are illustrated in figure~\ref{fig:bigandlittleg}.
As an immediate consequence of computing such tight bounds on the fixed point, we gain rigorous bounds on the universal constants $a=G^{*}(1)$ and $\alpha=a^{-1}$ to over $400$ significant digits each (see appendix \ref{d2summary}) with the first 20 given as:
\begin{align}
a=G^{*}(1)&=-0.39953\,52805\,23134\,48985...\\
\alpha= a^{-1}&=-2.50290\,78750\,95892\,8222...
\end{align}
\section{Eigenfunctions of the linearised operator at the fixed point}
\label{sec:DTG}
\begin{figure}[ht!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{figure3a}
&
\includegraphics[width=7.5cm]{figure3b}
\\
(a) & (b)\\
\includegraphics[width=7.5cm]{figure3c}
&
\includegraphics[width=7.5cm]{figure3d}
\\
(c) & (d)
\end{tabular}
\caption{Rigorous coverings of the graphs of (a,c) the eigenfunction $V^{*}(X)$ of the linearisation $DT(G^{*})$, of $T$ at the renormalisation fixed point $G^{*}$ and (b,d) the eigenfunction $v^{*}(x)$ of $DR(g^{*})$, corresponding to eigenvalue $\delta$: (a,b) plotted for $X\in\Omega\cap\mathbb{R}$ and $x$ in the preimage (for $X\ge 0$) under $Q$, respectively, computed using a covering of the interval by $10$ (dashed lines) and $1,000$ (solid lines) subintervals, and (c,d) extended to larger domains by making use of the fixed-point equation and eigenproblem equation recursively. In (a,b) dotted lines demonstrate how the eigenvalue $\delta\simeq 4.669$ is encoded via a coordinate functional $\delta=\varphi(V^{*})$; here, $\delta=V^{*}(c)$, with $c=1$, corresponding to the constant term of the relevant power series, equation~(\ref{eqn:powerseries}), expanded with respect to the disc $\Omega=D(c,r)$; recall that $V^{*}\in\mathcal{A}(\Omega)$.\label{fig:deltaeigenfunction}}
\end{center}
\end{figure}
Compactness of $DT(G)$ implies that its spectrum consists of $0$ together with countably-many isolated eigenvalues, each of finite multiplicity, accumulating at $0$.
We note that the spectra of $DT(G^{*})$ and $DR(g^{*})$ are related, with $\alpha^2\approx 6.264547$ and $\delta\approx 4.669201$ the only eigenvalues of $DT(G^{*})$ outside the closed unit disc, and $\alpha$ and $\delta$ the only eigenvalues of $DR(g^{*})$ outside the closed unit disc. We recall that $\alpha$ controls universal scaling in the state variable and $\delta$ in the parameter for families of unimodal maps in the corresponding universality class.
We use a method (introduced in \cite{Bur20}), novel in the context of bounding the spectrum of derivatives of renormalisation operators, where we write an eigenvalue, $\lambda$, as a linear functional of the corresponding eigenfunction, $\lambda = \varphi(V)$, thereby expressing the eigenproblem for $(V,\lambda)$ as the following nonlinear problem for $V\in\mathcal{A}(\Omega)$:
\begin{equation}
DT(G)V-\varphi(V) V=0,\label{eqn:deltaprob}
\end{equation}
and then adapt the techniques of section~\ref{sec:renormfp} to establish that a suitably-chosen Newton-like operator $\hat{\Phi}$ for this modified problem is a contraction mapping on a ball $B(V^0,\hat{\rho})$ centred on an approximate eigenfunction $V^0$. Note that, in equation~(\ref{eqn:deltaprob}), $G$ is taken to range over the ball $B(G^0,\rho)$, proven to contain the fixed point $G^{*}$. This necessarily places a restriction on the tightness of bounds on the eigenfunction $V$ (and the corresponding eigenvalue $\lambda$) that can ultimately be achieved by this approach.
The linear functional $\varphi$ is chosen as the coordinate functional that extracts the first non-zero power series coefficient of the desired eigenfunction (when expanded with respect to $\Omega$). Our choice of norm~(\ref{eqn:powerseries},\ref{eqn:norm}) thus implies that $\hat{\rho}$ provides bounds on the eigenvalue $\lambda$ directly via~$\lambda\in[\varphi(V^0)-\hat{\rho},\ \varphi(V^0)+\hat{\rho}]$.
For the eigenfunction $V^{*}$ corresponding to the universal constant $\delta$, we were able to achieve $\hat\rho=10^{-403}$, giving $\|\hat{\Phi}(V^0)-V^0\|\le\hat\varepsilon<1.2\times 10^{-404}$, and $\|D\hat{\Phi}(V)\|\le\hat\kappa<2.8\times 10^{-100}$ for all $V\in B(V^0,\hat{\rho})$. As in section~\ref{sec:renormfp}, care must be taken to minimise dependencies on the argument in the corresponding derivatives $D\hat{\Phi}(V)\delta V$ when bounding the action of $D\hat{\Phi}(V)$ on high-order perturbations in the computation of $\hat{\kappa}$.
The eigenfunction (with our chosen normalisation $\delta=V^{*}(1)$) is illustrated in figure~\ref{fig:deltaeigenfunction}.
As a result, we find rigorous bounds on $\delta$ that confirm $403$ significant digits
(see appendix \ref{d2summary}), with the first 20 given as:
\begin{equation}
\delta = 4.66920\,16091\,02990\,6718\ldots
\end{equation}
We note that the same method may be employed to bound the eigenfunction corresponding to $\alpha^2$ and hence to bound $\alpha$ itself, albeit in a less direct way (and less tightly) than in section~\ref{sec:renormfp}. The results confirm previous numerical estimates.
\section{Critical scaling of added uncorrelated noise}
\label{sec:LW}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{figure4a}
&
\includegraphics[width=7.5cm]{figure4b}
\\
(a) & (b)
\end{tabular}
\caption{Rigorous coverings of the graph of $w^{*}(x)=W^{*}(Q(x))$ where $W^{*}$ is the eigenfunction of the operator $L$ corresponding to the eigenvalue $\gamma^2$ of largest absolute value. The function $W^{*}$ (resp. $w^{*}$) is extended to domains larger than $\Omega\cap\mathbb{R}$ (resp.its preimage for $X\ge 0$) by using the fixed-point and eigenproblem equations recursively. In (a) dotted lines demonstrate how the eigenvalue $\gamma\simeq 6.619$ is encoded via a coordinate functional $\gamma=\varphi(W^{*})$; here, $\gamma=W^{*}(c)$ with $c=1$, corresponding to the constant term of the relevant power series, equation~(\ref{eqn:powerseries}); recall that $W^{*}\in\mathcal{A}(\Omega)$.\label{fig:noiseeigenfunction}}
\end{center}
\end{figure}
We further apply the techniques of sections~\ref{sec:renormfp}--\ref{sec:DTG} to find tight rigorous bounds on the eigenfunction, $w^{*}$, and eigenvalue, $\gamma$, controlling the universal scaling of added uncorrelated noise. We must first adapt the formulation to our modified operator $T$. Following \cite{Cru81}, we consider the iteration of a prototypical one-parameter family, $f_{\mu},$ modified by the addition of independently identically distributed random variables $\xi_n$ at each iterate, to give
\begin{equation}
x_{n+1}=F_{\mu,n}(x_n):= f_{\mu}(x_n)+\varepsilon\xi_n
\end{equation}
The relevant operator controlling universal scaling of the noise, obtained by considering the limits $\varepsilon\to 0$ and $g\to g^{*}$, is given by
\begin{equation}
N(g)w(x)=a^{-2}\left[(g'(g(ax)))^2w(ax)+w(g(ax))\right],
\end{equation}
with eigenproblem written as
\begin{equation}
N(g)w(x)= \lambda^2 w(x),
\end{equation}
in which the eigenvalue $\gamma^2$ of largest absolute value controls universal scaling of the overall variance of the added noise, with the corresponding eigenfunction $w(x)$ controlling the distribution in $x$.
Firstly, we note that the relevant operator, written in terms of $G$ and $X$, corresponding to our modified renormalisation operator $T$ is given by
\begin{equation}
LW(X)=a^{-2}(G'(Q(G(a^2X)))\cdot 2G(a^2X))^2W(a^2X)+a^{-2}W(Q(G(a^2X)))
\label{LK}
\end{equation}
where we have taken $W\circ Q=w$. The first and second terms on the right hand side of equation~(\ref{LK}) correspond to those terms in the expression for the Fr\'echet derivative $DT(G)$ that don't involve variations in $a$, namely terms~(\ref{eq:term3}) and~(\ref{eq:term2}), respectively. (We note that the terms corresponding to~(\ref{eq:term1}) and~(\ref{eq:term4}) are removed in the derivation in~\cite{Cru81} as a consequence of the independence of the $\xi_n$, and that the correlated case is dealt with in~\cite{Sch81}.)
We adapt our procedure from section~\ref{sec:DTG} for $DT(G)$, writing the eigenvalue as a linear coordinate functional of the eigenfunction to give the (nonlinear in $W\in\mathcal{A}(\Omega)$) problem
$$
LW(X)-\varphi(W)^2W=0,
$$
and use a contraction mapping argument for an appropriate Newton-like operator $\tilde{\Phi}$ on a ball $B(W^0,\tilde{\rho})$. Note, again, that in formulating the Newton-like operator corresponding to equation~(\ref{LK}) we allow $G$ to vary over the ball $B(G^0,\rho)$, proven to contain the renormalisation fixed point $G^{*}$. We thereby establish rigorous bounds on both the eigenfunction $W^{*}$ and eigenvalue $\gamma=\varphi(W^{*})$.
Using a ball of radius $\tilde\rho=10^{-401}$, we gain bounds $\|\tilde{\Phi}(W^0)-W^0\|\le\tilde\varepsilon<8.9\times 10^{-402}$, and $\|D\tilde{\Phi}(W)\|\le\tilde\kappa<7.4\times 10^{-101}$ for all $W\in B(W^0,\tilde{\rho})$. The eigenfunction (with our chosen normalisation) is illustrated in figure~\ref{fig:noiseeigenfunction}.
This yields $401$ correct significant digits for $\gamma$
(see appendix \ref{d2summary}), with the first 20 given as
\begin{equation}
\gamma = 6.61903\,65108\,17928\,0453...
\end{equation}
\clearpage
\appendix
\section{Appendix}
\subsection{Tight rigorous bounds on universal constants}
\label{d2summary}
Below we state the digits proven correct of $a=\alpha^{-1}=G^{*}(1)$ ($409$ digits), $\alpha$ ($408$ digits), $\delta=\varphi(V^{*})=V^{*}(1)$ ($403$ digits), and $\gamma=\varphi(W^{*})=W^{*}(1)$ ($401$ digits) obtained from a proof with truncation degree $N=640$ for the polynomial part of the space for $G^*$, $V^*$, and $W^*$ (corresponding to degree $2N = 1280$ for $g^*$, $v^*$, and $w^*$, respectively). The numbers listed here are found by computing safely rounded decimal interval bounds, containing the corresponding dyadic rational computer-representable bounds, on the relevant constants, and quoting only those digits guaranteed accurate as a result.
\begin{center}
\begin{tabular}{rlllll}
$a={}$
-0.
&3995352805 &2313448985 &7580468633 &6937194335 &4428046695\\ &2727517073 &0449124380 &1660883804 &2981844594 &8741812667\\
&6179406484 &6838366714 &0945404846 &1643643736 &0947557018\\
&4545976789 &4023268702 &2548579773 &5028209746 &4775103925\\
&5797877507 &3697474932 &3269755137 &3492308212 &2088541722\\
&2413083309 &4802739189 &0574703944 &6460416066 &9938415778\\
&2298900077 &7299013544 &2121397192 &4552385259 &4449033723\\
&7697553775 &0905488329 &7544336726 &9368114050 &5788840461\\
&79344018\ldots\\
$\alpha={}$
-2.
&5029078750 &9589282228 &3902873218 &2157863812 &7137672714\\ &9977336192 &0567792354 &6317959020 &6703299649 &7464338341\\ &2959523186 &9995854723 &9421823777 &8544517927 &2863314993\\ &3725781121 &6359487950 &3744781260 &9973805986 &7123971173\\ &7328927665 &4044010306 &6983138346 &0009413932 &2364490657\\ &8899512205 &8431725078 &7337746308 &7853424285 &3519885875\\ &0004235824 &6918740820 &4281700901 &7148230518 &2162161941\\ &3199856066 &1293827426 &4970984408 &4470100805 &4549677936\\ &7608881\ldots\\
$\delta={}$
+4.
&6692016091 &0299067185 &3203820466 &2016172581 &8557747576\\ &8632745651 &3430041343 &3021131473 &7138689744 &0239480138\\ &1716598485 &5189815134 &4086271420 &2793252231 &2442988890\\ &8908599449 &3546323671 &3411532481 &7142199474 &5564436582\\ &3793202009 &5610583305 &7545861765 &2222070385 &4106467494\\ &9428498145 &3391726200 &5687556659 &5233987560 &3825637225\\ &6480040951 &0712838906 &1184470277 &5854285419 &8011134401\\ &7500242858 &5382498335 &7155220522 &3608725029 &1678860362\\ &67\ldots\\
$\gamma={}$
+6.
&6190365108 &1792804532 &3808905147 &4666014364 &4298809101\\ &1980889058 &1539120755 &2294388390 &1250134543 &0103013791\\ &0116621507 &6680991461 &7111062123 &4676765967 &2263346641\\ &5349015651 &5469980646 &2621251411 &3242709973 &9377082075\\ &2957874751 &6962711711 &6928533607 &9067798211 &8951469414\\ &0224500385 &6708624240 &5473933494 &6093414214 &2285269246\\ &7145643730 &8826640353 &2825154865 &5386124267 &3930589439\\ &2213420488 &3953151516 &3766198410 &1165280871 &0270346725\\ &\ldots
\end{tabular}
\end{center}
\subsection{Computational Considerations}
The computations were performed via two independent implementations of the rigorous framework: (1) written in the high-performance language Julia, making use of IEEE754-2008 compliant binary multi-precision arithmetic with rigorous directed rounding, and (2) written in the language Python, using IEEE754-2008 compliant decimal multi-precision arithmetic with rigorous directed rounding. In both cases, directed rounding modes are respected at the process level, and multiprocessing rather than threads was therefore used to ensure safety during parallel computations.
| 150,654
|
The Chicago area is lucky enough to have not one, but two award-winning zoos. Our Lincoln Park neighborhood is home to one of the oldest and largest free zoos in the country, set in a scenic lakefront park. And in nearby Brookfield, you’ll find another large zoo known for their family-friendly events and cutting-edge animal care and conservation.
Here’s where you can take a walk on the wild side at Chicago’s best zoos.
Lincoln Park Zoo
Opened in 1868, Lincoln Park Zoo is among the oldest zoos in the United States. It’s open 365 days a year and is one of the few zoos across the country that offer free admission.
Those aren’t the only things that makes Lincoln Park Zoo special. The zoo is home to more than 1,100 animals from across the globe, including mammals, primates, birds, reptiles, amphibians, fish, insects, many rare and endangered species. Make sure to check out the award-winning Regenstein African Journey exhibit, with houses giraffes, rhinos, and pygmy hippos.
Attractions at the Lincoln Park Zoo include:
- The Nature Boardwalk, a planned prairie ecosystem with native plants and wildlife
- The Patio at Café Brauer, which offers al fresco dining with stunning views
- The Farm-in-the-Zoo, which recreates a Midwestern farm, with goats to pet, cows to feed, and ponies to greet
- The AT&T Endangered Species Carousel and Lionel Train Adventure
- Sea Explorer 5-D, which takes you on a virtual-reality submarine adventure
And then there’s the setting — the zoo is nestled on 49 acres within scenic Lincoln Park. Step just outside the zoo to explore the Lincoln Park Conservatory, the Alfred Caldwell Lily Pool, and the Peggy Notebaert Nature Museum. On either end of the zoo, North Pond and South Pond are natural havens where you’ll find scenic paths and sky-line photo opps.
The zoo also hosts a bunch of great events for both kids and adults, including wine tastings, live music, craft beer festivals, and the holiday favorite ZooLights, when the grounds are transformed into a winter wonderland with millions of twinkling lights.
Lincoln Park Zoo, 2001 N. Clark St. Hours: 10 a.m. – 5 p.m. daily (hours subject to change). Free admission.
Getting there
By bus
#151 and #156 stop at the zoo’s West Gate (Stockton & Webster) and near the Farm-in-the-Zoo (Stockton & Armitage); #22 stops near the zoo’s West Gate (exit at Clark & Webster) and Café Brauer Gate (exit at Clark & Armitage); #36 stops near the zoo’s West Gate (exit at Clark & Webster) and Café Brauer Gate (exit at Clark & Armitage).
By train
Take the Brown Line or Purple Line ‘L’ train to the Armitage Station, or the Red Line ‘L’ train to the Fullerton Station, both of which are approximately one mile west of the zoo. For more information, call (312) 836-7000 or use the CTA’s Quick Trip-Planner.
By car
The zoo is located off Lake Shore Drive at the Fullerton Parkway exit. From I-94, exit at either Fullerton Parkway or North Avenue and go east. The zoo’s parking lot entrance is located at Fullerton Parkway and Cannon Drive (2400 North Cannon Drive). Note: parking fees apply.
Brookfield Zoo
Venture beyond Chicago city limits to Brookfield Zoo, a 216-acre oasis that’s home to over 2,000 animals. Located on the grounds of the Forest Preserves of Cook County, the zoo is open all year long and welcomes more than two million guests annually.
At Brookfield Zoo, you can get up close and personal with some of the animal kingdom’s most awe-inspiring creatures, like tigers, dolphins, kangaroos, zebras, bald eagles, snow leopards, penguins, and so much more. Don’t miss feeding the African giraffes by hand and watching polar bears dive for food from an underwater viewing area.
Attractions at Brookfield Zoo include:
- Dolphins in Action, a captivating dolphin show that shows off their intelligence and agility
- Butterflies!, a seasonal butterfly house
- Hamill Family Play Zoo, where kids can run wild
- Hamill Family Wild Encounters, where you can touch and feed animals
- The Carousel, featuring 72 hand-carved animals
You can also go behind the scenes with Backstage Adventures, which gives you the chance to feed bears, meet marsupials, encounter apes, and a bunch more animal-care experiences alongside the zoo’s experts.
The zoo hosts tons of special events for both grown-ups and kids, including champagne brunches, ZooBrew for craft beer lovers, Summer Nights Concerts, and Boo at the Zoo. There are also sleepover safaris, zoo camps, and other education-focused programs for all ages.
Brookfield Zoo, 8400 W. 31st St., Brookfield, IL. Hours: 10 a.m. – 5 p.m. Monday – Friday, 10 a.m. – 6 p.m. Saturday and Sunday. Tickets: adults $21.95, seniors $15.95, children ages 3 – 11 $15.95. Tip: Save $1 on ticket prices when you order online. (Hours and ticket prices subject to change.)
Getting there
By train
From downtown Chicago, take Metra’s Burlington Northern Santa Fe (BNSF) Line. Exit at the Zoo Stop (Hollywood Station) and then walk two blocks northeast. Call (312) 322-6777 or visit metrarail.com for more information.
By car
The to the zoo’s main entrance and zoo parking (fees apply).
| 165,945
|
With sustainability and quality in mind, our units are made from mass timber. Chosen for it’s strength and durability, and environmental impact in production compared to traditional building methods. We also utilized renewal building materials in the finishings, with natural wood interiors for a beautiful aesthetic and cozy modern style.
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| 239,152
|
\begin{document}
\begin{frontmatter}
\title{Constructive description of Hardy-Sobolev spaces in $\Cn$}
\author[mymainaddress]{Alexander Rotkevich}
\ead{rotkevichas@gmail.com}
\address[mymainaddress]{Department of Mathematical analysis, Mathematics and Mechanics Faculty, St. Petersburg State University, 198504, Universitetsky prospekt, 28, Peterhof, St. Petersburg, Russia}
\begin{abstract}
In this paper we study the polynomial approximations in
Hardy-Sobolev spaces on for convex domains. We use the method of
pseudoanalytical continuation to obtain the characterization of
these spaces in terms of polynomial approximations.
\end{abstract}
\begin{keyword} Hardy-Sobolev Spaces \sep polynomial approximations,
pseudoanalytical continuation \sep Cauchy-Leray-Fantappi\`{e}
integral
\MSC[2010] 32E30\sep 41A10
\end{keyword}
\end{frontmatter}
\section{Introduction\label{intro}}
The purpose of this paper is to give an alternative
characterizations of Hardy-Sobolev (see. \cite{AB88}) spaces
\begin{equation} \label{def:Hardy-Sobolev}
H^l_p(\O) =\{f\in H(\O): \norm{f}_{H^p(\O)} + \suml_{\abs{\alpha}\leq l}\norm{\partial^{\alpha} f}_{H^p(\O)}<\infty\}
\end{equation} on strongly convex domain $\O\subset\Cn.$
We continue the research started in \cite{R13} and devoted to
description of basic spaces of holomorphic functions of several
variables in terms of polynomial approximations and pseudoanalytical
continuation. In particular, we show that for $1<p<\infty$ and
$l\geq 1$ a holomorphic on a strongly convex domain $\O$ function
$f$ is in the Hardy-Sobolev space $H^l_p(\O)$ if and only if there
exist a sequence of $2^k-$degree polynomials $P_{2^k}$ such that
\begin{equation}
\intl_\dom d\sigma(z) \left(\SK \abs{f(z)-P_{2^k}(z)}^2 2^{2l k}
\right)^{p/2} < \infty. \label{eq:SobSp_cond}
\end{equation}
In the one variable case this condition follows from the
characterization obtained by E.M. Dynkin \cite{D81} for Radon
domains.
The paper is divided into five sections with one appendix. In
section~\ref{notations} we give main definitions and preliminaries
of this work. Section~\ref{CLF} is devoted to the
Cauchy-Leray-Fantappi\`{e} integral formula, the polynomial
approximations and estimates of its kernel. We also define internal
and external Kor\'{a}nyi regions, the multidimensional analog of
Lusin regions. In section~\ref{PAC} we introduce the method of
pseudoanalytical continuation and three constructions of the
continuation with different estimates. We use these constructions to
obtain the characterization of Hardy-Sobolev spaces in terms of
estimates of the pseudoanalytical continuation. To prove this result
we use the special analog of the Krantz-Li area-integral
inequality~\cite{KL97} for external Kor\'{a}nyi regions established
in appendix~\ref{Area_int}. Finally, section~\ref{Poly_Sobolev}
contains the proof of characteristics~(\ref{eq:SobSp_cond}).
\section{Main notations and definitions \label{notations}}
Let $\Cn$ be the space of $n$ complex variables, $n\geq 2,$ $z =
(z_1,\ldots, z_n),\ z_j = x_j + i y_j;$
$$\partial_j f =\frac{\partial f}{\partial z_j} = \frac{1}{2}\left( \frac{\partial f}{\partial x_j} - i \frac{\partial f}{\partial y_j}\right), \quad \bar\partial_j f = \frac{\partial f}{\partial\bar{z}_j} = \frac{1}{2}\left( \frac{\partial f}{\partial x_j} + i \frac{\partial f}{\partial y_j}\right),$$
$$\partial f = \suml_{k=1}^{n} \frac{\partial f}{\partial z_k} dz_k,\quad \bar{\partial} f = \suml_{k=1}^{n} \frac{\partial f}{\partial\bar{z}_k} d\bar{z}_k,\quad df=\partial f+ \bar{\partial} f.$$
\noindent The notation
$$\scp{\partial f(z)}{w} = \suml_{k=1}^{n} \frac{\partial f(z)}{\partial z_k} w_k.$$
is used to indicate the action of $\partial f$ on the vector
$w\in\Cn,$ and $$|\bar\partial f| = \abs{\frac{\partial f}{\partial
z_1}} + \ldots+\abs{\frac{\partial f}{\partial z_n}}.$$
The euclidean distance form the point $z\in\Cn$ to the set
$D\subset\Cn$ we denote as $\dist{z}{D} = \inf\{\abs{z-w}:w\in D\}.$
Lebesgue measure in $\Cn$ we denote as $d\mu.$
For a multiindex $\alpha =
(\alpha_1,\ldots,\alpha_n)\in\mathds{N_0}^n$ we set
$\abs{\alpha}~=~\alpha_1~+~\ldots~+~\alpha_n$ and
$\alpha!~=\alpha_1!\ldots\alpha_2!,$
also $z^\alpha =
z_1^{\alpha_1}\ldots z_n^{\alpha_n}$ and $\partial^\alpha f =
\frac{\partial^{\abs{\alpha}} f}{\partial
\bar{z}_1^{\alpha_1}\ldots\partial \bar{z}_n^{\alpha_n}}.$
Let $\Om$ be a strongly convex domain with a $C^3$-smooth defining
function. We need to consider a family of domains $$\Omega_t =
\left\{z\in\Cn : \rho(z)<t \right\}$$ that are also strongly convex
for each $|t|<\eps,$ where $\eps>0$ is small enough, that is $d^2\rho(z)$ is positive definite when $|\rho(z)|\leq\eps.$ For $z\in \O_{\eps}\setminus\O_{-\eps}$ we denote the nearest point on $\dom$ as $\pr_{\dom}(z).$ Then the mapping
$$\pr_{\dom} : \O_{\eps}\setminus\O_{-\eps}\to \dom$$
is well defined, $C^2-$smooth on $\O_{\eps}\setminus\O$ and $|z-\pr_{\dom}(z)|=\dist{z}{\dom}.$
For $\xi\in\dom_t$ we define the complex tangent space
$$T_\xi = \left\{ z\in\Cn : \scp{\partial{\rho}(\xi)}{\xi-z} = 0 \right\}. $$
The space of holomorphic functions we denote as $H(\O)$ and consider
the Hardy space (see~\cite{S76},~\cite{FS72}) $$H^p(\O):=\left\{f\in
H(\O):\ \norm{f}_{H^p(\O)}^p=\sup\limits_{-\eps<t<0} \intl_{\dom_t} |f(z)|^p d\sigma_t
(z) <\infty\right\},$$ where $d\sigma_t$ is induced Lebesgue measure
on the boundary of $\O_t.$ We also denote $d\sigma = d\sigma_0.$
Hardy-Sobolev spaces $H^l_p(\O)$ are defined
by~(\ref{def:Hardy-Sobolev}).
Throughout this paper we use notations $\lesssim,\ \asymp.$ We let
$f\lesssim g$ if $f\leq c g$ for some constant $c>0,$ that doesn't
depend on main arguments of functions $f$ and $g$ and usually depend
only on dimension $n$ and domain $\O.$ Also $f\asymp g$ if $c^{-1}
g\leq f\leq c g$ for some $c>1.$
\section{Cauchy-Leray-Fantappi\`{e} formula \label{CLF}}
In the context of theory of several complex variables there is no
unique reproducing formula formula, however we could use the Leray
theorem, that allows us to construct holomorphic reproducing kernels
(\cite{AYu79}, \cite{L59}, \cite{Ra86}). For convex domain $\Om$
this theorem brings us Cauchy-Leray-Fantappi\`{e} formula, and for
$f\in H^1(\Omega)$ and $z\in\O$ we have
\begin{equation} \label{eq:CLF}
f(z) = K_\O f(z) = \cn \int\limits_{\dom} \frac{f(\xi) \partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n} = \int\limits_{\dom} f(\xi) K(\xi,z) \omega(\xi),
\end{equation}
where $\omega(\xi) = \cn
\partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1},$ and
$K(\xi,z) =\scp{\partial\rho(\xi)}{\xi-z}^{-n}.$
The $(2n-1)$-form $\omega$ defines on $\dom_t$ Leray-Levy measure
$dS,$ that is equivalent to Lebesgue surface measure $d\sigma_t$
(for details see \cite{AYu79}, \cite{LS13}, \cite{LS14}). This
allows us to identify Lebesgue, Hardy and Hardy-Sobolev spaces
defined with respect to measures $d\sigma_t$ and $dS$. Also note,
that measure $dV$ defined by the $2n$-form
$d\omega=(\partial\dbar\rho)^{n}$ is equivalent to Lebesgue measure
$d\mu$ in $\Cn.$
By \cite{R12} the integral operator $K_\O$ defines a bounded mapping
on $L^p(\dom)$ to $H^p(\Omega)$ for $1<p<\infty.$
The function $d(w,z) = \abs{\V{w}{z}}$ defines on $\dom$
quasimetric, and if $B(z,\delta) = \{w\in\dom: d(w,z)<\delta\}$ is a
quasiball with respect to $d$ then $\sigma(B(z,\delta))\asymp
\delta^n,$ see for example \cite{R12}. Therefore $\{\dom,d,\sigma\}$
is a space of homogeneous type.
Note also the crucial role in the forthcoming considerations of the
following estimate that is proved in \cite{R13}.
\begin{lemma} \label{lm:QM_est1} Let $\O$ be strongly convex, then
$$d(w,z) \asymp \rho(w) + d(\pr_{\dom}(w),z),\ w\in\CO,\ z\in\dom.$$
\end{lemma}
\subsection{The polynomial approximation of Cauchy-Leray-Fantappi\'{e} kernel\label{CLF_approx}}
In lemma~\ref{lm:CLF_approx} here we construct a polynomial
approximations of Cauchy-Leray-Fantappi\'{e} kernel based on theorem
by V.K. Dzyadyk about estimates of Cauchy kernel on domains on
complex plane (theorem~1 in part~1 of section~7 in~\cite{Dz77}). The
approximation is choosed similarly to \cite{Sh89}.
This construction allows us in theorem~\ref{thm:Poly_Sobolev} to get
polynomials that approximate holomorphic function with desired
speed.
\begin{lemma}
Let $\O$ be a strongly convex domain with $0\in\O,$ then for every
$\xi\in\OO$ the value of $\lambda =
\frac{\scp{\partial\rho(\xi)}{z}}{\scp{\partial\rho(\xi)}{\xi}}$ for
$z\in\O$ lies in domain $L(t),$ bounded by the bigger arc of the
circle $\abs{\lambda}=R=R(\O)$ and the chord $\{\lambda\in\C :
\lambda = 1 + e^{it}s,\ s\in\R,\ \abs{\lambda}\leq R \},$ where $t =
\frac{\pi}{2} - \arg(\scp{\partial\rho(\xi)}{\xi}).$
\end{lemma}
\begin{proof}
For $\xi\in\dom$ define
$$\Lambda(\xi) = \left\{\lambda\in\C : \lambda = \frac{\scp{\partial\rho(\xi)}{z}}{\scp{\partial\rho(\xi)}{\xi}},\ z\in\O\right\}. $$
The convexity of $\O$ with $0\in\O$ implies that
\begin{equation} \label{eq1}
\abs{\scp{\partial\rho(\xi)}{\xi}} \gtrsim \abs{\partial\rho(\xi)} \abs{\xi} \gtrsim 1,
\end{equation}
\begin{equation} \label{eq2}
\RE \scp{\partial\rho(\xi)}{z-\xi} \leq 0,\quad z\in\bar{\O},\ \xi\in\OO.
\end{equation}
The domain $\Lambda(\xi)\subset\C$ is also convex and contains 0, thus the
equality
$$ \frac{\scp{\partial\rho(\xi)}{z}}{\scp{\partial\rho(\xi)}{\xi}} = 1 + \frac{\scp{\partial\rho(\xi)}{z-\xi}}{\scp{\partial\rho(\xi)}{\xi}} $$
with estimates (\ref{eq1}), (\ref{eq2}) completes the proof of the
lemma. \qed
\end{proof}
\begin{lemma} \label{lm:CLF_approx}
Let $\O$ be a strongly convex domain and $r>0.$ Then for every
$k\in\N$ there exist function $K^{glob}_k(\xi,z)$ defined for $\xi\in\OO$
and polynomial in $z\in\O$ with $\deg K_k(\xi,\cdot)\leq k$ and
following properties:
\begin{equation} \label{eq:CLF_approx1}
\abs{K(\xi,z) - K^{glob}_k(\xi,z)} \lesssim \frac{1}{k^{r}} \frac{1}{d(\xi,z)^{n+r}},\quad d(\xi,z)\geq \frac{1}{k};
\end{equation}
\begin{equation} \label{eq:CLF_approx2}
\abs{K^{glob}_k(\xi,z)}\lesssim k^n,\quad d(\xi,z)\leq\frac{1}{k}.
\end{equation}
\end{lemma}
\begin{proof}
Due to \cite{Dz77} and \cite{Sh93} for any $j\in\N$ there exists a
function $T_j(t,\lambda)$ polynomial in $\lambda$ with $\deg
T_j(t,\cdot)\leq j$ such that
\begin{equation} \label{eq:Cauchy_approx1}
\left\vert \frac{1}{1-\lambda} - T_j(t,\lambda) \right\vert \lesssim
\frac{1}{j^r}\frac{1}{\abs{1-\lambda}^{1+r}}
\end{equation} for $\lambda\in
L(t)\setminus\left\{\lambda : \abs{1-\lambda}<\frac{1}{j}\right\}$
and coefficients of polynomials $T_j(t,\lambda)$ continuously depend
on $t.$ Note also that by maximum principle
\begin{equation} \label{eq:Cauchy_approx2}
T_j(t,\lambda) \lesssim j,\quad \lambda\in
L(t)\bigcap\left\{\lambda : \abs{1-\lambda}<\frac{1}{j}\right\}.
\end{equation}
Let $t(\xi) = \frac{\pi}{2} - \arg(\scp{\partial\rho(\xi)}{\xi}$ and for $j\in\N$ and $(j-1)<k\leq jn$ define $$ K_k^{glob}(\xi,z) =
K_{jn}^{glob}(\xi,z) = \frac{1}{\scp{\partial\rho(\xi)}{\xi}^n}
T_j^n\left(t(\xi),\frac{\scp{\partial\rho(\xi)}{z}}{\scp{\partial\rho(\xi)}{\xi}}\right).$$
Due to definition of $T_j$ polynomials $K_{k}^{glob}(\xi,\cdot)$
satisfy relations~(\ref{eq:CLF_approx1}), (\ref{eq:CLF_approx2}). \qed
\end{proof}
\subsection{Kor\'{a}nyi regions}
For $\xi\in\dom$ and $\eps>0$ we define the {\it inner Kor\'{a}nyi
region} as
$$D^i(\xi,\eta,\eps) = \{\tau\in\O : \pr_{\dom}(\tau) \in B(\xi,-\eta\rho(\tau)),\ \rho(\tau)>-\eps \}. $$
The strong convexity of $\O$ implies that area-integral inequality
by S.~Krantz and S.Y.~Li~\cite{KL97} for $f\in H^p(\O),\
0<p<\infty,$ could be expressed as
\begin{equation} \label{ineq:Luzin_internal}
\intl_\dom d\sigma(z) \left( \intl_{D^i(z,\eta,\eps)} \abs{\partial f(\tau)}^2
\frac{d\mu(\tau)}{(-\rho(\tau))^{n-1}}\right)^{p/2} \leq c(\O,p) \intl_\dom \abs{f}^p d\sigma.
\end{equation}
Consider the decomposition of vector $\tau\in\Cn$ as $\tau = w +
t n(\xi),$ where $w\in T_\xi,\ t\in\C,$ and
$n(\xi)=\frac{\bar\partial\rho(\xi)}{\abs{\bar\partial\rho(\xi)}}$ is a
complex normal vector at $\xi$. We define the {\it external
Kor\'{a}nyi region} as
\begin{multline}\label{df:KoranyExt}
D^e(\xi,\eta,\eps) = \{\tau\in\CO : \tau = w + t n(\xi),\\ w\in T_\xi,\
t\in\C,\ \abs{w}<\sqrt{\eta\rho(\tau)},\ \abs{\IM(t)}<\eta\rho(\tau),\
\rho(\tau)<\eps \}.
\end{multline}
In appendix~\ref{Area_int} we will proof the area-integral
inequality similar to~(\ref{ineq:Luzin_internal}) for external
regions~$D^e(\xi,\eta,\eps).$
We point out two rules for integration over regions $D^e(\xi,\eta,\eps).$
First, for every function $F$ we have
$$\intl_{\O_\eps\setminus\O} \abs{F(z)} d\mu(z) \asymp \intl_\dom d\sigma(\xi)
\intl_{D^e(\xi,\eta,\eps)} \abs{F(\tau)}
\frac{d\mu(\tau)}{\rho(\tau)^{n}}.
$$
Second, if $F(w) = \tilde{F}(\rho(w))$ then
$$ \intl_{D^e(\xi,\eta,\eps)} \abs{F(\tau)} d\mu(\tau) \asymp \intl_0^\eps \abs{\tilde{F}(t)} t^{n} dt. $$
Similar rules are valid for regions $D^i(\xi,\eta,\eps).$
We could clarify the estimate of $d(\tau,w)$ in
lemma~\ref{lm:QM_est1} for $\tau\in D^e(z,\eta,\eps).$
\begin{lemma} \label{lm:QM_est2} Let $\O$ be a strongly convex domain and $\eps,\eta>0,$
then
\begin{equation} \label{ineq:QM_est2}
d(\tau,w) \asymp \rho(\tau) + d(z,w),\quad z,w\in\dom,\ \tau\in
D^e(z,\eta,\eps).
\end{equation}
\end{lemma}
\begin{proof}
For $\tau\in D^e(z,\eta,\eps)$ we denote $\hat{\tau}=\pr_{\dom}(\tau),$ then $d(\hat{\tau},z)\lesssim
\eta\rho(\tau)$ and by lemma~\ref{lm:QM_est1}
\begin{equation*}
d(\tau,w)\lesssim \rho(\tau)+d(\hat{\tau}\lesssim
\rho(\tau)+d(\hat{\tau},z) +d(z,w) \lesssim \rho(\tau) +d(z,w).
\end{equation*}
On the other hand,
\begin{multline*} \rho(\tau) +d(z,w) \lesssim \rho(\tau) +
(d(z,\hat{\tau})+d(\hat{\tau},w))\lesssim
(1+\eta)\rho(\tau)+d(\hat{\tau},w)\\
\lesssim \rho(\tau)+d(\hat{\tau},w) \lesssim d(\tau,w).
\end{multline*}
\qed
\end{proof}
\section{The method of pseudoanalytical continuation \label{PAC}}
\subsection{Definition of pseudoanalytical continuation}
The main tool of this paper is the method of continuation of
function $f\in H(\O)$ outside the domain $\Omega.$ Let $f\in
H^1(\Omega)$ and let the boundary values of $f$ almost everywhere
coincide with the boundary values of some function $\f\in
C_{loc}^1(\COO)$ such that $ \abs{\bar{\partial}\f}\in L^1(\CO).$
Then by Stokes formula for $z\in\O$ we have
\begin{multline*}
f(z) = \lim\limits_{r\to 0+ } \cn \int\limits_{\dom_r} \frac{\f(\xi) \partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n} = \\
\lim\limits_{r\to 0+ } \cn \int\limits_{\CO_r}
\frac{\bar{\partial}\f(\xi)\wedge
\partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n}\\
= \cn \int\limits_{\CO} \frac{\bar{\partial}\f(\xi) \wedge
\partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n},
\end{multline*}
since (for details see \cite{Ra86})
$$ d_\xi\left( \frac{\partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n} \right) = 0,\quad z\in\O,\ \xi\in\CO. $$
This formula allows us to study properties of function $f\in H(\O)$
relying on estimates of its continuation.
\begin{definition} We call the function $\f\in
C_{loc}^1(\Cn\setminus\overline{\O})$ {\it the pseudoanalytic continuation of the function} $f\in H(\O)$ if
\begin{equation} \label{eq:PAC}
f(z) = \cn \int\limits_{\CO} \frac{\bar{\partial}\f(\xi) \wedge
\partial\rho(\xi)\wedge(\bar{\partial}\partial\rho(\xi))^{n-1}}{\scp{\partial\rho(\xi)}{\xi-z}^n},\ z\in\O.
\end{equation}
\end{definition}
Note that it is not
necessary for the function~$\f$ to be a continuation in terms of
coincidence of boundary values.
\subsection{Continuation by symmetry}
For $z\in\O_\eps\setminus\O$ we define the symmetric along $\dom$ point $z^*\in\O$ by $$z^* - z = 2(\pr_{\dom}(z)-z).$$
\begin{theorem} \label{thm:PAC_sym}
Let $f\in H^1_p(\O)$ and $1<p<\infty,\ m\in\N.$ There exist a pseudoanalytical continuation
$\f\in C_{loc}^1(\COO)$ of function $f$ such that
$\supp{\f}\subset\O_\eps,$
$\abs{\dbar \f(z)}\in L^p(\O_\eps\setminus\overline{\O})$ and
\begin{equation} \label{ineq:PAC_sym1}
\abs{\dbar \f(z)} \lesssim \max\limits_{\abs{\alpha}=m}
\abs{\partial^{\alpha}f(z^*)}
\rho(z)^{m-1},\quad z\in\O_\eps\setminus\O.
\end{equation}
\end{theorem}
\begin{proof}
Define
\begin{equation} \label{ineq:PAC_sym0}
\f_0(z) = \sum\limits_{\abs{\alpha}\leq m-1} \partial^{\alpha}f(z^*)
\frac{(z-z^*)^\alpha}{\alpha !},\ z\in\OO.
\end{equation}
Let $\alpha\pm e_k=(\alpha_1,\ldots,\alpha_k\pm 1,\alpha_n)$ and
define $(z-z^*)^{\alpha-e_k}=0$ if $\alpha_k=0.$ In these notations
we have
\begin{multline}
\dbar_j \f_0=\SK \sum\limits_{\abs{\alpha}\leq m-1} \left(\partial^{\alpha+e_k}f(z^*)
\frac{(z-z^*)^\alpha}{\alpha !} - \partial^{\alpha}f(z^*)
\frac{(z-z^*)^{\alpha-e_k}}{(\alpha-e_k)!}\right)\dbar_j z^*_k\\
= \SK \sum\limits_{\abs{\alpha}=m-1} \partial^{\alpha+e_k}f(z^*)
\frac{(z-z^*)^\alpha}{\alpha !} \dbar_j z^*_k,
\end{multline}
hence, $$ \abs{\dbar \f_0(z)} \lesssim \max\limits_{\abs{\alpha}=m}
\abs{\partial^{\alpha}f(z^*)}
\rho(z)^{m-1},\quad z\in\CO. $$
Consider function $\chi\in C^\infty(0,\infty)$ such that $\chi(t)=1$
for $t\leq \eps/2$ and $\chi(t)=0$ for $t\geq \eps.$ The function
$\f(z) = \f_0(z)\chi(\rho(z))$ satisfies the condition
(\ref{ineq:PAC_sym0}) and $\supp\f~\subset~\O_\eps.$
Let $d = \dist{z^*}{\dom}/10,$ then for every mutiindex $\alpha$
such that $\abs{\alpha}=m$ by Cauchy maximal inequality we have
$$\abs{\partial^\alpha f(z^*)} \lesssim d^{-m+1} \sup\limits_{\abs{\tau-z^*}<d}
\abs{\partial f(\tau)} \lesssim \rho(z)^{-m+1} \sup\limits_{\tau\in
D^i(\pr_{\dom}(z),c_0d,\eps)} \abs{\partial f(\tau)}, $$ for some $c_0>0.$
Finally, by theorem~2.1 from \cite{KL97} we get
\begin{multline*}
\int_{\OO} \abs{\dbar \f(z)}^p d\mu(z) \lesssim \int_{\O\setminus\O_{-\eps}} d\mu(z) \left(\sup\limits_{\tau\in
D^i(\pr_{\dom}(z),c_0d,\eps)} \abs{\partial f(\tau)}\right)^p\\
\lesssim \norm{\partial f}^p_{H^p(\O)}<\infty.
\end{multline*}
\qed
\end{proof}
\subsection{Continuation by global approximations.\label{PAC_glob}}
Let $f\in H^1(\Omega)$ and consider a polynomial sequence
$P_1,P_2,\ldots$ converging to $f$ in $L^1(\dom).$ Define
$$ \lambda(z) = \rho(z)^{-1} \abs{P_{2^{k+1}}(z) - P_{2^{k}}(z)},\quad 2^{-k}<\rho(z)\leq 2^{-k+1}.$$
\begin{theorem} \label{thm:PAC_glob}
Assume that $\lambda\in L^p(\Cn\setminus\Omega)$ for some $p\geq 1.$ Then there exist a pseudoanalytical continuation $\f$ of function $f$
such that
\begin{equation}\label{eq:PAC_la}
\abs{\bar{\partial} \mathbf{f}(z)} \lesssim \lambda(z),\quad
z\in\Cn\setminus\Omega.
\end{equation}
\end{theorem}
\begin{proof}
Consider function $\chi\in C^\infty(0,\infty)$ such that $\chi(t)=1$
for $t\leq \frac{5}{4}$ and $\chi(t)=0$ for $t\geq \frac{7}{4}.$ We let
$$ \mathbf{f}_0(z) = P_{2^{k}}(z) + \chi(2^k\rho(z)) (P_{2^{k+1}}(z) - P_{2^{k}}(z)), \quad 2^{-k}<\rho(z)<2^{-k+1},\ k\in\mathbb{N},$$
and define the continuation of a function $f$ by formula $\f=\chi(2\rho(z))\f_0(z).$
Now $\mathbf{f}$ is $C^1$-function on $\Cn\setminus\overline{\Omega}$ and
$\abs{\bar{\partial} \mathbf{f}(z)} \lesssim \lambda(z).$ We define
a function $F_k(z)$ as $F_k(z)= \mathbf{f}(z)$ for $\rho(z)>2^{-k}$
and as $F_k(z) = P_{2^{k+1}}(z)$ for $\rho(z)<2^{-k}.$ The the
function $F_k$ is smooth and holomorphic in $\Omega_{2^{-k}},$ and
$\abs{\dbar F_k(z)}\lesssim\lambda(z)$ for
$z\in\Cn\setminus\Omega_{2^{-k}}.$ Thus similarly to \ref{eq:PAC} we
get
$$P_{2^{k+1}}(z) = F_k(z) = \cn \int\limits_{\CO} \frac{\bar{\partial}F_k(\xi) \wedge\dS{\xi} }{\V{\xi}{z}^n},\ z\in\Omega,$$
We can pass to the limit in this formula by the dominated
convergence theorem; hence, function~$\f$ satisfies the
formula~(\ref{eq:PAC}) and is a pseudoanalytical continuation of
function~$f$. \qed
\end{proof}
\subsection{Pseudoanalytical continuation of Hardy-Sobolev spaces\label{PAC_Sobolev}}
\begin{theorem} \label{thm:PAC_Sobolev}
Let $\O$ be a strongly convex domain, $1<p<\infty,$ $l\in\N$ and
$f\in H^p(\O).$ Then $f\in H_p^l(\O)$ if and only if there exists
such pseudoanalytical continuation $\f$ that for some $\eta>0$
\begin{equation} \label{ineq:PAC_Sobolev}
\intl_\dom d\sigma(z) \left(\intl_{D^e(z,\eta,\eps)} \abs{\dbar \f(\tau) \rho(\tau)^{-l}}^2
d\nu(\tau)
\right)^{p/2} <\infty,
\end{equation}
where $d\nu(\tau)=\frac{d\mu(\tau)}{\rho(\tau)^{n-1}}.$
\end{theorem}
\begin{proof}
Let $f\in H^l_p(\O).$ By theorem~\ref{thm:PAC_sym} we could
construct pseudoanalytical continuation~$\f$ such that
$$\abs{\dbar \f(z)} \lesssim \max\limits_{\abs{\alpha}=l+1} \abs{\partial^\alpha
f(z^*)}
\rho(z)^{l},\quad z\in\CO.$$
Note that the symmetry $(z\mapsto z^*)$ with respect to $\dom$ maps the external
sector $D^e(z,\eta,\eps)$ into some internal Kor\'{a}nyi sector. Indeed,
for every $\eta>0$ there exists $\eta_1,\eps_1>0$ such that
$$\{\tau^*:\tau\in D^e(z,\eta,\eps)\} \subseteq D^i(z,\eta_1,\eps_1).$$
Applying area-integral inequality~(\ref{ineq:Luzin_internal}) we
obtain
\begin{multline*}
\intl_\dom d\sigma(z) \left(\intl_{D^e(z,\eta,\eps)} \abs{\dbar \f(\tau)
\rho(\tau)^{-l}}^2
d\nu(\tau)
\right)^{p/2}\\
\lesssim \max\limits_{\abs{\alpha}=l+1} \intl_\dom d\sigma(z) \left(\intl_{D^e(z,\eta,\eps)} \abs{\partial^{\alpha}f(\tau^*)}^2
d\nu(\tau)
\right)^{p/2}\\
\lesssim \max\limits_{\abs{\alpha}=l+1} \intl_\dom d\sigma(z) \left(\intl_{D^i(z,\eta_1,\eps_1)} \abs{\partial^{\alpha}f(\tau) }^2
\frac{d\mu(\tau)}{(-\rho(\tau))^{n-1}}
\right)^{p/2} <\infty
\end{multline*}
To prove the sufficiency, assume that function $f\in H^1(\O)$ admits
the pseudoanalytical continuation~$\f$ with the estimate~(\ref{ineq:PAC_Sobolev}.) We will prove that for every function
$g\in L^{p'}(\dom),\ \frac{1}{p}+\frac{1}{p'} =1,$ and every
multiindex $\alpha,\ \abs{\alpha}\leq l,$
$$ \abs{ \intl_\dom g(z) \partial^{\alpha}f(z) dS(z)} \leq c(f)\norm{g }_{L^{p'}(\dom)}.$$
Assume, without loss of generality, that $\alpha=(l,0,\ldots,0).$ By
representation~(\ref{eq:PAC}) we have
$$ f(z) = \intl_{\CO} \frac{\bar{\partial}\f(\xi) \wedge \omega(\xi)}{\scp{\partial\rho(\xi)}{\xi-z}^n}$$
and with $C_{nl} = \frac{(n+l-1)!}{(n-1)!}$
\begin{multline*}
\intl_\dom g(z)\partial^{\alpha}f(z) dS(z)\\
= C_{nl} \intl_\dom g(z) \left( \intl_{\CO} \left(\frac{\partial\rho(\xi)}{\partial\xi_1}\right)^{l} \frac{\bar{\partial}\f(\xi) \wedge \omega(\xi)}{\scp{\partial\rho(\xi)}{\xi-z}^{n+l}}\right)
dS(z)\\
= C_{nl} \intl_{\CO} \left(\frac{\partial\rho(\xi)}{\partial\xi_1}\right)^{l} \bar{\partial}\f(\xi) \wedge
\omega(\xi) \intl_\dom
\frac{g(z)dS(z)}{\scp{\partial\rho(\xi)}{\xi-z}^{n+l}}.
\end{multline*}
Define $\Phi_l(\xi) = \intl_\dom
\frac{g(z)dS(z)}{\scp{\partial\rho(\xi)}{\xi-z}^{n+l}}. $
Applying H\"{o}lder inequality twice we have
\begin{multline*}
\abs{\intl_\dom g(z) \partial^\alpha f(z) dS(z)} \lesssim
\intl_{\CO} \abs{\dbar \f(\xi)} \abs{\Phi_l(\xi)} d\mu(\xi)\\
\lesssim \intl_\dom dS(\xi) \intl_{D^e(\xi,\eta,\eps)} \abs{ \dbar \f(\tau) }
\abs{\Phi_l(\tau)} \frac{d\mu(\tau)}{\rho(\tau)^n} \\
\lesssim \intl_\dom d\sigma(\xi) \left(\intl_{D^e(\xi,\eta,\eps)} \abs{\dbar \f(\tau)
}^2\rho(\tau)^{-2l} \frac{d\mu(\tau)}{\rho(\tau)^{n-1}}\right)^{1/2} \times\\
\times\left(\intl_{D^e(\xi,\eta,\eps)} \abs{\Phi_l(\tau)}^2 \rho(\tau)^{2l-2}
\frac{d\mu(\tau)}{\rho(\tau)^{n-1}}\right)^{1/2}\\
\lesssim \left( \intl_\dom dS(\xi) \left(\intl_{D^e(\xi,\eta,\eps)} \abs{\dbar \f(\tau)}^2\rho(\tau)^{-2l}
d\nu(\tau)\right)^{p/2}\right)^{1/p} \times\\ \times
\left( \intl_\dom dS(\xi) \left(\intl_{D^e(\xi,\eta,\eps)} \abs{\Phi_l(\tau)}^2 \rho(\tau)^{2l-2}
d\nu(\tau)\right)^{p'/2}\right)^{1/p'}.
\end{multline*}
The first product term is bounded by~(\ref{ineq:PAC_Sobolev}), and
the second one by the area-integral inequality~(\ref{est:area_int}),
that we will prove in the appendix~\ref{Area_int} in
theorem~\ref{thm:area_int}. \qed
\end{proof}
\section{Constructive description of Hardy-Sobolev spaces \label{Poly_Sobolev}}
\begin{theorem} \label{thm:Poly_Sobolev} Let $f\in H^1(\O)$ and $1<p<\infty,\
l\in\N.$ Then $f\in H^l_p(\O)$ iff there exists sequence of $2^k$-degree polynomials $P_{2^k}$ such that
\begin{equation} \label{ineq:Poly_Sobolev}
\intl_{\dom} d\sigma(z) \left(\SK \abs{f(z)-P_{2^k}(z)}^2 2^{2l k}
\right)^{p/2} < \infty.
\end{equation}
\end{theorem}
\begin{proof}
Assume that condition~(\ref{ineq:Poly_Sobolev}) holds, then
polynomials $P_{2^k}$ converge to function $f$ in $L^p(\dom)$ and by
the theorem~\ref{thm:PAC_glob} we could construct pseudoanalytical
continuation~$\f$ such that
$$ \abs{\dbar \f(z)} \lesssim \abs{P_{2^{k+1}}(z)-P_{2^k}(z)}
\rho(z)^{-1},\quad z\in \CO,\ 2^{-k}\leq \rho(z)<2^{-k+1}.$$
Consider the decomposition of region $D^e(z,\eta,\eps)$ to sets $D_k(z) =
\{\tau\in D^e(z,\eta,\eps):\ 2^{-k}\leq \rho(\tau) < 2^{-k+1}\},$ and
define functions
\begin{align*}
a_k(z) &= \abs{P_{2^{k+1}}(z)-P_{2^k}(z)} 2^{-kl },\\
b_k (z) &= \left(\intl_{D_k(z)} \abs{\dbar
\f(\tau)\rho(\tau)^{-l}}^2 d\nu(\tau) \right)^{1/2}, z\in\dom.
\end{align*}
\begin{lemma}\label{lm:Poly_Sobolev} $b_k(z) \lesssim M a_k (z),$
where $Ma_k$ is the maximal function with respect to centred quasiballs on $\dom$ $$Ma_k(z) = \sup\limits_{r>0}\frac{1}{\sigma(B(z,r))} \int\limits_{B(z,r)} |a_k(\xi)| d\sigma(\xi).$$
\end{lemma}
Assume, that this lemma holds, then by Fefferman-Stein maximal
theorem (see \cite{GLY04}, \cite{FS72}) we have
$$ \intl_\dom \left(\SK b_k(z)^2\right)^{p/2} d\sigma(z) \lesssim \intl_\dom \left(\SK a_k(z)^2\right)^{p/2}d\sigma(z).$$
The right-hand side of this inequality is finite by the
condition~(\ref{ineq:Poly_Sobolev}), also we have
$$ \SK b_k(z)^2 = \intl_{D^e(z,\eta,\eps)} \abs{\dbar \f(\xi) \rho(\xi)^{-l}}^2 d\nu(\xi), $$
which completes the proof of the sufficiency in the theorem.
Prove the necessity. Now $f\in H^l_p(\O)$ with $1<p<\infty$ and
$l\in\N.$ By theorem~\ref{thm:PAC_Sobolev} we could construct
continuation~$\f$ of function~$f$ with
estimate~(\ref{ineq:PAC_Sobolev}). Applying the approximation of
Cauchy-Leray-Fantappi\`{e} kernel from lemma~\ref{lm:CLF_approx} to
function $\f$ we define polynomials
$$ P_{2^k}(z) = \int_{\CO}
\bar{\partial} \f(\xi)\wedge\omega(\xi) K^{glob}_{2^k}(\xi,z).$$ We
will prove that these polynomials satisfy the
condition~(\ref{ineq:Poly_Sobolev}). From lemma~\ref{lm:CLF_approx}
we obtain
\begin{multline*}
\abs{f(z)-P_{2^k}(z)} \lesssim \int_{\CO} \abs{\bar{\partial}\f(\xi)} \abs{\frac{1}{\scp{\partial{\rho(\xi)}}{\xi-z}^d} - K^{glob}_{2^k}(\xi,z) } d\mu(\xi)\\ \lesssim U(z) + V(z) + W_1(z) +
W_2(z),
\end{multline*}
where
\begin{align*}
U(z) &= \int\limits_{d(\tau,z)<2^{-k}} \frac{ \abs{\bar{\partial}\f(\tau)} }{ \abs{ \scp{\partial{\rho(\tau)}}{\tau-z} }^n }d\mu(\tau), \\
V(z) &= 2^{kn} \int\limits_{d(\tau,z)<2^{-k}} \abs{ \bar{\partial}\f(\tau) } d\mu(\tau), \\
W_1(z) &= 2^{-kr} \int\limits_{\substack{d(\tau,z)>2^{-k}\\ \rho(\tau)<2^{-k}}} \frac{ \abs{\bar{\partial}\f(\tau)} }{ \abs{\scp{\partial{\rho(\tau)}}{\tau-z}}^{n+r} }d\mu(\tau),\\
W_2(z) &= 2^{-kr} \int\limits_{\rho(\tau)>2^{-k}} \frac{
\abs{\bar{\partial}\f(\tau)} }{
\abs{\scp{\partial{\rho(\tau)}}{\tau-z}}^{n+r} }d\mu(\tau).
\end{align*}
The parameter $r>0$ will be chosen later.
Note that $ V(z) \lesssim c U(z)$ and estimate the contribution of
$U(z)$ to the sum. For some $c_1,c_2>0$ we have
\begin{multline*}
U(z) \leq \intl_{\substack{d(w,z)<c_1 2^{-k}\\ w\in\dom}} d\sigma(w)
\sum\limits_{j>c_2 k} \intl_{D_j(w)} \frac{ \abs{\dbar \f(\tau)} }{
\abs{\scp{\partial{\rho(\tau)}}{\tau-z}}^n }
\frac{d\nu(\tau)}{\rho(\tau)}\\
\leq \intl_{\substack{d(w,z)<c_1 2^{-k}\\ w\in\dom}} d\sigma(w)
\sum\limits_{j>c_2 k} \left(\intl_{D_j(w)} \abs{\dbar
\f(\tau)\rho(\tau)^{-l}}^2 d\nu(\tau)\right)^{1/2}\times\\
\times\left(\intl_{D_j(w)} \frac{ \rho(\tau)^{2(l-1)} d\nu(\tau) }{ \abs{\scp{\partial{\rho(\tau)}}{\tau-z}}^n }\right)^{1/2}
= \sum\limits_{j>c_2 k} \intl_{d(w,z)<c_1 2^{-j}} b_j(w) m_j(w)
d\sigma(w)
\end{multline*}
Consider the integral~$m_j(w).$ Since $\tau\in D_j(w)$ then by
estimates from lemma~\ref{lm:QM_est2} $d(\tau,z) \asymp \rho(\tau) +
d(w,z)> 2^{-j}$ and
\begin{equation}
m_j(w) = \left(\intl_{D_j(w)} \frac{ \rho(\tau)^{2(l-1)} d\nu(\tau) }{ \abs{\V{\tau}{z}}^n }\right)^{1/2} \lesssim \frac{2^{-j(l-1)}}{2^{-jn}} 2^{-j} = 2^{jn-jl}.
\end{equation}
Thus
\begin{equation} \label{est_U}
2^{kl} U(z) \lesssim \sum\limits_{j>c_1 k} 2^{-(j-k)l}
2^{jn}\intl_{d(w,z)< c_2 2^{j} } b_j(w) d\sigma(w)
\lesssim \sum\limits_{j>c_1 k} 2^{-(j-k)l} Mb_j(z).
\end{equation}
Now estimate the value $W_1(z).$ Similarly to the previous we have
\begin{multline*}
W_1(z) \leq 2^{-kr} \suml_{j>k} \intl_{ d(w,z)\geq c_1 2^{-k} }
b_j(w) m_j^r(w) d\sigma(w)\\
\leq 2^{-kr} \suml_{j>k} \suml_{t=c_2}^k \intl_{ c_12^{-t}\leq d(w,z)\leq c_1 2^{-t+1} }
b_j(w) m_j^r(w) d\sigma(w),
\end{multline*}
where
\begin{equation*}
m_j^r(w) =\left( \intl_{ D_j(w) } \frac{ \rho(\tau)^{ 2(l-1) } d\nu(\tau) }{ \abs{\V{\tau}{z}}^{2(n+r)} } \right)^{1/2}.
\end{equation*}
Applying the estimate $d(\tau,z) \asymp \rho(\tau) + d(w,z)\gtrsim
2^{-t},$ we obtain
\begin{equation*}
m_j^r(w) \lesssim 2^{-jl+t(n+r)} .
\end{equation*}
Finally
\begin{equation*}
\suml_{t=c_2}^k \intl_{ d(w,z)\leq c_1 2^{-t+1} }
b_j(w) m_j^r(w) d\sigma(w) \lesssim \suml_{t=c_2}^k 2^{-jl+tr}Mb_j(z)\lesssim 2^{-jl+kr} Mb_j(z)
\end{equation*}
and
\begin{equation} \label{est_W1}
2^{kl} W_1(z) \lesssim \sum\limits_{j>k} 2^{-l(j-k)} Mb_j(z).
\end{equation}
Similarly, estimating the contribution of $W_2(z),$ we obtain
\begin{equation}
2^{kl}W_2(z) \lesssim 2^{-k(r-l)} \sum\limits_{j=0}^k \intl_\dom
b_j(w) m_j^r(w) d\sigma(w).
\end{equation}
Since $d(\tau,z) \gtrsim 2^{-j}+d(w,z)$ for $\tau\in\dom,\ \tau\in
D_j(z)$ then
$$m_j^r(w) \lesssim \frac{2^{-jl}}{(2^{-j} + d(w,z))^{n+r}}\leq \min\left(2^{j(n+r-l)}, 2^{-jl}d(w,z)^{-n-r}\right) .$$
Thus
\begin{multline*}
\intl_\dom b_j(w) m_j^r(w) d\sigma(w) \lesssim \intl_{d(w,z)\leq
2^{-j}} \frac{ 2^{-jl} }{2^{-j(n+r)}} b_j(w) d\sigma(w)\\ +\sum\limits_{t=1}^{j-1}\intl_{2^{-t-1}\leq d(w,z)\leq
2^{-t}} \frac{ 2^{-jl} }{2^{-t(n+r)}} b_j(w) d\sigma(w)\\
\lesssim \sum\limits_{t=1}^{j} 2^{-jl} 2^{tr} M b_j(z) \lesssim
2^{-jl} 2^{j r} M b_j(z).
\end{multline*}
Choosing $r=2l,$ we have
\begin{equation} \label{est_W2}
W_2(z) 2^{kl} \lesssim \sum\limits_{j=1}^k 2^{-(k-j)(r-l)} M
b_j(z)\leq\sum\limits_{j=1}^k 2^{-(k-j)l} M
b_j(z).
\end{equation}
Combining the estimates~(\ref{est_U},~\ref{est_W1},~\ref{est_W2}) we
finally obtain
$$\abs{f(z)-P_{2^k}(z)}2^{kl} \lesssim \sum\limits_{j=1}^k 2^{-(k-j)l} M
b_j(z) + \sum\limits_{j>k} 2^{-(j-k) l} M b_j(z), $$
which similarly to \cite{D81} implies
$$ \SK \abs{f(z)-P_{2^k}(z)}^2 2^{2kl} \lesssim \SK (M b_k(z))^2. $$
Then, by Fefferman-Stein theorem
\begin{multline*}\intl_{\dom} d\sigma(z) \left(\SK \abs{f(z)-P_{2^k}(z)}^2 2^{2l k}
\right)^{p/2}\leq \intl_{\dom} \left(\SK b^2_k(z)\right)^{p/2}d\sigma(z)\\ \leq \intl_{\dom} d\sigma(z) \left( \intl_{D^e(z,\eta,\eps)} \abs{\dbar \f(\xi) \rho(\xi)^{-l}}^2 d\nu(\xi) \right)^{p/2}<\infty.
\end{multline*}
This completes the proof of the theorem and it remains to prove
lemma~\ref{lm:Poly_Sobolev}. \qed
\end{proof}
\begin{proof}[of the lemma~\ref{lm:Poly_Sobolev}]. Define
$g_k(z) := 2^{-kl}(P_{2^{k+1}}(z) - P_{2^k}(z)).$
Let $z\in\dom$ and $\tau\in S_k(z).$ Consider complex normal vector
$n(z) = \frac{\bar\partial\rho(z)}{ \abs{\bar\partial\rho(z)} }$ at~$z,$
complex tangent hyperplane $T_z=\{w\in\C^n:
\scp{\partial\rho(z)}{w-z}=0\}$ and complex plane
$T_{z,\tau}^{\perp}$, orthogonal to $T_z$ and containing the point
$\tau$
$$T_{z,\tau}^{\perp} := \{\tau+s n(z): s\in\C\}.$$
Projection of vector $\tau\in\C$ to $\dom\bigcap T_{z,\tau}^{\perp}$
we will denote as $\pi_z(\tau).$
Define $\O_{z,\tau} = \O\bigcap T_{z,\tau}^{\perp}$ and
$\gamma_{z,\tau}=\dom_{z,\tau}.$ There exist a conformal map\\
$\varphi_{z,\tau}:T_{z,\tau}^{\perp}\setminus\O_{z,\tau} \to
\C\setminus\{w\in\C: \abs{w}=1\}$ such that $\
\varphi_{z,\tau}(\infty)=\infty,\ \varphi_{z,\tau}'(\infty)>0, $ and
we could consider analytical in
$T_{z,\tau}^{\perp}\setminus\O_{z,\tau} $ function
$G_k(s):=\frac{g_k(s)}{\varphi_{z,\tau}^{2^{k+1}}(s)}.$
Applying to function $G_k$ Dyn'kin maximal estimate from~\cite{D77}
for domain $T_{z,\tau}^{\perp}~\setminus~\O(z,\tau)$ we obtain the
estimate
$$ \abs{G_k(\tau)}\lesssim \frac{1}{\rho(\tau)} \intl_{s\in I_{z,\tau}}\abs{G_k(s)} \abs{ds} + \intl_{\dom_{z,\tau}\setminus I_{z,\tau}} \abs{G_k(s)}\frac{\rho(\tau)^m}{\abs{s-\pi_z(\tau)}^{m+1}} \abs{ds}, $$
where $I_{z,\tau} = \{s\in\gamma_{z,\tau}: \abs{s-\pi_z(\tau)}<
\dist{\tau}{\dom_{z,\tau}}/2\},$ and $m>0$ could be chosen arbitrary
large.
Note that $\abs{\varphi_{z,\tau}(s)}-1\asymp
\dist{s}{\dom_{z,\tau}}\asymp 2^{-k}, $ thus $\abs{g_k(s)}\asymp
\abs{G_k(s)}$ for $s\in D_k(z)\bigcap T_{z,\tau}^{\perp}.$ Hence,
\begin{equation} \label{lm:Poly_Sobolev_ineq1}
\abs{g_k(\tau)} \lesssim \suml_{j=1}^\infty 2^{-jm}
\frac{1}{2^j\rho(\tau)} \intl_{\substack{s\in\dom_{z,\tau}\\
\abs{s-\pi_z(\tau)}<2^j\rho(\tau)}} \abs{g_k(s)} \abs{ds}.
\end{equation}
Since the boundary of the domain $\O$ is $C^3$-smooth, we can assume
that the constant in this inequality~(\ref{lm:Poly_Sobolev_ineq1})
does not depend on $z\in\dom$ and $\tau\in\OO.$
Note that function $g_k(\tau+z-w)$ is holomorphic in $w\in T_z,$
then estimating the mean we obtain
\begin{multline}
\abs{g_k(\tau)}\leq \frac{1}{\rho(\tau)^{n-1}} \intl_{\abs{w-z}<\sqrt{\rho(\tau)}} \abs{g_k(\tau+z-w)} d\mu_{2n-2}(w)\\
\lesssim \suml_{j=1}^\infty 2^{-jm} \frac{1}{\rho(\tau)^{n-1}} \intl_{\abs{w-z}<\sqrt{\rho(\tau)}} \frac{d\mu_{2n-2}(w) }{2^j\rho(\tau)} \intl_{\substack{s\in\dom_{z,\tau}\\ \abs{s-\pi_z(\tau+z-w)}<2^j\rho(\tau)}} \abs{g_k(s)} \abs{ds}\\ \lesssim \suml_{j=1}^\infty
2^{-j(m-n+1)} \intl_{B(z,2^j\rho(\tau))} \abs{g_k(w)}d\sigma(w),
\end{multline}
where $d\mu_{2n-2}$ is Lebesgue measure in $T_z$
Assume that $m>n-1,$ then $ \abs{g_k(\tau)} \lesssim M g_k(z),\
z\in\dom,\ \tau\in D_k(z).$ Finally,
\begin{multline*}
b_k(z) = \intl_{D_k(z)} \abs{\dbar \f(\tau)\rho(\tau)^{-l}}^2
d\nu(\tau)
\lesssim \intl_{D_k(z)} \abs{
g_k(\tau)\rho(\tau)^{-l-1}}^2 d\nu(\tau)\\ \lesssim \left(M
a_k(z)\right)^{2} \intl_{D_k(z)} \frac{d\nu(\tau)}{\rho(\tau)^2}
\lesssim \left(Ma_k(z)\right)^{2}
\end{multline*}
and the lemma is proved.\hfill $\Box$
\end{proof}
\begin{appendix}
\renewcommand*{\thesection}{\Alph{section}}
\section{Area-integral inequality for external Kor\'{a}nyi region
}\label{Area_int} \numberwithin{theorem}{section}
Let $\O\subset\Cn$ be a strongly convex domain and $\eta,\eps>0$. For
function $g\in L^1(\dom)$ and $l\in\N$ we define a function
\begin{equation} \label{eq:area_int}
I_l(g,z) = \left(\ \intl_{D^e(z,\eta,\eps)} \abs{\ \intl_\dom \frac{ g(w)
dS(w)}{\V{\tau}{w}^{n+l}} }^2 d\nu_l(\tau) \right)^{1/2},
\end{equation}
where $dS(w) =\cn
\partial\rho(w)\wedge(\bar{\partial}\partial\rho(w))^{n-1}$ (see (\ref{CLF})) and $d\nu_l(\tau) = \frac{d\mu_{2n}(\tau)}{\rho(\tau)^{n-2l-1}}.$
\begin{theorem}\label{thm:area_int}
Let $\O$ be strongly convex domain and $g\in L^p(\dom),\
1<p<\infty,$ Then
\begin{equation} \label{est:area_int}
\intl_\dom I_l(g,z)^p d\sigma(z) \lesssim \intl_\dom \abs{g(z)}^p
d\sigma(z).
\end{equation}
\end{theorem}
Note that in the one-variable case the integral~(\ref{eq:area_int})
gives the holomorphic function and the result of the theorem follows
from~\cite{D81}.
\begin{definition} Assume, that defining function~$\rho$ for strongly convex domain $\O$
has the following form near $0\in\dom$
\begin{equation} \label{eq:rhostandart}
\rho(z) = 2\RE(z_n) + \suml_{j,k=1}^n A_{jk} z_j \bar{z}_k + O(\abs{z}^3)
\end{equation}
with positive definite form $A_{jk} z_j \bar{z}_k.$ We define a set
\begin{multline}
D_0(\eta,\eps) = \{ \tau\in\CO : \abs{\tau_1}^2+\ldots+\abs{\tau_{n-1}}^2 < \eta \RE(\tau_n),\\ \abs{\IM(\tau_n)} <\eta\RE(\tau_n),\ \abs{\RE(\tau_n)}<\eps\}.
\end{multline}
\end{definition}
\begin{lemma}\label{lemma:rhostandart}
Suppose, that $\rho$ has the form (\ref{eq:rhostandart}). There exist constants $c,\eps_0>0$ such that
\begin{equation*}
D^e(0,\eta,\eps)\subset D_0(c\eta,c\eps),\ D_0(\eta,\eps)\subset D^e(0,c\eta,c\eps)\ \text{for}\ 0<\eta,\eps<\eps_0.
\end{equation*}
\end{lemma}
\begin{proof} For the function
$\rho$ of the form~(\ref{eq:rhostandart}) the Kor\'{a}nyi sector (\ref{df:KoranyExt})
could be expressed as follows
\begin{multline*}
D^e(0,\eta,\eps) = \{\tau\in \CO: \abs{\tau_1}^2+\ldots+\abs{\tau_{n-1}}^2\leq \eta\rho(\tau),\\ \abs{\IM(\tau_n)}\leq\eta\rho(\tau),\ \rho(\tau)<\eps\}
\end{multline*}
and
\begin{multline*}
\rho(\tau)\leq 2\RE(\tau_n) + c_0\left( \abs{\tau_1}^2+\ldots+\abs{\tau_{n-1}}^2 + \IM(\tau_n)^2 + \RE(\tau_n)^2\right)\\
\leq (2+ c_0 \RE(\tau_n))\RE(\tau_n) + c_0(1+ \eta\rho(\tau))\eta\rho(\tau), \ \tau\in D^e(0,\eta,\eps) .
\end{multline*}
Thus for $\eta<\eta_0=\frac{1}{8c_0}$ we have $\rho(\tau)\leq c \RE(\tau_n).$
It is easy to see, that $|\tau|\to 0$ when $\rho(\tau)\to 0, \tau\in D^e(0,\eta,\eps).$ Then by convexity of $\O$
$$2\RE(\tau_n) = \rho(\tau) - \suml_{j,k=1}^n A_{jk} \tau_j \bar{\tau}_k + O(\abs{\tau}^3)\leq \rho(\tau),\ \tau\in D^e(0,\eta,\eps_0) $$
for some $\eps_0\in(0,\eta_0).$
Finally $D^e(0,\eta,\eps)\subset D_0(c\eta,\eps)$ and analogously $D_0(\eta,\eps)\subset D^e(0,\eta,\eps)$ for $0<\eta,\eps<\eps_0.$ \qed
\end{proof}
\begin{theorem} \label{thm:rhostandart}
There exists such covering of the set
$\overline{\O}_\eps\setminus\O_{-\eps}$ by open sets $\Gamma_j$ such
that for every $\xi\in\Gamma_j$ we can find a holomorphic change of
coordinates $\varphi_j(\xi,\cdot) : \Cn \to \Cn $ such that
\begin{enumerate}
\item[{\rm{1.}}] The mapping $\varphi_j(\xi,\cdot)$
transforms function $\rho$ to the type (\ref{eq:rhostandart})
and could be expressed as follows
\begin{equation}\label{thm:rhostandart:cond1}
\varphi_j(\xi,z) = \Phi_j(\xi) (z-\xi) + (z-\xi)^\perp B_j(\xi) (z-\xi) e_n,
\end{equation}
where matrices $\Phi_j(\xi), B_j(\xi)$ are $C^1$-smooth on
$\Gamma_j,$ and $e_n=(0,\ldots,0,1).$
\item[{\rm{2.}}] Let $\psi_j(\xi,\cdot)$ be an inverse map of $\varphi_j(\xi,\cdot),$ and let $J_j(\xi,\cdot)$ be a complex Jacobian of $\psi_j$.
Then
\begin{align} \label{thm:rhostandart:cond2}
\sup\limits_{\tau\in\O_\eps\setminus\overline{\O}_\eps}
\abs{J_j(\xi,\cdot) - J_j(\xi',\cdot)} &\lesssim \abs{\xi-\xi'},\\
\sup\limits_{\tau\in\O_\eps\setminus\overline{\O}_\eps}
\abs{\psi_j(\xi,\cdot) - \psi_j(\xi',\cdot)} &\lesssim \abs{\xi-\xi'}.\
\end{align}
Note that real Jacobian is then equal to $\abs{J_j(\xi,\cdot)}^2 = J_j(\xi,\cdot)\overline{J_j(\xi,\cdot)}.$
\item[{\rm{3.}}] There exist constants $c,\eps_0>0$
such that for $0<\eta,\eps<\eps_0$
\begin{equation} \label{thm:rhostandart:cond3}
\varphi_j(\xi,D^e(\xi,\eta,\eps))\subseteq D_0(c\eta,c\eps),\quad \psi_j(\xi,D_0(\eta,\eps)\subseteq
D^e(\xi,c\eta,c\eps).
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\xi\in\dom,$ by linear change of coordinates $z' =
(z-\xi)\Phi(\xi)$ we could obtain the following form for function
$\rho$
\begin{multline*}
\rho(z) = \rho(\xi+ \Phi^{-1}(\xi)z')\\ = 2\RE(z'_n) + \suml_{j,k=1}^n A^1_{jk}(\xi) z'_j\bar{z}'_k + \RE\suml_{j,k=1}^n A_{jk}^2(\xi) z'_j z'_k +O(\abs{z'}^3).
\end{multline*}
Setting $z''_n =z'_n + A_{jk}^2 z'_j z'_k$ and $z''_j = z'_j,\
1\leq j\leq n-1,$ we have (see \cite{Ra86})
$$\rho(z'')= 2\RE(z'_n) + \suml_{j,k=1}^n A^1_{jk}(\xi) z''_j\bar{z}''_k +O(\abs{z''}^3). $$
Denote $B(\xi) = \Phi(\xi)^\perp A^2(\xi) \Phi(\xi),$ then
$$\varphi(\xi,z) = \Phi(\xi) (z-\xi) + (z-\xi)^\perp B(\xi)
(z-\xi) e_n.$$
We choose $\Gamma_j$ such that the matrix $\Phi(\xi)$ could be
defined on $\Gamma_j$ smoothly, this choice we denote as $\Phi_j,$
and the change corresponding to this matrix as $\varphi_j$
$$\varphi_j(\xi,z) = \Phi_j(\xi) (z-\xi) + (z-\xi)^\perp B_j(\xi)
(z-\xi)e_n.$$ Thus mappings $\varphi_j$ satisfy the first condition.
Easily, the second condition also holds.
The last condition (\ref{thm:rhostandart:cond3}) follows immediately from lemma~\ref{lemma:rhostandart}. This ends the
proof of the theorem. \qed
\end{proof}
Further we will assume, that the covering
$\overline{\O}_\eps\setminus\O_{-\eps} \subset
\bigcup\limits_{j=1}^N\Gamma_j$ and maps $\varphi_j, \psi_j$ are
chosen by the theorem~\ref{thm:rhostandart}. For covering
$\{\Gamma_j\}$ we consider a smooth decomposition of identity on
$\dom:$
$$\chi_j \in C^{\infty}(\Gamma_j),\ 0\leq\chi_j\leq 1,\ \supp{\chi_j}\subset\Gamma_j,\ \suml_{j=1}^N \chi_j(z) = 1,\ z\in\dom.$$
Fix parameters $0<\eps,\eta<\eps_0,$ denote $D_0 =D_0(\eta,\eps).$ Then by
(\ref{thm:rhostandart:cond3})
$$D^e(z)=\varphi_j(z,D^e(z,\eta/c,\eps/c))\subset D_0$$ and
\begin{multline} \label{eq:Luzin_decomposition}
I_l (g,z)^2\\ = \suml_{j=1}^N \chi_j(z)
\intl_{D^e(z)} \abs{\ \intl_\dom \frac{
g(w)J_j(z,\tau) dS(w)}{\V{\psi_j(z,\tau)}{w}^{n+l}} }^2
\frac{d\mu(\tau)}{\RE(\tau_n)^{n-2l+1}} \\
\lesssim \suml_{j=1}^N \intl_{D_0} \abs{\ \intl_\dom \frac{ g(w)\chi_j^{1/2}(z) J_j(z,\tau)
dS(w)}{\V{\psi_j(z,\tau)}{w}^{n+l}} }^2
\frac{d\mu(\tau)}{\RE(\tau_n)^{n-2l+1}}.
\end{multline}
\noindent We will consider the function
\begin{equation}
K_j(z,w) (\tau) = \frac{\chi_j^{1/2}(z)J_j(z,\tau)}{\V{\psi_j(z,\tau)}{w}^{n+1}}
\end{equation}
as a map $\dom\times\dom\to \mathscr{L}(\C,L^2(D_0,d\nu_l)),$ such
that its values are operator of multiplication from $\C$ to
$L^2(D_0,d\nu_l),$ where
$d\nu_l(\tau)=\frac{d\mu(\tau)}{\IM(\tau_n)^{n-2l+1}}$ is a measure
on the region $D_0.$ Throughout the proof of the
theorem~\ref{thm:area_int} $j,l$ will be fixed integers and the norm
of function $F$ in the space $L^2(D_0,d\nu_l)$ will be denoted as~$\norm{F}.$
We will show that integral operator defined by kernel $K_j$ is
bounded on $L^p.$ To prove this we apply $T1$-theorem for
transformations with operator-valued kernels formulated by
Hyt\"{o}nen and Weis in \cite{HW05}, taking in account that in our
case concerned spaces are Hilbert. Some details of the proof are
similar to the proof of the boundedness of operator
Cauchy-Leray-Fantappi\`{e} $K_\O$ for lineally convex domains
introduced in \cite{R12}. Below we formulate the $T1$-theorem,
adapted to our context.
\begin{definition}
We say that the function $f\in C^\infty_0(\dom)$ is a normalized
bump-function, associated with the quasiball $B(w_0,r)$ if
$\supp{f}\subset B(z,r),$ $\abs{f}\leq 1,$ and
$$\abs{f(\xi)-f(z)}\leq \frac{d(\xi,z)^{\gamma}}{r^{\gamma}}.$$
The set of bump-functions associated with $B(w_0,r)$ is denoted as
$A(\gamma,w_0,r).$
\end{definition}
\begin{theorem} \label{thm:T1}
Let $K:\dom\times\dom\to \mathscr{L} (\C, L^2(D_0,d\nu_l))$ verify
the estimates
\begin{align}
&\norm{K(z,w)} \lesssim \frac{1}{d(z,w)^n}; \label{KZ1}\\
&\norm{K(z,w)-K(\xi,w)} \lesssim \frac{d(z,\xi)^\gamma}{d(z,w)^{n+\gamma}},\quad d(z,w)> C d(z,\xi); \label{KZ2}\\
&\norm{K(z,w)-K(z,w')} \lesssim \frac{d(w,w')^\gamma}{d(z,w)^{n+\gamma}},\quad d(z,w)> C d(w,w').\label{KZ3}
\end{align}
Assume that operator $T:
\mathscr{S}(\dom)\to\mathscr{S}'(\dom,\mathscr{L} (\C,
L^2(D_0,d\nu_l)))$ with kernel $K$ verify the following conditions.
\begin{itemize}
\item $T1,\ T'1\in\BMO(\dom,L^2(D_0,d\nu_l)),$ where $T'$ is
formally adjoint operator.
\item Operator $T$ satisfies the weak boundedness property, that is
for every pair of normalized bump-functions $f,g\in A(\gamma,w_0,r)$ we have $$\norm{\scp{g}{Tf}}\leq C
r^{-n}.$$
\end{itemize}
Then $T\in\mathscr{L} (L^p(\dom), L^p(\dom,L^2(D_0,d\nu_l))$ for
every $p\in (1,\infty).$
\end{theorem}
\medskip
In the following three lemmas we will prove that kernels $K_j$ and
corresponding operators $T_j$ satisfy the conditions of the
$T1$-theorem.
\begin{lemma} \label{lm:T1_1} The kernel $K_j$ verify estimates (\ref{KZ1}-\ref{KZ3}).
\end{lemma}
\begin{proof} By lemma \ref{lm:QM_est2} we have
$\abs{\V{\tau}{w}}\asymp\rho(\tau) + \abs{\V{z}{w}},$ $z,w\in\dom,\
\tau\in D^e(z,c\eta,c\eps). $
Thus
\begin{multline*}
\norm{ K_j(z,w)}^2 = \intl_{D_0} \abs{K_j(z,w)(\tau)}^2 d\nu_l(\tau)
\lesssim\intl_{D^e(z,c\eta,c\eps)}
\frac{d\nu_l(\tau)}{ \abs{ \V{\tau}{w} }^{2n+2l} } \\
\lesssim\intl_{D^e(z,c\eta,c\eps)}\frac{1}{( \rho(\tau) + \abs{\V{z}{w}} )^{2n+2l}}
\frac{d\mu(\tau)}{\rho(\tau)^{n-2l+1}}\\
\lesssim \intl_0^\infty\frac{t^{2l-1} dt}{(t+\abs{\V{z}{w}})^{2n+2l}} \lesssim
\frac{1}{\abs{\V{z}{w}}^{2n}}\lesssim \frac{1}{d(z,w)^{2n}}.
\end{multline*}
Similarly,
\begin{multline*}\norm{K_j(z,w) - K_j(z,w')}^2\\
\lesssim\intl_{D^e(z,c\eta,c\eps)} \abs{\frac{1}{\V{\tau}{w}^{n+l}} -
\frac{1}{\V{\tau}{w'}^{n+l}}}^2 d\nu_l(\tau)
\end{multline*}
\noindent Denote $\hat{\tau} =\pr_{\dom}(\tau),$ then
\begin{multline*} \abs{\V{\tau}{w}} \lesssim \rho(\tau) +
\abs{\V{\hat{\tau}}{w}}\\ \lesssim \rho(\tau) + \abs{\V{z}{w}} +
\abs{\V{\hat{\tau}}{z}} \lesssim \rho(\tau) + \abs{\V{z}{w}},
\end{multline*}
which combined with lemma \ref{lm:QM_est2} and condition $$d(w,w')=\abs{\V{w}{w'}}<
C \abs{\V{z}{w}}=C d(z,w)$$ implies
\begin{multline*}
\abs{\V{\tau}{w}} \asymp \rho(\tau) + \abs{\V{z}{w}} \asymp \rho(\tau) + \abs{ \V{z}{w'} }\\ \asymp \abs{\V{\tau}{w'}}.
\end{multline*}
Next, we have
\begin{multline*}
\abs{ \V{\tau}{w'}-\V{\tau}{w} } = \abs{ \scp{\partial\rho(\tau)}{\hat{\tau}-w}-\scp{\partial\rho(\tau)}{\hat{\tau}-w'} }\\
\leq \abs{ \scp{\partial\rho(\tau)-\partial\rho(\hat{\tau})}{w-w'} } +
\abs{ \scp{\partial\rho(\hat{\tau})}{\hat{\tau}-w}-\scp{\partial\rho(\hat{\tau})}{\hat{\tau}-w'} }\\
\lesssim \rho(\tau) \abs{\V{w}{w'}}^{1/2} +
\abs{\V{\hat{\tau}}{w}}^{1/2}\abs{\V{w}{w'}}^{1/2}\\ \lesssim \abs{\V{\tau}{w}}^{1/2} \abs{\V{w}{w'}}^{1/2}
\end{multline*}
Hence,
\begin{multline*}\norm{K_j(z,w) - K_j(z,w')}^2\lesssim
\intl_{D^e(z,c\eta,c\eps)} \frac{ \abs{\V{w}{w'}} }{ \abs{\V{\tau}{w}}^{2n+2l+1} } d\nu_l(\tau)\\
\lesssim \intl_0^\infty \frac{\abs{\V{w}{w'}} t^{2l-1}
dt}{(t+\abs{\V{z}{w}})^{2n+2l+1}} \lesssim \frac{ \abs{\V{w}{w'}} }{
\abs{ \V{z}{w} }^{2n+1} }=\frac{d(w,w')}{d(z,w)^{n+1}}.
\end{multline*}
The last inequality (\ref{KZ3}) is a bit harder to prove.
Let $z,\xi,w\in\dom,\ Cd(z,\xi)<d(z,w),$ and estimate the value
$$A = \abs{\V{\psi_j(z,\tau)}{w} - \V{\psi_j(\xi,\tau)}{w} }. $$
Denote $\tau_z = \psi_j(z,\tau),\ \tau_\xi =\psi_j(\xi,\tau),$ then by (\ref{thm:rhostandart:cond1})
\begin{multline*}
\tau = \Phi(z)(\tau_z-z)+i (\tau_z-z)^{T}B(z)(\tau_z-z)e_n\\ = \Phi(\xi)(\tau_\xi-\xi)+i
(\tau_\xi-\xi)^{T}B(\xi)(\tau_\xi-\xi)e_n,
\end{multline*} whence denoting $\Psi(z) = \Phi(z)^{-1}$ and introducing $L(z,\xi,\tau)$ we obtain
\begin{align*}
\tau_z &= z+ \Psi(z)\tau - (\tau_z-z)^{T}B(z)(\tau_z-z)\Psi(z)e_n ,\\
\tau_\xi &= \xi+ \Psi(\xi)\tau - (\tau_\xi-\xi)^{T}B(\xi)(\tau_\xi-\xi)\Psi(\xi)e_n ,\\
\tau_z-\tau_\xi &= z-\xi + (\Psi(z)-\Psi(\xi))\tau + L(z,\xi,\tau)e_n .
\end{align*}
Note, that norms of matrices $\norm{\Psi(\xi)}$ are bounded, thus
\begin{multline*}
\abs{L(z,\xi,\tau)} \leq \abs{(\tau_z-z)^{T}B(z)(\tau_z-z)(\Psi(z)-\Psi(\xi))}\\ + \abs{(\tau_z-z)^{T}B(z)(\tau_z-z) -
(\tau_\xi-\xi)^{T} B(\xi)(\tau_\xi-\xi)} \norm{\Psi(\xi)}\\
\lesssim \abs{z-\xi}\abs{\tau_z-z}^2 + \abs{(\tau_z-z-\tau_\xi+\xi)^T
B(z)(\tau_z-z)}\\
+ \abs{(\tau_\xi-\xi)^T B(z)(\tau_z-z) - (\tau_\xi-\xi)^T B(\xi)(\tau_\xi-\xi)}\\
\lesssim \abs{z-\xi}\abs{\tau_z-z}^2 + \abs{z-\xi}\abs{\tau} + \abs{ ((\Psi(z)-\Psi(\xi))\tau+L(z,\xi,\tau)e_n)^T
B(z)(\tau_z-z)}\\ + \abs{(\tau_\xi-\xi)^T(B(z)-B(\xi))(\tau_z-z)} + \abs{(\tau_\xi-\xi)^T B(\xi)
(\tau_z-z-\tau_\xi-\xi)}\\
\lesssim \abs{z-\xi}\abs{\tau_z-z}^2 + \abs{z-\xi}\abs{\tau} +\abs{\tau}
\abs{L(z,\xi,\tau)}
+\abs{z-\xi}\abs{\tau}^2 + \abs{\tau} L(z,\xi,\tau).
\end{multline*}
Choosing $\eps>0$ small enough we get
$\abs{\tau}\leq\eta\abs{\IM(\tau_n)}+(1+\eta)\abs{\IM(\tau_n)}\leq3{\eps}$
and
$\abs{L(z,\xi,\tau)} \lesssim d(z,\xi)^{1/2}\abs{\tau},$ for $\tau\in D_0=D_0(\eta,\eps).$ Hence,
\begin{multline*}
A\leq
\abs{\scp{\partial\rho(\tau_z)-\partial\rho(\tau_\xi)}{\tau_z-w}} +
\abs{\scp{\partial\rho(\tau_\xi)}{\tau_z-w}}\\
\lesssim \abs{\tau_z-\tau_\xi}(\rho(\tau_z)+d(z,w)^{1/2}) +
\abs{\scp{\partial\rho(\tau_z)-\partial\rho(\tau_\xi)}{z-\xi}} +
\abs{\V{z}{\xi}}\\
+ \abs{\scp{\partial\rho(\tau_\xi)}{(\Psi(z)-\Psi(\xi))\tau}} + \abs{\scp{\partial\rho(\tau_\xi)}{L(z,\xi,\tau)}} \lesssim d(z,\xi)^{1/2}d(\tau_z,w) +\\
\abs{\tau_z-\xi}\abs{z-\xi}+d(z,\xi)+\abs{z-\xi}\abs{\tau}+\abs{L(z,\xi,\tau)} \lesssim d(z,\xi) + d(z,\xi)^{1/2}d(z,w)^{1/2}\\
\lesssim d(z,\xi)^{1/2}d(z,w)^{1/2}
\end{multline*}
Combining this estimate with inequality
$\abs{\V{\tau_z}{w}}\asymp\abs{\V{\tau_\xi}{w}}$ we obtain
\begin{multline*}
\norm{K_j(z,w) - K_j(\xi,w)}^2 \lesssim
\intl_{D^e(z,c\eta,c\eps)}
\frac{\abs{\chi_j(z)^{1/2}-\chi_j(\xi)^{1/2}}^2}{\abs{\V{\tau}{w}}^{2n+2l}} \frac{d\mu(\tau)}{\rho(\tau)^{n-2l+1}}\\
+ \chi_j(\xi) \intl_{D_0}
\frac{ \abs{\V{z}{\xi}}\abs{\V{z}{w}} }{ \abs{\V{\tau_z}{w}}^{2n+4} }\frac{d\mu(\tau)}{\RE(\tau_n)^{n-2l+1}}\\
\lesssim \frac{ \abs{\V{z}{\xi}} }{ \abs{\V{z}{w}}^{2n} } + \frac{
\abs{\V{z}{\xi}} }{ \abs{\V{z}{w}}^{2n+1} } \lesssim \frac{
\abs{\V{z}{\xi}} }{ \abs{\V{z}{w}}^{2n+1} }\\
\lesssim \frac{d(z,\xi)}{d(z,w)^{2n+1}}.
\end{multline*}\qed
\end{proof}
\begin{lemma} \label{lm:T1_2} $T_j(1) =0$ and $\norm{T_j'(1)} \lesssim 1.$
\end{lemma}
\begin{proof}
Introduce the notation $\tau_z = \psi_j(z,\tau).$ The function
$\V{\tau_z}{w}$ is holomorphic in $\O$ with respect to $w,$ then the
form $\V{\tau_z}{w}^{-n-l}dS(w)$ is closed in $\O$ and
$$T_j(1)(\tau) = \chi_j(z)^{1/2} J_j(z,\tau) \intl_\dom \frac{dS(w)}{\V{\tau_z}{w}^{n+l}} = 0. $$
It remains to estimate the value of formally-adjoint operator $T'$
on $f\equiv1$.
\begin{multline*}
T'_j(1)(w)(\tau)= \intl_\dom \frac{\chi_j(z)^{1/2}
J_j(z,\tau) dS(z)}{\V{\tau_z}{w}^{n+l}}\\ = \intl_\dom
\frac{\chi_j(z)^{1/2} J_j(z,\tau)
(dS(z)-dS(\tau_z))}{\V{\tau_z}{w}^{n+l}} + \intl_\dom
\frac{\chi_j(z)^{1/2} J_j(z,\tau) dS(\tau_z)}{\V{\tau_z}{w}^{n+l}} =
L_1+L_2.
\end{multline*}
Note that $ \abs{z-\tau_z}\lesssim \RE(\tau_n),$ therefore
$\abs{dS(z)-dS(\psi(z,\tau))} \lesssim \RE(\tau_n)d\sigma(z)$ and
\begin{multline*}
\abs{L_1} \lesssim \intl_\dom \frac{\RE(\tau_n)d\sigma(z)}{
\abs{\V{\tau_z}{w}}^{n+l} } \lesssim
\frac{\RE(\tau_n)d\sigma(z)}{(\RE(\tau_n)+ \abs{\V{z}{w}})^{n+l} }\\
\lesssim \intl_0^\infty \frac{\RE(\tau_n)
v^{n-1}dv}{(\RE(\tau_n)+v)^{n+l}}\lesssim\frac{1}{\RE(\tau_n)^{l-1}}.
\end{multline*}
Thus we get
\begin{equation} \label{T'1_I1}
\intl_{D_0} \abs{L_1}^2 d\nu_l(\tau) \lesssim \intl_{D_0}
\frac{1}{\RE(\tau_n)^{2l-2}}\frac{d\mu(\tau)}{\RE(\tau_n)^{n-2l+1}}
\lesssim\intl_0^{\eps} \frac{t^ndt}{t^{n-1}} \lesssim 1
\end{equation}
To estimate $L_2$ we recall that $d_\xi
\frac{dS(\xi)}{\V{\xi}{z}^{n}} =0,\ z\in\dom,\ \xi\in\CO,$ and
consequently
\begin{multline*}
d \frac{dS(\xi)}{\V{\xi}{z}^{n+l}} =
\frac{(\dbar\partial\rho(\xi))^n}{\V{\xi}{z}^{n+l}}\\
- (n+l)\frac{(\dbar_\xi\left(\V{\xi}{z}\right)\wedge\dbar\partial\rho(\xi))^{n-1}}{\V{\xi}{z}^{n+l}}
= -\frac{l}{n} \frac{dV(\xi)}{\V{\xi}{z}^{n+l}}.
\end{multline*}
By Stokes' theorem we obtain
\begin{multline*}
L_2= \intl_\dom \frac{\chi_j(z)^{1/2} J_j(z,\tau)
dS(\tau_z)}{\V{\tau_z}{w}^{n+l}}\\ = \intl_{\O_{\eps_1}\setminus\O}
\frac{\dbar_z\left(\chi_j(z)^{1/2} J_j(z,\tau)\right)\wedge
dS(\tau_z)}{\V{\tau_z}{w}^{n+l}} -\frac{l}{n}
\intl_{\O_{\eps_1}\setminus\O} \frac{\chi_j(z)^{1/2} J_j(z,\tau)
dV(\tau_z)}{\V{\tau_z}{z}^{n+l}}
\end{multline*}
Analogously to lemma \ref{lm:QM_est2} we have $\abs{\V{\tau_z}{w}}
\asymp \IM(\tau_n) + \rho(z) + \abs{\V{\hat{z}}{w}},$ where $\hat{z}=\pr_{\dom}(z).$ Hence,
\begin{multline*}
\abs{L_2}\lesssim \intl_{\O_{\eps_1}\setminus\O}
\frac{d\mu(z)}{ \abs{\V{\tau_z}{w}}^{n+l} }\\
\lesssim\intl_0^{\eps}
dt\intl_{\dom_t}\frac{d\sigma_t}{(t+\IM(\tau_n)+ \abs{\V{\hat{z}}{w}} )^{n+l}}\\
\lesssim \intl_0^{\eps} dt \intl_0^\infty
\frac{v^{n-1}dv}{(t+\RE(\tau_n)+v)^{n+l}}\lesssim \intl_0^{\eps}
\frac{dt}{(t+\RE(\tau_n))^l}\\ \lesssim (\RE(\tau_n))^{1-l}
\ln{\left(1+\frac{1}{\RE(\tau_n)}\right)},
\end{multline*}
and
\begin{multline*}
\intl_{D_0} \abs{L_2}^2d\nu_l(\tau) \lesssim \intl_{D_0} (\RE(\tau_n))^{2-2l}\ln^2\left(1+\frac{1}{\RE(\tau_n)}\right)d\nu_l(\tau)\\ \lesssim\intl_0^{\eps} \ln^2{\left(1+\frac{1}{s}\right)} s ds\lesssim 1,
\end{multline*}
which with the estimate (\ref{T'1_I1}) completes the proof of the
lemma. \qed
\end{proof}
\begin{lemma} \label{lm:T1_3} Operator $T_j$ is weakly bounded.
\end{lemma}
\begin{proof}
Let $f,g\in A(\frac{1}{2},w_0,r),$ denote again
$\tau_z=\psi_j(z,\tau),$ then
$$\norm{\scp{g}{Tf}}^2 \lesssim \intl_{D_0} d\nu_l(\tau) \left( \intl_{B(w_0,r)} \abs{g(z)} dS(z) \abs{\intl_{B(w_0,r)}\frac{f(w) dS(w)}{\V{\tau_z}{w}^{n+1}}
} \right)^2.$$
Denote $t:=\inf\limits_{w\in\dom} \abs{\V{\tau_z}{w}} $ and
introduce the set
$$W(z,\tau,r) :=\left\{ w\in\dom: \abs{\V{\tau_z}{w}}<t+r\right\}.$$
Note that $B(w_0,r)\subset W(z,\tau,cr) \subset B(z,c^2r)$ for some
$c>0,$ therefore,
\begin{multline*}
\abs{\intl_{B(w_0,r)}\frac{f(w) dS(w)}{\V{\tau_z}{w}^{n+l}} }\\
=\abs{\intl_{W(z,\tau,cr)}\frac{f(w)
dS(w)}{\V{\tau_z}{w}^{n+l}} } \lesssim
\intl_{W(z,\tau,cr)}\frac{\abs{f(z)-f(w)} dS(w)}{
\abs{\V{\tau_z}{w}}^{n+l} }\\ + \abs{f(z)} \abs{\intl_{\dom\setminus
W(z,\tau,cr)}\frac{dS(w)}{\V{\tau_z}{w}^{n+l}} } = L_1(z,\tau) +
\abs{f(z)} L_2(z,\tau).
\end{multline*}
It follows from the estimate $|f(z)-f(w)|\leq \sqrt{v(w,z)/r}$ that
\begin{multline*}
L_1(z,\tau)\lesssim \frac{1}{\sqrt{r}}\intl_{B(z,c^2r)}
\frac{v(w,z)^{1/2}}{(\RE(\tau_n) + v(w,z))^{n+l}}\lesssim
\frac{1}{\sqrt{r}} \intl_0^{c^2r} \frac{t^{n-1/2}dt}{(\RE(\tau_n) + t)^{n+l}}\\
\lesssim \frac{1}{\sqrt{r}} \intl_0^{c^2r}\frac{dt}{(\RE(\tau_n) +
t)^{l+1/2}} \lesssim
\frac{1}{\sqrt{r}}\left(\frac{1}{\RE(\tau_n)^{l-1/2}}-
\frac{1}{(\RE(\tau_n)+r)^{l-1/2}}\right)\\ = \frac{1}{\sqrt{r}}
\frac{(\RE(\tau_n)+r)^{l-1/2} - r^{l-1/2}}{\RE(\tau_n)^{l-1/2}
(\RE(\tau_n)+r)^{l-1/2} } \lesssim \frac{1}{\sqrt{r}}
\frac{(\RE(\tau_n)+r)^{2l-1} - r^{2l-1}}{
\IM(\tau_n)^{l-1/2}(\RE(\tau_n)+r)^{2l-1}}\\ \lesssim
\frac{1}{\sqrt{r}} \frac{r \RE(\tau_n)^{2l-2} +
r^{2l-1}}{\RE(\tau_n)^{l-1/2}(\RE(\tau_n)+r)^{2l-1}}.
\end{multline*}
Estimating the $L^2(D_0,d\nu_l)-$norm of the function $L_1(z,\tau),$
we obtain
\begin{multline} \label{est:I1}
\intl_{D_0(\tau)} L_1(z,\tau)^2 d\nu_l(\tau)\\ \lesssim
\intl_{D_0(\tau)} \left( \frac{r
\RE(\tau_n)^{2l-3}}{(\RE(\tau_n)+r)^{4l-2}} +
\frac{r^{4l-3}}{\RE(\tau_n)^{2l-1}(\RE(\tau_n)+r)^{4l-2}} \right)
\frac{d\mu(\tau)}{\RE(\tau_n)^{n-2l+1}}\\
\lesssim r \intl_0^\infty \frac{s^{4l-4}}{(s+r)^{4l-2}} ds +
r^{4l-3} \intl_0^\infty \frac{ds}{(s+r)^{4l-2}}\lesssim 1
\end{multline}
To estimate the second summand $L_2$ we apply the Stokes theorem to the
domain
$$\left\{ w\in\O:
\abs{\V{\tau_z}{w}} > t+c r\right\}$$ and to the closed in this
domain form $\frac{dS(w)}{\V{\tau_z}{w}^{n+l}}$
\begin{multline*}
\intl_{\dom\setminus W(z,\tau,cr)}\frac{dS(w)}{\V{\tau_z}{w}^{n+l}}
= - \intl_{\substack{w\in\O\\ \abs{v(\tau_z,w)} = t+cr}
}\frac{dS(w)}{\V{\tau_z}{w}^{n+l}}\\ = -\frac{1}{(t+cr)^{2n+2l}}
\intl_{\substack{w\in\O\\ \abs{v(\tau_z,w)} =
t+cr}}\overline{\V{\tau_z}{w}}^{n+l} dS(w).
\end{multline*}
Applying Stokes' theorem again, now to the domain $$\left\{ w\in\O:
\abs{\V{\tau_z}{w}}<t+cr\right\},$$ we obtain
\begin{multline*}
L_3:=\intl_{\substack{w\in\O\\
\abs{v(\tau_z,w)}=t+cr}}\overline{\V{\tau_z}{w}}^{n+l} dS(w)\\ =
-\intl_{\substack{w\in\dom\\ \abs{v(\tau_z,w)}<t+cr}}\overline{\V{\tau_z}{w}}^{n+l} dS(w)\\
+ \intl_{\substack{w\in\O\\
\abs{v(\tau_z,w)}<t+cr}}
\dbar_w\left(\overline{\V{\tau_z}{w}}^{n+l}\right)\wedge dS(w)\\ +
\intl_{\substack{w\in\O\\
\abs{v(\tau_z,w)}<t+cr}} \overline{\V{\tau_z}{w}}^{n+l} dV(w).
\end{multline*}
Since $ \abs{\
\dbar_w\left(\overline{\V{\tau_z}{w}}^{n+l}\right)\wedge dS(w)}
\lesssim \abs{\V{\tau_z}{w}}^{n+l-1}$ we get
$$
\abs{L_3} \lesssim \intl_t^{t+cr} (s^{n+l}s^{n-1} + s^{n+l}s^{n} +
s^{n+l-1}s^n) ds\lesssim \intl_t^{t+cr} s^{2n+l-1} ds \lesssim r
(t+r)^{2n+l-1}.
$$
Note that $t\asymp \rho(\tau_z)\asymp \IM(\tau_n)$ and consequently
\begin{multline} \label{est:I2}
\intl_{D_0} L_2(z,\tau)^2 d\nu_l(\tau) \lesssim
\intl_{D_0} \left(\frac{r
(\RE(\tau_n)+r)^{2n+l-1}}{(\RE(\tau_n)+r)^{2n+2l}}\right)^2
d\nu_l(\tau)\\ \lesssim \intl_0^\infty \frac{r^2}{(t+r)^{2l+2}}
\frac{t^n dt}{t^{n-2l+1}} = r^2\intl_0^\infty
\frac{t^{2l-1}}{(t+r)^{2l+2}} \frac{t^n dt}{t^{n-2l+1}} \lesssim r^2
\intl_0^\infty \frac{dt}{(r+t)^3} \lesssim 1.
\end{multline}
Summarizing estimates (\ref{est:I1}, \ref{est:I2}) and condition
$\abs{f(z)}\leq 1,\ z\in\dom,$ we obtain
\begin{multline*}
\norm{\scp{g}{Tf}}^2 \leq \intl_{D_0} d\nu_l(\tau) \left(
\intl_{B(w_0,r)} \abs{g(z)} (L_1(z,\tau) + L_2(z,\tau)|f(z)|) dS(z)\right)^2 \\
\lesssim \norm{g}_{L^1(\dom)}^2
\sup\limits_{z\in\dom}\intl_{D_0}\left(L_1(z,\tau)^2 + L_2(z,\tau)^2\right)
d\nu_l(\tau)\\ \lesssim \norm{g}_{L^1(\dom)}^2\lesssim
\abs{B(w_0,r)}^2.
\end{multline*}
The last estimate implies weak boundedness of operator $T$ and
completes the proof of the lemma. \qed
\end{proof}
\begin{proof}[of the theorem \ref{thm:area_int}] Since operators $T_j$ with kernels $K_j$
verify the conditions of $T1$-theorem, we have $T_j\in\mathscr{L}
(L^p(\dom), L^p(\dom,L^2(D_0,d\nu_l))$ and
\begin{multline*}
\suml_{j=1}^N \intl_\dom \norm{T_j g(z)}^p dS(z)\\ =
\suml_{j=1}^N \intl_\dom dS(z) \left(\ \intl_{D_0} \abs{\ \intl_\dom
\frac{ g(w)\chi_j^{1/2}(z) J_j(z,\tau)
dS(w)}{\V{\psi_j(z,\tau)}{w}^{n+1}} }^2
\frac{d\mu(\tau)}{\RE(\tau_n)^{n-1}}\right)^p\\ \lesssim
\norm{g}^p_{L^p(\dom)}.
\end{multline*}
Thus by decomposition (\ref{eq:Luzin_decomposition}) $\intl_\dom
I_l(g,z)^p\ d\sigma(z) \lesssim \intl_\dom \abs{g(z)}^p\ d\sigma(z),$
which proves the theorem. \qed
\end{proof}
\end{appendix}
\section*{References}
| 218,592
|
\begin{document}
\title[On $L_{p}$-estimates of singular integrals]{On $L_{p}$-estimates of
some singular integrals related to jump processes}
\author{R. Mikulevicius and H. Pragarauskas}
\address{University of Southern California, Los Angeles\\
Institute of Mathematics and Informatics, Vilnius University}
\date{March 14, 2012}
\subjclass{60H15}
\keywords{$L_{p}$-estimates of singular integrals, SPDEs, L\'{e}vy
processes, Zakai equation}
\begin{abstract}
We estimate fractional Sobolev and Besov norms of some singular integrals
arising in the model problem for the Zakai equation with discontinuous
signal and observation.
\end{abstract}
\maketitle
\section{ Introduction}
In a complete probability space $(\Omega ,\mathcal{F},\mathbf{P})$ with a
filtration of $\sigma $-algebras $\mathbb{F}=(\mathcal{F}_{t})$ satisfying
the usual conditions, the following linear stochastic integro-differential
parabolic equation of the fixed order $\alpha \in (0,2]$ was considered in H
\"{o}lder classes (see \cite{mikprag1}):
\begin{equation}
\left\{
\begin{array}{ll}
du(t,x)=\bigl(A^{({\scriptsize \alpha )}}u(t,x)+f(t,x)\bigr)
dt+\int_{U}g(t,x,v)q(dt,dv) & \quad \text{in }{E}_{0,T}, \\
u(0,x)=u_{0}(x) & \quad \text{in }\mathbf{R}^{d},
\end{array}
\right. \label{intr1}
\end{equation}
where ${E}_{0,T}=\left[ 0,T\right] \times \mathbf{R}^{d},\,f$ is an $\mathbb{
F}$-adapted measurable real-valued function on $\mathbf{R}^{d+1}$,
\begin{eqnarray*}
&&A^{({\scriptsize \alpha )}}u(t,x) \\
&=&\int_{\mathbf{R}_{0}^{d}}[u(t,x+y)-u(t,x)-(\nabla u(t,x),y)\chi ^{(
{\scriptsize \alpha )}}{(y)}]m^{({\scriptsize \alpha )}}(t,y)\frac{dy}{
|y|^{d+{\scriptsize \alpha }}} \\
&&\quad +{}\bigl(b(t),\nabla u(t,x)\bigr)1_{{\scriptsize \alpha =1}
}+\sum_{i,j=1}^{d}B^{ij}(t)\partial _{ij}^{2}u(t,x)1_{{\scriptsize \alpha =2}
},\quad (t,x)\in \mathbf{R}^{d+1},
\end{eqnarray*}
$\chi ^{({\scriptsize \alpha )}}{(y)}=1_{{\scriptsize \alpha >1}
}+1_{|y|\leqslant 1}1_{{\scriptsize \alpha =1}},m^{({\scriptsize \alpha )}
}(t,y)$ is a bounded measurable real-valued function homogeneous in $y$ of
order zero, $\mathbf{R}_{0}^{d}=\mathbf{R}^{d}\backslash \{0\},$ $
b(t)=(b^{1}(t),\ldots ,b^{d}(t))$ is a bounded measurable function and $
B(t)=(B^{ij}(t))$ is a bounded symmetric non-negative definite measurable
matrix-valued function;
\begin{equation*}
q(dt,d\upsilon )=p(dt,d\upsilon )-\Pi (d\upsilon )dt
\end{equation*}
is a martingale measure on a measurable space $([0,\infty )\times U,\mathcal{
B}([0,\infty ))\otimes \mathcal{U})$ ($p(dt,d\upsilon )$ is a Poisson point
measure on $([0,\infty )\times U,\mathcal{B}([0,\infty ))\otimes \mathcal{U}
) $ with the compensator $\Pi (d\upsilon )dt)$ and $g$ is an $\mathbb{F}$
-adapted measurable real-valued function on $\mathbf{R}^{d+1}\times U.$ It
is the model problem for the Zakai equation (see \cite{za}) arising in the
nonlinear filtering problem with discontinuous observation (see \cite
{mikprag1}). Let us consider the following example.
\begin{example}
\label{exa1}\textit{Assume that the signal process }$X_{t}$\textit{\ in }$
R^{d}$\textit{\ is defined by }
\begin{equation*}
X_{t}=X_{0}+\int_{0}^{t}b(X_{s})ds+W_{t}^{\alpha },t\in \lbrack 0,T],
\end{equation*}
\textit{where }$b(x)=(b^{i}(x))_{1\leq i\leq d}$\textit{,}$x\in R^{d},$
\textit{\ are measurable and bounded }$W_{t}^{\alpha }$\textit{\ is a }$d$
\textit{-dimensional }$\alpha $\textit{-stable (}$\alpha \in (1,2)$\textit{)
L\'{e}vy process. Suppose}
\begin{equation*}
W^{\alpha }{}_{t}=\int_{0}^{t}\int \upsilon \lbrack p(ds,d\upsilon )-m\left(
\frac{\upsilon }{|\upsilon |}\right) \frac{d\upsilon ds}{|\upsilon
|^{d+\alpha }}],
\end{equation*}
\textit{where }$m(\frac{\upsilon }{|\upsilon |})$\textit{\ is a smooth
bounded function (it characterizes the intensity of the jumps of }$W^{\alpha
}$\textit{\ in in the direction }$\frac{\upsilon }{|\upsilon |}$\textit{)
and }$p(ds,d\upsilon )$\textit{\ is a Poisson point measure on }$[0,\infty
)\times R_{0}^{d}$\textit{\ with }
\begin{equation*}
\mathbf{E}p(ds,d\upsilon )=m(\frac{\upsilon }{|\upsilon |})\frac{d\upsilon ds
}{|\upsilon |^{d+\alpha }}.
\end{equation*}
\textit{Assume }$X_{0}$\textit{\ has a density function }$u_{0}\left(
x\right) ,$\textit{\ and the observation }$Y_{t}$\textit{\ is discontinuous,
with jump intensity depending on the signal, such that}
\begin{equation*}
Y_{t}=\int_{0}^{t}\int_{|y|>1}y\hat{p}(ds\,dy)+\int_{0}^{t}\int_{|y|
\leqslant 1}y\hat{q}(ds,dy),
\end{equation*}
\textit{where }$\hat{p}(ds,dy)$\textit{\ is a point measure on }$[0,\infty
)\times R_{0}^{d}$\textit{\ not having common jumps with }$W^{\alpha }$
\textit{\ with a compensator }$\rho (X_{t},y)\pi (dy)$\textit{\ and }$\hat{q}
(dt,dy)=\hat{p}(dt,dy)-\pi (dy)dt$\textit{. Assume }$C_{1}\geqslant \rho
(x,y)\geqslant c_{1}>0,\pi (dy)$\textit{\ is a measure on }$R_{0}^{d}$
\textit{\ such that}
\begin{equation*}
\int |y|^{2}\wedge 1\pi (dy)<\infty ,
\end{equation*}
\textit{and }$\int [\rho (x,y)-1]^{2}\pi (dy)$\textit{\ is bounded. Then for
every function }$\varphi $\textit{\ such that }$E[\varphi
(X_{t})^{2}]<\infty ,$\textit{\ the optimal mean square estimate for }$
\varphi \left( X_{t}\right) ,\,t\in \left[ 0,T\right] $\textit{, given the
past of \ the observations }$F_{t}^{Y}=\sigma (Y_{s},s\leqslant t),$\textit{
\ is of the form }
\begin{equation*}
\hat{\varphi}_{t}=\mathbf{E}\bigl[\varphi (X_{t})|\mathcal{F}_{t}^{Y}\bigr]=
\frac{\mathbf{\widetilde{E}}\big[\varphi (X_{t})\zeta _{t}|\mathcal{F}
_{t}^{Y}\big]}{\mathbf{\widetilde{E}}\big[\zeta _{t}|\mathcal{F}_{t}^{Y}\big
]},
\end{equation*}
\textit{where }$\zeta _{t}$\textit{\ is the solution of the linear equation}
\begin{equation*}
d\zeta _{t}=\zeta _{t-}\int [\rho (X_{t-},y)-1]\hat{q}(dt,dy)
\end{equation*}
\textit{and }$d\widetilde{P}=\zeta \left( T\right) ^{-1}dP.$\textit{\ Under
assumptions of differentiability, one can easily show that if }$v(t,x)$
\textit{\ is an }$F=(F_{t+}^{Y})$\textit{-adapted unnormalized filtering
density function }
\begin{equation}
\mathbf{\widetilde{E}}\left[ \varphi \left( X_{t}\right) \zeta _{t}|\mathcal{
F}_{t}^{Y}\right] =\int v\left( t,x\right) \psi \left( x\right) \,dx,
\label{prf0}
\end{equation}
\textit{then it is a solution of the Zakai equation }
\begin{eqnarray}
&&dv(t,x) \label{prf1} \\
&=&v(t,x)\int [\rho (x,y)-1]\hat{q}(dt,dy)+\bigg\{-\partial _{i}\bigl(
b^{i}(x)v(t,x)\bigr) \notag \\
&&+\int_{\mathbf{R}_{0}^{d}}[v(t,x+y)-v(t,x)-(\nabla v(t,x),y)]m(\frac{-y}{
|y|})\frac{dy}{|y|^{d+\alpha }}\bigg\}, \notag \\
v(0,x) &=&u_{0}(x). \notag
\end{eqnarray}
\textit{Since }$Y_{t},t\geqslant 0,$\textit{\ and }$X_{t},t\geqslant 0,$
\textit{\ are independent with respect to }$\widetilde{P},$\textit{\ for }$
u\left( t,x\right) =v\left( t,x\right) -u_{0}\left( x\right) $\textit{\ we
have an equation whose model problem is of the type given by (\ref{intr1}).
Indeed, according to \cite{grig1}, for any infinitely differentiable
function }$\varphi $\textit{\ on }$\mathbf{R}^{d}$\textit{\ with compact
support, the conditional expectation }$\pi _{t}(\varphi )=\widetilde{E}\left[
\varphi \left( X_{t}\right) \zeta _{t}|\mathcal{F}_{t}^{Y}\right] $\textit{\
satisfies the equation}
\begin{eqnarray*}
d\pi _{t}(\varphi ) &=&\int \pi _{t}\big(\varphi \lbrack \rho (\cdot ,y)-1]
\big)\hat{q}(dt,dy)+\pi _{t}\bigg\{(b,\nabla \varphi ) \\
&&+\int_{\mathbf{R}_{0}^{d}}\big[\varphi (\cdot +y)-\varphi -(\nabla \varphi
,y)\chi ^{(\alpha )}(y)\big]m(t,\frac{y}{|y|})\frac{dy}{|y|^{d+\alpha }}
\bigg\}dt.
\end{eqnarray*}
\textit{Assuming (\ref{prf0}) and integrating by parts, we obtain (\ref{prf1}
).}
\end{example}
In terms of Fourier transform,
\begin{equation*}
A^{({\scriptsize \alpha )}}v(x)=\mathcal{F}^{-1}\left[ \psi ^{({\scriptsize
\alpha )}}(t,\xi )\mathcal{F}v(\xi )\right] (x),
\end{equation*}
with
\begin{eqnarray*}
{}\psi ^{({\scriptsize \alpha )}}(t,\xi ) &=&i(b(t),\xi )1_{{\scriptsize
\alpha =1}}-\sum_{i,j=1}^{d}B^{ij}(t)\xi _{i}\xi _{j}1_{{\scriptsize \alpha
=2}} \\
&&-C\int_{S^{d-1}}|(w,\xi )|^{{\scriptsize \alpha }}\Big[1-i\Big(\tan \frac{
\alpha \pi }{2}\mbox{sgn}(w,\xi )1_{{\scriptsize \alpha \neq 1}} \\
&&-\frac{2}{\pi }\mbox{sgn}(w,\xi )\ln |(w,\xi )|1_{{\scriptsize \alpha =1}}
\Big)\Big]m^{(\alpha )}(t,w)dw,
\end{eqnarray*}
where $C=C(\alpha ,d)$ is a positive constant, $S^{d-1}$ is the unit sphere
in $\mathbf{R}^{d}$ and $dw$ is the Lebesgue measure on it. It was shown in
\cite{mikprag1} that in H\"{o}lder classes the solution of (\ref{intr1}) can
be represented as
\begin{equation}
u(t,x)=Rf(t,x)+\widetilde{R}g(t,x)+T_{t}u_{0}(x), \label{maf1}
\end{equation}
where
\begin{eqnarray}
Rf(t,x) &=&\int_{0}^{t}G_{s,t}\ast f(s,x)ds, \notag \\
\widetilde{R}g(t,x) &=&\int_{0}^{t}\int_{U}G_{s,t}\ast g(s,x,\upsilon
)q(ds,d\upsilon ), \label{rez} \\
T_{t}u_{0}(x) &=&G_{0,t}\ast u_{0}(x), \notag
\end{eqnarray}
with
\begin{equation*}
G_{s,t}(x)=\mathcal{F}^{-1}\left( \exp \left\{ \int_{s}^{t}\psi ^{(
{\scriptsize \alpha )}}(r,\xi )dr\right\} \right) x,\quad s\leq t,x\in
\mathbf{R}^{d},
\end{equation*}
and $\ast $ denoting the convolution with respect to the space variable $
x\in \mathbf{R}^{d}.$ According to \cite{SaT94}, $G_{s,t}$ is the density
function of an $\alpha $- stable distribution, and $A^{(\alpha )}$ is the
fractional Laplacian if $b=0$ and $m^{(a)}=1$. \
In order to estimate the $L_{p}$-norm of the fractional derivative
\begin{equation*}
\partial ^{{\scriptsize \alpha }}u(t,x)=-\mathcal{F}^{-1}[|\xi |^{
{\scriptsize \alpha }}\mathcal{F}u(t,\xi )]
\end{equation*}
of $u$ in (\ref{maf1}), we need the estimates for $\partial ^{\alpha
}Rf,\partial ^{\alpha }\tilde{R}g$ and $\partial ^{\alpha }T_{t}u_{0}.$ It
was derived in \cite{MiP922}, that
\begin{equation*}
|\partial ^{\alpha }Rf|_{L_{p}}\leq C|f|_{L_{p}}.
\end{equation*}
According to Corollary \ref{cor5} below (it provides two-sided estimates for
the moments of a martingale),
\begin{equation*}
\mathbf{E}|\partial ^{\alpha }\widetilde{R}g|_{L_{p}}^{p}\leq C[\mathbf{E}
I_{1}+\mathbf{E}I_{2}],
\end{equation*}
where
\begin{eqnarray}
I_{1} &=&\int_{0}^{T}\int_{\mathbf{R}^{d}}\left\{ \int_{0}^{t}\int_{U}\left[
\partial ^{{\scriptsize \alpha }}G_{s,t}\ast g(s,x,\upsilon )\right] ^{2}\Pi
(d\upsilon )ds\right\} ^{p/2}dxdt, \label{nf1} \\
I_{2} &=&\int_{0}^{T}\int_{0}^{t}\int_{\mathbf{R}^{d}}\int_{U}|\partial ^{
{\scriptsize \alpha }}G_{s,t}\ast g(s,x,\upsilon )|^{p}\Pi (d\upsilon
)dxdsdt. \label{nf11}
\end{eqnarray}
In this paper, we estimate the singular integrals of $I_{1}$- and $I_{2}$
-types related to $\widetilde{R}g(t,x)$ in (\ref{rez}) in Sobolev and Besov
spaces. If $\alpha =2$ and $B$ is $d{\times }d$-identity matrix, the
estimate of $I_{1}$-type was proved in \cite{kry1}. This estimate for (\ref
{nf1}) was generalized in \cite{kim1} for the case $m^{({\scriptsize \alpha )
}}=1$, $b=0$ $($in this case $A^{(a)}$ is the fractional Laplacian). Our
derivation of an estimate for (\ref{nf1}) follows a slightly different idea
communicated by N.V. Krylov. The problem cannot be reduced to a case with
fractional Laplacian. In fact, $m^{(\alpha )}$ can be zero on a substantial
set (see Remark \ref{r1}). The operator $\tilde{R}g$ in H\"{o}lder-Zygmund
classes was estimated in \cite{mikprag1}. The results of this paper were
applied in \cite{mikprag2} to solve the model problem above in the
fractional Sobolev spaces.
The paper consists of five sections. In Section 2, we introduce the notation
and state the main results. In Section 3, we derive the two-sided $p$-moment
estimates of discontinuous martingales that explain the need to consider (
\ref{nf1}) and (\ref{nf11}). In the last two sections, we present the proofs
of the main results.
\section{Notation, function spaces and main results}
\subsection{Notation}
The following notation will be used in the paper.
Let $\mathbf{N}_{0}=\{0,1,2,\ldots \},\mathbf{R}_{0}^{d}=\mathbf{R}
^{d}\backslash \{0\}.$ If $x,y\in \mathbf{R}^{d},$\ we write
\begin{equation*}
(x,y)=\sum_{i=1}^{d}x_{i}y_{i},\,|x|=\sqrt{(x,x)}.
\end{equation*}
We denote by $C_{0}^{\infty }(\mathbf{R}^{d})$ the set of all infinitely
differentiable functions on $\mathbf{R}^{d}$ with compact support.
We denote the partial derivatives in $x$ of a function $u(t,x)$ on $\mathbf{R
}^{d+1}$ by $\partial _{i}u=\partial u/\partial x_{i}$, $\partial
_{ij}^{2}u=\partial ^{2}u/\partial x_{i}\partial x_{j}$, etc.;$\,\partial
u=\nabla u=(\partial _{1}u,\ldots ,\partial _{d}u)$ denotes the gradient of $
u$ with respect to $x$; for a multiindex $\gamma \in \mathbf{N}_{0}^{d}$ we
denote
\begin{equation*}
\partial _{x}^{{\scriptsize \gamma }}u(t,x)=\frac{\partial ^{|{\scriptsize
\gamma |}}u(t,x)}{\partial x_{1}^{{\scriptsize \gamma _{1}}}\ldots \partial
x_{d}^{{\scriptsize \gamma _{d}}}}.
\end{equation*}
For $\alpha \in (0,2]$ and a function $u(t,x)$ on $\mathbf{R}^{d+1}$, we
write
\begin{equation*}
\partial ^{{\scriptsize \alpha }}u(t,x)=-\mathcal{F}^{-1}[|\xi |^{
{\scriptsize \alpha }}\mathcal{F}u(t,\xi )](x),
\end{equation*}
where
\begin{equation*}
\mathcal{F}h(t,\xi )=\int_{\mathbf{R}^{d}}\,\mathrm{e}^{-i({\scriptsize \xi
,x)}}h(t,x)dx,\mathcal{F}^{-1}h(t,\xi )=\frac{1}{(2\pi )^{d}}\int_{\mathbf{R}
^{d}}\,\mathrm{e}^{i({\scriptsize \xi ,x)}}h(t,\xi )d\xi .
\end{equation*}
The letters $C=C(\cdot ,\ldots ,\cdot )$ and $c=c(\cdot ,\ldots ,\cdot )$
denote constants depending only on quantities appearing in parentheses. In a
given context the same letter will (generally) be used to denote different
constants depending on the same set of arguments.
\subsection{Function spaces\label{test}}
Let $\mathcal{S}(\mathbf{R}^{d})$ be the Schwartz space of smooth
real-valued rapidly decreasing functions. Let $V$ be a Banach space with a
norm $\vert \cdot\vert_V $. The space of $V$-valued tempered distributions
we denote by $\mathcal{S}^{\prime }(\mathbf{R}^{d},V)$ ($f\in \mathcal{S}
^{\prime }(\mathbf{R}^{d},V)$ is a continuous $V$-valued linear functional
on $\mathcal{S}(\mathbf{R}^{d})$).
For a $V$-valued measurable function $h$ on $\mathbf{R}^{d}$ and $
p\geqslant1 $ we denote
\begin{equation*}
|h|_{V,p}^{p}=\int_{\mathbf{R}^{d}}|h(x)|_{V}^{p}dx.
\end{equation*}
Further, for a characterization of our function spaces we will use the
following construction (see \cite{BeL76}). By Lemma 6.1.7 in \cite{BeL76},
there exists a function $\phi \in C_{0}^{\infty }(\mathbf{R}^{d})$ such that
$\mathrm{supp}\,\phi =\{\xi :\frac{1}{2}\leqslant |\xi |\leqslant 2\}$, $
\phi (\xi )>0$ if $2^{-1}<|\xi |<2$ and
\begin{equation*}
\sum_{j=-\infty }^{\infty }\phi (2^{-j}\xi )=1\quad \text{if }\xi \neq 0.
\end{equation*}
Define the functions $\varphi _{k}\in \mathcal{S}(\mathbf{R}^{d}),$ $
k=1,\ldots ,$ by
\begin{equation*}
\mathcal{F}\varphi _{k}(\xi )=\phi (2^{-k}\xi ),
\end{equation*}
and $\varphi _{0}\in \mathcal{S}(\mathbf{R}^{d})$ by
\begin{equation*}
\mathcal{F}\varphi _{0}(\xi )=1-\sum_{k\geqslant 1}\mathcal{F}\varphi
_{k}(\xi ).
\end{equation*}
Let $\beta\in \mathbf{R}$ and $p\geqslant1$. We introduce the Besov space $
B_{pp}^{{\scriptsize \beta }}=B_{pp}^{{\scriptsize \beta }}(\mathbf{R}
^{d},V) $ of generalized functions $f\in \mathcal{S}^{\prime }(\mathbf{R}
^{d},V)$ with finite norm
\begin{equation*}
|f|_{B_{pp}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)}=\Bigg\{
\sum_{j=0}^{\infty }2^{j{\scriptsize \beta} p}|\varphi _{j}\ast f|_{V,p}^{p}
\Bigg\}^{1/p},
\end{equation*}
the Sobolev space $H_{p}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)$ of $f\in
\mathcal{S}^{\prime }(\mathbf{R}^{d},V)$ with finite norm
\begin{eqnarray}
|f|_{H_{p}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)} &=&|\mathcal{F}
^{-1}((1+|\xi |^{2})^{{\scriptsize \beta /2}}\mathcal{F}f)|_{V,p}
\label{nf5} \\
&=&\left\vert (I-\Delta )^{{\scriptsize \beta /2}}f\right\vert _{V,p},
\notag
\end{eqnarray}
where $I$ is the identity map and $\Delta $ is the Laplacian in $\mathbf{R}
^{d}$, and the space $\tilde{H}_{p} ^{{\scriptsize \beta }}(\mathbf{R}
^{d},V) $ of $f\in \mathcal{S}^{\prime }(\mathbf{R}^{d},V)$ with finite norm
\begin{equation}
|f|_{\tilde{H}_{p}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)}=\left\{\int_{
\mathbf{R}^{d}}\left( \sum_{j=0}^{\infty }2^{2{\scriptsize \beta} j}|\varphi
_{j}\ast f(x)|_{V}^{2}\right) ^{p/2}dx \right\}^{1/p}. \label{nf0}
\end{equation}
Similarly we introduce the corresponding spaces of generalized functions on $
E_{a,b}=[a,b]\times \mathbf{R}^{d}$ and $\tilde{E}_{a,b}=\{ (s,t,x)\in
\mathbf{R}^{d+2}:a\leq s\leq t\leq b,x\in \mathbf{R}^{d}\} . $
The spaces $B_{pp}^{{\scriptsize \beta }}(E_{a,b},V)$, $H_{p}^{{\scriptsize
\beta }}(E_{a,b},V)$ and $\tilde{H}_{p}^{{\scriptsize \beta }}(E_{a,b},V)$
consist of all measurable $\mathcal{S}^{\prime}(\mathbf{R}^{d},V)$-valued
functions on $[a,b]$ with finite corresponding norms:
\begin{eqnarray}
|f|_{B_{pp}^{{\scriptsize \beta }}(E_{a,b},V)}= \Bigg\{\int_{a}^{b}|
f(t,\cdot )|_{B_{pp}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)}^{p}dt\Bigg\}
^{1/p} , \notag \\
|f|_{H_{p}^{{\scriptsize \beta }}(E_{a,b},V)}= \Bigg\{\int_{a}^{b}|
f(t,\cdot )|_{H_{p}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)}^{p}dt\Bigg\}
^{1/p} \label{norm11}
\end{eqnarray}
and
\begin{equation}
|f|_{\tilde{H}_{p}^{{\scriptsize \beta }}(E_{a,b},V)}= \Bigg\{\int_{a}^{b}|
f(t,\cdot )|_{\tilde H_{p}^{{\scriptsize \beta }}(\mathbf{R}^{d},V)}^{p}dt
\Bigg\}^{1/p}. \label{norm2}
\end{equation}
The spaces $B_{pp}^{{\scriptsize \beta }}(\tilde E_{a,b},V)$, $H_{p}^{
{\scriptsize \beta }}(\tilde E_{a,b},V)$ and $\tilde{H}_{p}^{{\scriptsize
\beta }}(\tilde E_{a,b},V)$ consist of all measurable $\mathcal{S}^{\prime}(
\mathbf{R}^{d},V)$-valued functions on $\{(s,t): a\leqslant s\leqslant
t\leqslant b\}$ with finite corresponding norms:
\begin{eqnarray}
|f|_{B_{pp}^{{\scriptsize \beta }}(\tilde E_{a,b},V)}= \Bigg\{
\int_{a}^{b}\int_{a}^{t}| f(s,t,\cdot )|_{B_{pp}^{{\scriptsize \beta }}(
\mathbf{R}^{d},V)}^{p}dsdt\Bigg\}^{1/p} , \notag \\
|f|_{H_{p}^{{\scriptsize \beta }}(\tilde E_{a,b},V)}= \Bigg\{
\int_{a}^{b}\int_{a}^{t}| f(s,t,\cdot )|_{H_{p}^{{\scriptsize \beta }}(
\mathbf{R}^{d},V)}^{p}dsdt\Bigg\}^{1/p} \label{norm11n}
\end{eqnarray}
and
\begin{equation}
|f|_{\tilde{H}_{p}^{{\scriptsize \beta }}(\tilde E_{a,b},V)}= \Bigg\{
\int_{a}^{b}\int_{a}^{t}| f(s,t,\cdot )|_{\tilde H_{p}^{{\scriptsize \beta }
}(\mathbf{R}^{d},V)}^{p}dsdt\Bigg\}^{1/p}. \label{norm2n}
\end{equation}
For the scalar functions the norms (\ref{nf5}) and (\ref{nf0}) are
equivalent (see \cite{Tri92}, p.~15). Therefore, the norms (\ref{norm11})
and (\ref{norm2}) as well as (\ref{norm11n}) and (\ref{norm2n}) are
equivalent.
If $V$ is a separable Hilbert space, we will also use the spaces $\bar
B_{pp}^{{\scriptsize \beta}}(\tilde E_{a,b},V)$ and $\bar H_{p}^{
{\scriptsize \beta}}(\tilde E_{a,b},V)$ consisting of measurable $S^{\prime
}(\mathbf{R}^d,V)$-valued functions on $\{(s,t):a\leq s\leq t\leq b\}$ with
finite norms
\begin{equation*}
\vert f\vert_{\bar B_{pp}^{{\scriptsize \beta}}(\tilde E_{a,b},V)} =
\biggl\{ \sum_{j=0}^{\infty}2^{j{\scriptsize \beta p}}\int_a^b\int_{\mathbf{R
}^d} \biggl( \int_a^t\big\vert \varphi_j\ast f(s,t,x)\big\vert^2_V ds \biggr)
^{p/2}dxdt \biggr\}^{1/p}
\end{equation*}
and
\begin{equation*}
\vert f\vert_{\bar H_{p}^{{\scriptsize \beta}}(\tilde E_{a,b},V)} = \biggl\{
\int_a^b\int_{\mathbf{R}^d} \biggl( \int_a^t\big\vert \mathcal{F}^{-1}\bigl(
(1+\vert \xi\vert^2 )^{{\scriptsize \beta/2}}\mathcal{F}f\bigr)(s,t,x)
\big\vert^2_V ds \biggr)^{p/2}dxdt \biggr\}^{1/p} .
\end{equation*}
\subsection{Main results}
Throughout the paper we assume that the functions $b=b(t),B=B(t)$ and $m^{(
{\scriptsize \alpha )}}(t,y)\geq 0$ are measurable, $m^{(2)}=0$ and
\begin{equation*}
\int_{S^{d-1}}wm^{(1)}(t,w)dw=0,t\in \mathbf{R}.
\end{equation*}
Also, we will need the following assumptions.
\ \textbf{A}. (i) The function $m=m(t,y)\geq 0$ is 0-homogeneous and
differentiable in $y$ up to $d_{0}=\left[ \frac{d}{2}\right] +1;$
(ii) There is a constant $K$ such that for each $\alpha \in (0,2)$ and $t\in
\mathbf{R}$
\begin{equation*}
|b(t)|+|B(t)|+\sup_{\substack{ |\gamma |\leq d_{0}, \\ |\xi |=1}}|\partial
_{y}^{{\scriptsize \gamma }}m^{({\scriptsize \alpha )}}(t,y)|\leq K.
\end{equation*}
\textbf{B.} There is a constant $\mu >0$ such that
\begin{equation*}
\sup_{t,|\xi |=1}\mathop{\mathrm{Re}}\psi ^{({\scriptsize \alpha )}}(t,\xi
)\leq -\mu .
\end{equation*}
\begin{remark}
\label{r1}The assumption \textbf{B} holds with certain $\mu >0$ if, for
example,
\begin{eqnarray*}
\inf_{t,|\xi |=1}(B(t)\xi ,\xi ) &>&0\text{, }\alpha =2, \\
\inf_{t,w\in \Gamma }m^{({\scriptsize \alpha )}}(t,w) &>&0,\alpha \in (0,2),
\end{eqnarray*}
for a measurable subset $\Gamma \subseteq S^{d-1}$ of a positive Lebesgue
measure.
\end{remark}
Given a measurable $\mathcal{S}^{\prime }(\mathbf{R}^{d},V)$-valued function
$g$ on $\mathbf{R}$, we consider a linear operator $\mathcal{I}$ that
assigns to it a $\mathcal{S}^{\prime }(\mathbf{R}^{d},V)$-valued function on
$\{(s,t):s\leq t\}:$
\begin{equation*}
\mathcal{I}g(s,t,x)=G_{s,t}\ast g(s,x),s\leq t,x\in \mathbf{R}^{d}.
\end{equation*}
The main results of the paper are the two propositions given below.
Proposition \ref{main1} in the case $V=L_p(U,\mathcal{U},\Pi)$ is related to
the integral $I_2$ in (\ref{nf11}) and Proposition \ref{main2} in the case $
V=L_2(U,\mathcal{U},\Pi)$ is related to the integral $I_1 $ in (\ref{nf1}).
\begin{proposition}
\label{main1}Let Assumptions $\mathbf{A}\mbox{ and }{B}$ hold, $p\geq
2,\beta \in \mathbf{R,}-\infty \leq a<b\leq \infty $. Then the operator $
\mathcal{I}:B_{pp}^{{\scriptsize \beta +\alpha -\frac{\alpha }{p}}
}(E_{a,b},V)\rightarrow \tilde{H}_{p}^{{\scriptsize \beta +\alpha }}(\tilde{E
}_{a,b},V)$ is bounded: there is a constant $C=C(\alpha ,K,\mu ,p,d)$ such
that
\begin{equation}
|\mathcal{I}g|_{\tilde{H}_{p}^{{\scriptsize \beta +\alpha }}(\tilde{E}
_{a,b},V)}\leq C|g|_{B_{pp}^{{\scriptsize \beta +\alpha -\frac{\alpha }{p}}
}(E_{a,b},V)},g\in B_{pp}^{{\scriptsize \beta +\alpha -\frac{\alpha }{p}}
}(E_{a,b},V). \label{nf2}
\end{equation}
\end{proposition}
Since for the scalar functions the norms (\ref{norm11n}) and (\ref{norm2n})
are equivalent, we have the following statement.
\begin{corollary}
\label{corn0}Let $V=L_{p}(U,\mathcal{U},\Pi )$. Then Proposition~\ref{main1}
holds with $\tilde{H}_{p}^{{\scriptsize \beta +\alpha }}(\tilde{E}_{a,b},V)$
replaced by $H_{p}^{{\scriptsize \beta +\alpha }}(\tilde{E}_{a,b},V)$.
\end{corollary}
\begin{proof}
Let $V=L_{p}(U,\mathcal{U},\Pi ).$ If $\mathcal{I}g\in H_{p}^{\beta +\alpha
}(\tilde{E}_{a,b},V)$, then $\Pi $ -a.e. $\mathcal{I}g(\cdot ,\cdot ,v)\in
H_{p}^{{\scriptsize \beta +\alpha }}(\tilde{E}_{a,b},\mathbf{R}).$ Since the
norms (\ref{norm11}) and (\ref{norm2}) are equivalent for the scalar
functions, we have
\begin{eqnarray*}
|\mathcal{I}g|_{H_{p}^{\beta +\alpha }(\tilde{E}_{a,b},V)}^{p}
&=&\int_{a}^{b}\int_{\mathbf{R}^{d}}\int_{U}|(I-\Delta )^{(\beta +\alpha )/2}
\mathcal{I}g(t,x,\upsilon )|^{p}\Pi (d\upsilon )dxdt \\
&\leq &C\int_{a}^{b}\int_{U}\int_{\mathbf{R}^{d}}\left( \sum_{j=0}^{\infty
}2^{2(\beta +\alpha )j}|\varphi _{j}\ast \mathcal{I}g(t,x,\upsilon
)|^{2}\right) ^{p/2}dx\Pi (d\upsilon )dt,
\end{eqnarray*}
and by Minkowski inequality
\begin{eqnarray*}
&&\int_{a}^{b}\int_{U}\int_{\mathbf{R}^{d}}\left( \sum_{j=0}^{\infty
}2^{2(\beta +\alpha )j}|\varphi _{j}\ast \mathcal{I}g(t,x,\upsilon
)|^{2}\right) ^{p/2}dx\Pi (d\upsilon )dt \\
&\leq &C\int_{a}^{b}\int_{\mathbf{R}^{d}}\left( \sum_{j=0}^{\infty
}2^{2(\beta +\alpha )j}|\varphi _{j}\ast \mathcal{I}g(t,x,\cdot
)|_{V}^{2}\right) ^{p/2}dxdt \\
&=&C|\mathcal{I}g|_{\tilde{H}_{p}^{\beta +\alpha }(\tilde{E}_{a,b},V)}^{p}
\end{eqnarray*}
and the statement follows by Proposition \ref{main1}.
\end{proof}
\begin{proposition}
\label{main2}Let Assumptions $\mathbf{A}$ (with $d_{0}$ replaced by $d_{0}+1$
) and $\mathbf{B}$ hold, $p\geq 2,\beta \in \mathbf{R},-\infty \leq a<b\leq
\infty $, and let $V$ be a separable Hilbert space.
Then there is a constant $C=C(\alpha ,K,\mu ,p,d)$ such that
\begin{equation*}
|\mathcal{\partial }^{{\scriptsize \alpha /2}}\mathcal{I}g|_{\bar H_{p}^{
{\scriptsize \beta }}(\tilde{E}_{a,b},V)}\leq C|g|_{H_{p}^{{\scriptsize
\beta }}(E_{a,b},V)},\quad g\in H^{{\scriptsize \beta}}_p(E_{a,b},V)
\end{equation*}
and
\begin{equation*}
|\partial ^{{\scriptsize \alpha /2}}\mathcal{I}g|_{\bar B_{pp}^{{\scriptsize
\beta }}(\tilde{E}_{a,b},V)}\leq C|g|_{B_{pp}^{{\scriptsize \beta }
}(E_{a,b},V)},\quad g\in B^{{\scriptsize \beta}}_{pp}(E_{a,b},V).
\end{equation*}
\end{proposition}
\section{Moment estimates of discontinuous martingales}
The following two-sided moment estimate for discontinuous martingales should
be well known (see e.g. \cite{PrT97} for this type of estimate from above).
For the sake of completeness we provide its proof. Let $p(dt,d\upsilon )$ be
a $\sigma $-finite point measure on $([0,\infty )\times U,\mathcal{B}
([0,\infty ))\otimes \mathcal{U})$ with a dual predictable projection
measure $\pi (dt,d\upsilon )$ such that $\pi \left( \{t\}\times U\right)
=0,t\geq 0$, and let $\mathcal{R}(\mathbb{F})$ be the progressive $\sigma $
-algebra on $[0,\infty )\times \Omega $ (see \cite{Jac79}). Denote by $
L_{loc}^{2}$ the space of all $\mathcal{R}(\mathbb{F})\otimes \mathcal{U}$
-measurable functions $g(t,\upsilon )=g(\omega ,t,\upsilon )$ such that $
\mathbf{P}$-a.s.
\begin{equation*}
\int_{0}^{t}\int_{U}g(s,\upsilon )^{2}\pi (ds,d\upsilon )<\infty
\end{equation*}
for all $t.$
\begin{lemma}
\label{hl0}Let $p\geq 2,g\in L_{loc}^{2}$ and
\begin{equation*}
Q_{t}=\int_{0}^{t}\int_{U}g(s,\upsilon )q(ds,d\upsilon ),t\geq 0.
\end{equation*}
Then there are constants $C=C(p)$ and $c=c(p)>0$ such that for any $\mathbb{F
}$-stopping time $\tau \leq T$
\begin{eqnarray}
&&c\mathbf{E}\bigg[\int_{0}^{{\scriptsize \tau }}\int_{U}|g(s,\upsilon
)|^{p}\pi (d\upsilon ,ds)+\bigg(\int_{0}^{{\scriptsize \tau }
}\int_{U}g(s,\upsilon )^{2}\pi (d\upsilon ,ds)\bigg)^{p/2}\bigg] \notag \\
&&\quad \leq \mathbf{E}\big[\sup_{t\leq {\scriptsize \tau }}|Q_{t}|^{p}\big]
\\
&&\quad \leq C\mathbf{E}\bigg[\int_{0}^{{\scriptsize \tau }
}\int_{U}|g(s,\upsilon )| ^{p}\pi(d\upsilon ,ds)+\bigg(\int_{0}^{
{\scriptsize \tau }}\int_{U}g(s,\upsilon )^{2}\pi (d\upsilon ,ds)\bigg)^{p/2}
\bigg] \notag \label{eight}
\end{eqnarray}
\end{lemma}
\begin{proof}
Let
\begin{equation*}
A_{t}=\int_{0}^{t}\int_{U}g(s,\upsilon )^{2}p(ds,d\upsilon ),\quad
L_{t}=\int_{0}^{t}\int_{U}g(s,\upsilon )^{2}\pi (d\upsilon ,ds),\quad t\geq
0.
\end{equation*}
By the Burkholder--Davis--Gundy inequality (see \cite{Jac79}), there are
positive constants $c_{p}$ and $C_{p}$ such that for each $\mathbb{F}$
-stopping time $\tau $
\begin{equation*}
c_{p}\mathbf{E}[A_{{\scriptsize \tau }}^{p/2}]\leq \mathbf{E}\big[
\sup_{t\leq {\scriptsize \tau }}|Q_{t}|^{p}\big]\leq C_{p}\mathbf{E}[A_{
{\scriptsize \tau }}^{p/2}].
\end{equation*}
Denoting $q=p/2\geq 1$, we have
\begin{equation*}
A_{{\scriptsize \tau }}^{q}=\sum_{s\leq {\scriptsize \tau }}\big[
(A_{s-}+\Delta A_{s})^{q}-A_{s-}^{q}\big]=\int_{0}^{{\scriptsize \tau }
}\int_{U}\big[(A_{s-}+g(s,\upsilon )^{2})^{q}-A_{s-} ^{q}\big]p(ds,d\upsilon
)
\end{equation*}
and
\begin{equation*}
\mathbf{E}[A_{{\scriptsize \tau }}^{q}]=\mathbf{E}\int_{0}^{{\scriptsize
\tau }}\int_{U}\big[(A_{s-}+g(s,\upsilon )^{2})^{q}-A_{s-}^{q}\big]\pi
(d\upsilon ,ds).
\end{equation*}
Since there are two positive constants $c,C$ such that for all non-negative
numbers $a,b$
\begin{equation*}
C\bigl(b^{q}+a^{q-1}b\bigr)\geq (a+b)^{q}-a^{q}\geq c\bigl(b^{q}+a^{q-1}b
\bigr),
\end{equation*}
we have
\begin{eqnarray}
&&C\mathbf{E}\int_{0}^{{\scriptsize \tau }}\int_{U}\big[|g(s,\upsilon
)|^{p}+A_{s-}^{q-1}g(s,\upsilon )^{2}\big]\pi (d\upsilon ,ds)\geq \mathbf{E}
[A_{{\scriptsize \tau }}^{q}] \label{hf0} \\
&&\qquad \geq c\mathbf{E}\int_{0}^{{\scriptsize \tau }}\int_{U}\big[
|g(s,\upsilon )|^{p}+A_{s-}^{q-1}g(s,\upsilon )^{2}\big]\pi (d\upsilon ,ds).
\notag
\end{eqnarray}
Hence,
\begin{eqnarray*}
&&c\mathbf{E}\int_{0}^{{\scriptsize \tau }}\int_{U}|g(s,\upsilon )|^{p}\pi
(d\upsilon ,ds) \leq \mathbf{E}[A_{{\scriptsize \tau }}^{q}] \\
&&\quad \leq C\mathbf{E}\bigg\{\int_{0}^{{\scriptsize \tau }
}\int_{U}|g(s,\upsilon )|^{p}\pi (d\upsilon ,ds) + A_{{\scriptsize \tau }
}^{q-1}L_{{\scriptsize \tau }}\bigg\}.
\end{eqnarray*}
On the other hand, for $q>1$,
\begin{equation*}
L_{{\scriptsize \tau }}^{q}=q\int_{0}^{{\scriptsize \tau }}L_{s}^{q-1}dL_{s}
\end{equation*}
and
\begin{equation*}
\mathbf{E}[L_{{\scriptsize \tau }}^{q}]=q\mathbf{E}\int_{0}^{{\scriptsize
\tau }}L_{s}^{q-1}dA_{s}\leq q\mathbf{E}[L_{{\scriptsize \tau }}^{q-1}A_{
{\scriptsize \tau }}].
\end{equation*}
According to Young's inequality, for each $\varepsilon >0$ there is a
constant $C_{{\scriptsize \varepsilon }}$ such that
\begin{eqnarray*}
A_{{\scriptsize \tau }}^{q-1}L_{{\scriptsize \tau }} &\leq &{}\varepsilon A_{
{\scriptsize \tau }}^{q}+C_{{\scriptsize \varepsilon }}L_{{\scriptsize \tau }
}^{q}, \\
L_{{\scriptsize \tau }}^{q-1}A_{{\scriptsize \tau }} &\leq &{}\varepsilon L_{
{\scriptsize \tau }}^{q}+C_{{\scriptsize \varepsilon }}A_{{\scriptsize \tau }
}^{q}.
\end{eqnarray*}
Therefore, there is a constant $C$ such that
\begin{eqnarray*}
\mathbf{E[}L_{{\scriptsize \tau }}^{q}] &\leq &C\mathbf{E[}A_{{\scriptsize
\tau }}^{q}], \\
\mathbf{E}[A_{{\scriptsize \tau }}^{q}] &\leq &C\mathbf{E}\bigg\{\int_{0}^{
{\scriptsize \tau }}\int_{U}|g(s,\upsilon )|^{p}\pi (d\upsilon ,ds)+L_{
{\scriptsize \tau }}^{q}\bigg\}, \\
\mathbf{E}[A_{{\scriptsize \tau }}^{q}] &\geq &\mathbf{E}\int_{0}^{
{\scriptsize \tau }}\int_{U} |g(s,\upsilon )|^{p}\pi (d\upsilon ,ds) ,
\end{eqnarray*}
and the statement follows.
\end{proof}
\begin{corollary}
\label{cor5}Let $p\geq 2,g=g(s,x,\upsilon )$ be such that $\mathbf{P}$-a.s.
\begin{equation*}
\int_{0}^{T}\int_{U}\int_{\mathbf{R}^{d}}g(s,x,\upsilon )^{2}\pi (d\upsilon
,ds)dx<\infty ,
\end{equation*}
and
\begin{equation*}
Q(t,x)=\int_{0}^{t}\int_{U}g(s,x,\upsilon )q(ds,d\upsilon ),0\leq t\leq T.
\end{equation*}
Then
\begin{eqnarray*}
\mathbf{E}\sup_{s\leq {\scriptsize \tau }}|Q(s,\cdot)|_{p}^{p} &\sim &
\mathbf{E}\Bigg\{\int_{0}^{{\scriptsize \tau }}\int_{U}|g(s,\cdot,\upsilon
)|_{p}^{p}\pi (d\upsilon ,ds)+ \\
&&+\bigg\vert \bigg[\int_{0}^{{\scriptsize \tau }}\int_U g(s,\cdot ,\upsilon
)^{2}\pi (d\upsilon ,ds)\bigg]^{1/2}\bigg\vert _{p}^{p}\Bigg\}
\end{eqnarray*}
and
\begin{eqnarray*}
\mathbf{E}\int_{0}^{T}|Q(s,\cdot)|_{p}^{p}ds &\sim& \mathbf{E}
\int_{0}^{T}\sup_{s\leq t}|Q(s,\cdot)|_{p}^{p}dt \\
&\sim & \mathbf{E}\Bigg\{\int_{0}^{T}\int_{0}^{t}\int_{U}|g(s,\cdot
,\upsilon )|_{p}^{p}\pi (d\upsilon ,ds)dt+ \\
&&\quad +\int_{0}^{T}\bigg\vert \bigg[\int_{0}^{t}\int_U g(s,\cdot ,\upsilon
)^{2}\pi (d\upsilon ,ds)\bigg]^{1/2}\bigg\vert _{p}^{p}dt\Bigg\} ,
\end{eqnarray*}
where$\vert f\vert^p_p=\int\vert f(x)\vert^pdx $ and $\sim $ denotes the
equivalence of norms.
\end{corollary}
\section{Proof of Proposition \protect\ref{main1}}
Let us introduce the functions
\begin{eqnarray*}
\widetilde{\varphi }_{0} &=&{}\varphi _{0}+\varphi _{1}, \\
\widetilde{\varphi }_{j} &=&{}\varphi _{j-1}+\varphi _{j}+\varphi
_{j+1},\quad j\geqslant 1,
\end{eqnarray*}
where $\varphi _{j},j\geq 0,$ are defined in Subsection \ref{test}. Let
\begin{equation*}
h_{s,t}^{j}(x)=\mathcal{F}^{-1} \biggl\{ \exp{\biggl\{ \int_s^t \psi^{(
{\scriptsize \alpha )}}(r,\xi)dr\biggr\} } \mathcal{F}\widetilde{\varphi }
_{j}(\xi)\biggr\}(x),\quad j\geqslant 0.
\end{equation*}
According to Lemma 12 in \cite{mikprag1} or inequality (36) and Lemma 16 in
\cite{MiP09}, there are constants $C$, $c>0$ such that for all $s\leq
t,j\geq 1,$
\begin{eqnarray}
\int \big\vert h_{s,t}^{j}(x)\big\vert dx &\leq &Ce^{-c2^{j{\scriptsize
\alpha }} (t-s)}\sum_{k\leq d_{0}}\big[2^{j{\scriptsize \alpha }}(t-s)\big]
^{k}, \label{ff1} \\
\int |h_{s,t}^{0}(x)|dx &\leq &C. \notag
\end{eqnarray}
For $g\in B_{pp}^{{\scriptsize \alpha -\frac{\alpha }{p}}}(E_{a,b},V),$ we
set
\begin{equation*}
g_{j}(t,\cdot )=g(t,\cdot )\ast \varphi _{j},\quad j\geqslant 0.
\end{equation*}
Obviously,
\begin{equation*}
{}{}\varphi _{j}\ast \mathcal{I}g(s,t,\cdot )= \mathcal{I}g_j(s,t,\cdot
),\quad j\geqslant 0.
\end{equation*}
Since $\varphi _{j}=\varphi _{j}\ast \widetilde{\varphi }_{j},j\geq 0,$ we
have
\begin{equation*}
\mathcal{I}g_{j}(s,t,x)=h_{s,t}^{j}\ast g_{j}(s,x).
\end{equation*}
Therefore, by Minkowski's inequality,
\begin{eqnarray*}
|\mathcal{I}g|_{\tilde{H}_{p}^{{\scriptsize \beta }}(\tilde{E}_{a,b},V)}^{p}
&=&\int_{a}^{b}\int_{a}^{t}\int \bigg( \sum_{j=0}^{\infty }2^{2{\scriptsize
\beta} j}\big\vert \varphi_j\ast \mathcal{I}g(s,t,x)\big\vert_V^{2}\bigg)
^{p/2}dxdsdt \\
&=&\int_{a}^{b}\int_{a}^{t}\int \bigg( \sum_{j=0}^{\infty }2^{2{\scriptsize
\beta} j}\big\vert h_{s,t}^{j}\ast g_{j}(s,x)\big\vert _V^{2}\bigg)
^{p/2}dxdsdt \\
&\leq &\int_{a}^{b}\int_{a}^{t}\Bigg( \sum_{j=0}^{\infty }2^{2{\scriptsize
\beta} j}\bigg\{\int \big\vert h_{s,t}^{j}\ast g_{j}(s,x)\big\vert _V^{p}dx
\bigg\}^{2/p}\Bigg) ^{p/2}dsdt.
\end{eqnarray*}
By (\ref{ff1}),
\begin{eqnarray*}
\bigg\{\int |h_{s,t}^{j}\ast g_{j}(s,x)|_{V}^{p}dx\bigg\}^{1/p} &\leq
&\int\vert h_{s,t}^{j}(x)\vert dx \vert g_{j}(s,\cdot)\vert _{V,p} \\
&\leq &Ce^{-c2^{{\scriptsize \alpha j}}(t-s)}|g_{j}(s,\cdot )|_{V,p},\quad
j\geq 0.
\end{eqnarray*}
So,
\begin{eqnarray}
|\mathcal{I}g|_{\tilde{H}_{p}^{{\scriptsize \beta }}(\tilde{E}_{a,b},V)}^{p}
&\leq &\int_{a}^{b}\int_{a}^{t}\Bigg( \sum_{j=0}^{\infty }2^{2{\scriptsize
\beta} j}\{\int |h_{s,t}^{j}\ast g_{j}(s,x)|_{V}^{p}dx\}^{2/p}\Bigg)
^{p/2}dsdt \notag \\
&\leq &C\int_{a}^{b}\int_{a}^{t}\Bigg( \sum_{j=0}^{\infty }e^{-c2^{
{\scriptsize \alpha j}}(t-s)}2^{2{\scriptsize \beta} j}|g_{j}(s,\cdot
)|_{V,p}^{2}\Bigg) ^{p/2}dsdt \label{ff2} \\
&=&C\int_{a}^{b}\int_{s}^{b}\Bigg( \sum_{j=0}^{\infty }e^{-c2^{{\scriptsize
\alpha j}}(t-s)}2^{2{\scriptsize \beta} j}|g_{j}(s,\cdot )|_{V,p}^{2}\Bigg)
^{p/2}dtds. \notag
\end{eqnarray}
If $p=2$, we have immediately
\begin{eqnarray*}
|\mathcal{I}g|_{H_{2}^{{\scriptsize \beta }}(\tilde{E}_{a,b},V)}^{2} &\leq
&C\int_{a}^{b}\int_{s}^{b}\sum_{j=0}^{\infty }e^{-c2^{{\scriptsize \alpha j}
}(t-s)}2^{2{\scriptsize \beta} j}|g_{j}(s,\cdot )|_{V,2}^{2}dtds \\
&\leq &C\int_{a}^{b}\sum_{j=0}^{\infty }2^{2{\scriptsize \beta} j}2^{-
{\scriptsize \alpha j}}|g_{j}(s,\cdot )|_{V,2}^{2}ds .
\end{eqnarray*}
If $p>2$, we split the sum in (\ref{ff2}) as follows:
\begin{eqnarray*}
\sum_{j=0}^{\infty }e^{-c2^{{\scriptsize \alpha j}}(t-s)}2^{2{\scriptsize
\beta} j}|g_{j}(s,\cdot)|_{V,p}^{2} &=&\sum_{j\in J}e^{-c2^{{\scriptsize
\alpha j}}(t-s)}2^{2{\scriptsize \beta} j}|g_{j}(s,\cdot)|_{V,p}^{2} \\
&&\quad +\sum_{j\in \mathbf{N}_0\setminus J}e^{-c2^{{\scriptsize \alpha j}
}(t-s)}2^{2{\scriptsize \beta} j}|g_{j}(s,\cdot)|_{V,p}^{2} = A(s,t)+B(s,t),
\end{eqnarray*}
where $J=\{j\in\mathbf{N}_0:2^{{\scriptsize \alpha }j}(t-s)\leq 1\}$.
Fix $\kappa \in (0,\frac{2\alpha }{p}).$ Using H\"{o}lder's inequality, we
get
\begin{eqnarray*}
A(s,t) &\leq &\sum_{j\in J}2^{2{\scriptsize \beta }j}2^{{\scriptsize \kappa j
}}2^{-{\scriptsize \kappa j}}|g_{j}(s,\cdot )|_{V,p}^{2} \\
&\leq &\bigg(\sum_{j\in J}2^{q{\scriptsize \kappa j}}\bigg)^{1/q}\bigg(
\sum_{j\in J}2^{p{\scriptsize \beta j}}2^{-p{\scriptsize \kappa j/2}
}|g_{j}(s,\cdot )|_{V,p}^{p}\bigg)^{2/p}
\end{eqnarray*}
with $q=\frac{p}{p-2}$. Since
\begin{equation*}
\sum_{j\in J}2^{q{\scriptsize \kappa j}}\leq C(t-s)^{-q{\scriptsize \kappa
/\alpha }},
\end{equation*}
we have
\begin{eqnarray*}
A(s,t) &\leq &C(t-s)^{-\frac{{\scriptsize \kappa }}{{\scriptsize \alpha }}}
\bigg(\sum_{j\in J}2^{p{\scriptsize \beta j}}2^{-p{\scriptsize \kappa j/2}
}|g_{j}(s,\cdot )|_{V,p}^{p}\bigg)^{2/p} \\
&=&C(t-s)^{-\frac{{\scriptsize \kappa }}{{\scriptsize \alpha }}}\bigg(
\sum_{j=0}^{\infty }1_{\left\{ (t-s)\leq 2^{-{\scriptsize \alpha j}}\right\}
}2^{p{\scriptsize \beta j}}2^{-p{\scriptsize \kappa j/2}}|g_{j}(s,\cdot
)|_{V,p}^{p}\bigg)^{2/p}.
\end{eqnarray*}
So,
\begin{eqnarray*}
\int_{a}^{b}\int_{s}^{b}A(s,t)^{p/2}dtds &\leq
&C\int_{a}^{b}\sum_{j=0}^{\infty }2^{p{\scriptsize \beta j}}2^{-p
{\scriptsize \kappa j/2}}|g_{j}(s,\cdot )|_{V,p}^{p}\int_{s}^{s+2^{-
{\scriptsize \alpha j}}}(t-s)^{-\frac{p{\scriptsize \kappa }}{2{\scriptsize
\alpha }}}dtds \\
&\leq &C\int_{a}^{b}\sum_{j=0}^{\infty }2^{-{\scriptsize \alpha j}}2^{p
{\scriptsize \beta j}}|g_{j}(s,\cdot )|_{V,p}^{p}ds=C|g|_{B_{pp}^{
{\scriptsize \beta -\frac{\alpha }{p}}}(E_{a,b},V)}^{p}.
\end{eqnarray*}
Let us consider the sum $B(s,t)$. By H\"{o}lder's inequality,
\begin{equation*}
B(s,t)\leq \bigg\{\sum_{j\in \mathbf{N}_{0}\setminus J}e^{-c2^{{\scriptsize
\alpha j}}(t-s)}\bigg\}^{\frac{1}{q}}\bigg\{\sum_{j\in \mathbf{N}
_{0}\setminus J}e^{-c2^{{\scriptsize \alpha j}}(t-s)}2^{{\scriptsize \beta pj
}}|g_{j}(s,\cdot )|_{V,p}^{p}\bigg\}^{\frac{2}{p}}
\end{equation*}
with $q=\frac{p}{p-2}$. Since $e^{-c2^{{\scriptsize \alpha j}}(t-s)}$ is
decreasing in $j$,
\begin{equation*}
\sum_{j\in \mathbf{N}_{0}\setminus J}e^{-c2^{{\scriptsize \alpha j}
}(t-s)}\leq \int_{\left\{ 2^{{\scriptsize \alpha r}}(t-s)\geq 1\right\}
}e^{-c2^{-{\scriptsize \alpha }}2^{{\scriptsize \alpha r}}(t-s)}dr\leq C.
\end{equation*}
Therefore,
\begin{eqnarray*}
\int_{a}^{b}\int_{s}^{b}B(s,t)^{p/2}dtds &\leq
&C\int_{a}^{b}\sum_{j=0}^{\infty }\int_{s}^{b}e^{-c2^{{\scriptsize \alpha j}
}(t-s)}dt2^{{\scriptsize \beta pj}}|g_{j}(s,\cdot )|_{V,p}^{p}ds \\
&\leq &C\int_{a}^{b}\sum_{j=0}^{\infty }2^{-{\scriptsize \alpha j}}2^{
{\scriptsize \beta pj}}|g_{j}(s,\cdot )|_{V,p}^{p}ds.
\end{eqnarray*}
Finally,
\begin{eqnarray*}
|\mathcal{I}g|_{\tilde{H}_{p}^{{\scriptsize \beta }}(\tilde{E}_{a,b},V)}^{p}
&\leq &C\bigg[\int_{a}^{b}\int_{s}^{b}A(s,t)^{p/2}dtds+\int_{a}^{b}
\int_{s}^{b}B(s,t)^{p/2}dtds\bigg] \\
&\leq &C\int_{a}^{b}\sum_{j=0}^{\infty }2^{-{\scriptsize \alpha j}}2^{
{\scriptsize \beta pj}}|g_{j}(s,\cdot )|_{V,p}^{p}ds\leq C|g|_{B_{pp}^{
{\scriptsize \beta -\frac{\alpha }{p}}}(E_{a,b},V)}^{p}.
\end{eqnarray*}
The proposition is proved.
\section{Proof of Proposition \protect\ref{main2}}
In the proof we follow an idea communicated by N.V. Krylov.
\subsection{Auxiliary results}
We start with
\begin{lemma}
\label{auxl2}Let $\delta \in (0,1),l\in (-d,\delta )$. Assume that a
function $F:\mathbf{R}_{0}^{d}\rightarrow \mathbf{R}$ satisfies the
inequalities
\begin{equation*}
|F(\xi )|\leq C|\xi |^{l},|\nabla F(\xi )|\leq C|\xi |^{l-1},\xi \in \mathbf{
R}_{0}^{d}.
\end{equation*}
Then
\begin{equation*}
|\partial ^{{\scriptsize \delta }}F(\xi )|\leq C|\xi |^{l-{\scriptsize
\delta }},\xi \in \mathbf{R}_{0}^{d}.
\end{equation*}
\end{lemma}
\begin{proof}
For any $\xi \in \mathbf{R}_{0}^{d},$
\begin{eqnarray*}
|\partial ^{{\scriptsize \delta }}F(\xi )| &=& C \bigg\vert \int [F(\xi
+y)-F(\xi )]\frac{dy}{|y|^{d+{\scriptsize \delta }}}\bigg\vert \\
&\leq & C \int_{|y|>\frac{1}{2}|\xi |}[|F(\xi +y)|+|F(\xi )|]\frac{dy}{
|y|^{d+{\scriptsize \delta }}} \\
&&+C\int_{|y|\leq \frac{1}{2}|\xi |}\int_{0}^{1}|\nabla F(\xi +sy)|\frac{
dsdy }{|y|^{d+{\scriptsize \delta -1}}},
\end{eqnarray*}
where the constant $C=C(\delta )$.
Changing the variable of integration, $y=|\xi |\bar{y},$ we have
\begin{eqnarray*}
\int_{|y|>\frac{1}{2}|\xi |}|F(\xi +y)|\frac{dy}{|y|^{d+{\scriptsize \delta }
}} &\leq &C\int |\xi +y|^{l}\frac{dy}{|y|^{d+{\scriptsize \delta }}} \\
&=&C|\xi |^{l-{\scriptsize \delta }}\int_{|\bar{y}|\geq \frac{1}{2}}~|\frac{
\xi }{|\xi |}+\bar{y}|^{l}\frac{d\bar{y}}{|\bar{y}|^{d+{\scriptsize \delta }}
} \\
&\leq &C|\xi |^{l-{\scriptsize \delta }}\sup_{|w|=1}\int_{|\bar{y}|\geq
\frac{1}{2}}~|w+\bar{y}|^{l}\frac{d\bar{y}}{|\bar{y}|^{d+{\scriptsize \delta
}}}.
\end{eqnarray*}
Obviously,
\begin{eqnarray*}
\int_{|y|\geq \frac{1}{2}|\xi |}|F(\xi )|\frac{dy}{|y|^{d+{\scriptsize
\delta }}} &\leq &C|\xi |^{l}\int_{|y|\geq \frac{1}{2}|{\scriptsize \xi |}}
\frac{dy}{|y|^{d+{\scriptsize \delta }}} \leq C|\xi |^{l-{\scriptsize \delta
}}.
\end{eqnarray*}
If $|y|\leq \frac{1}{2}|\xi |,s\in (0,1)$, then $|\xi +sy|\geq |\xi
|-s|y|\geq \frac{1}{2}|\xi |$ and
\begin{eqnarray*}
\int_{|y|\leq \frac{1}{2}|{\scriptsize \xi |}}\int_{0}^{1}|\nabla F(\xi +sy)|
\frac{dsdy}{|y|^{d+{\scriptsize \delta -1}}} &\leq &C\int_{|y|\leq \frac{1}{2
}|{\scriptsize \xi |}} \int_{0}^{1}|\xi +sy|^{l-1}\frac{dsdy}{|y|^{d+
{\scriptsize \delta -1}}} \\
&\leq &C\int_{|y|\leq \frac{1}{2}|{\scriptsize \xi |}}|\xi |^{l-1}\frac{dy}{
|y|^{d+{\scriptsize \delta -1}}}\leq C|\xi |^{l-{\scriptsize \delta }}.
\end{eqnarray*}
\end{proof}
We will need some facts about maximal and sharp functions as well (see \cite
{stein2}).
For each $(s,z)\in \mathbf{R}^{d+1}$ and $\delta >0$ we consider a family of
open sets $B(s,z;\delta )$ of the form
\begin{equation*}
B(s,z;\delta )=(s-\delta ^{\alpha },s+\delta ^{\alpha })\times (z_{1}-\delta
,z_{1}+\delta )\times \ldots \times (z_{d}-\delta ,z_{d}+\delta ).
\end{equation*}
Let $\mathbb{Q}_{\delta }$ be the family of all $B(s,z;\delta ),(s,z)\in
\mathbf{R}^{d+1},$ and $\mathbb{Q=\cup }_{\delta >0}\mathbb{Q}_{\delta }$.
The collection $\mathbb{Q}$ satisfies the basic assumptions in \cite{stein2}
(see I.2.3 in \cite{stein2}).
Let $h\in L_{1}(\mathbf{R}^{d+1})$. For the rectangle $B\in \mathbb{Q}$ we
set
\begin{eqnarray*}
h_{B} &=&\frac{1}{\text{mes}\,B}\int_{B}h(s,y)dsdy, \\
h_{B}^{\#} &=&\frac{1}{\text{mes}\,B}\int_{B}|h(s,y)-h_{B}|dsdy.
\end{eqnarray*}
Let
\begin{eqnarray*}
{M}h(t,x) &=&\sup_{\delta >0}\frac{1}{\text{mes}\,B(t,x;\delta )}
\int_{B(t,x;\delta )}|h(s,y)|dsdy, \\
h^{\#}(t,x) &=&\sup_{B\in \mathbb{Q},(t,x)\in B}h_{B}^{\#},(t,x)\in \mathbf{R
}^{d+1}.
\end{eqnarray*}
In the definition of $h^{\#}$ the supremum is taken over all $B\in \mathbb{Q}
=\cup _{\delta >0}\mathbb{Q}_{\delta }$ such that $(t,x)\in B$. The
functions ${M}h$ and $h^{\#}$ are called the maximal and sharp functions of $
h$.
By H\"{o}lder's inequality for $h\in L_{2}(\mathbf{R}^{d+1}),$
\begin{equation}
\left( h_{B}^{\#}\right) ^{2}\leq \frac{1}{\text{mes}B}
\int_{B}h^{2}(s,y)dsdy, \label{2.2}
\end{equation}
\begin{equation}
\left( h_{B}^{\#}\right) ^{2}\leq \frac{1}{(\text{mes}~B)^{2}}
\int_{B}\int_{B}(h(s,y)-h(u,z))^{2}dudzdsdy. \label{2.3}
\end{equation}
We will also use the maximal functions defined by
\begin{equation*}
\mathcal{M}f(x)=\sup_{r>0}\frac{1}{\mbox{mes}\,B_{r}(0)}
\int_{B_{r}(x)}|f(y)|dy,
\end{equation*}
where $B_{r}(x)=\{y\in \mathbf{R}^{d}:|y-x|<r\}$.
As it is well known (\cite{stein2}, Theorem IV.2.2, ), for $h\in L_{p}(
\mathbf{R}^{d+1})$, $p>1$, the following norms are equivalent:
\begin{equation}
|h|_{p}\sim |Mh|_{p}\sim |h^{\#}|_{p}. \label{eq19}
\end{equation}
Also, for $h\in L_{p}(\mathbf{R}^{d})$, $p>1$,
\begin{equation}
|h|_{p}\sim |\mathcal{M}h|_{p}. \label{eq20}
\end{equation}
\begin{lemma}
\label{r2} Let $f\in C_{0}^{\infty }(\mathbf{R}^{d})$ and $v$ be a
continuously differentiable function on $\mathbf{R}^{d}$ such that $
\lim_{|z|\rightarrow \infty }|v(z)|=0$. Let $R,R_{1}\geq 0,\ x,y\in \mathbf{R
}^{d}$, $|x-y|\leq R_{1}$ and $f(z)=0$ if $|y-z|\leq R$.
Then
\begin{equation*}
|(f\ast v)(y)|\leq C\bigl[\mathcal{M}f^{2}(x)\bigr]^{\frac{1}{2}
}\int_{R}^{\infty }(R_{1}+\rho )^{d}\Phi (\rho )d\rho ,
\end{equation*}
where the constant $C=C(d)$ and
\begin{equation*}
\Phi (\rho )=\biggl(\int_{|w|=1}\bigl(\nabla v(\rho w),w\bigr)^{2}dw\biggr)^{
\frac{1}{2}},
\end{equation*}
where $dw$ is the counting measure on $\left\{ -1,1\right\} $ if $d=1,$ and $
dw$ is the Lebesgue measure if $d\geq 2.$
\end{lemma}
\begin{proof}
Integrating by parts, we have
\begin{eqnarray*}
\int f(y-z)v(z)dz &=&\int_{R}^{\infty }\int_{|w|=1}f(y-\rho w)v(\rho w)\rho
^{d-1}dwd\rho \\
&=&\int_{R}^{\infty }\int_{|w|=1}v(\rho w)\frac{d}{d\rho }\int_{R}^{
{\scriptsize \rho }}f(y-rw)r^{d-1}drdwd\rho \\
&=&\int_{|w|=1}\bigg[v(\rho w)\int_{R}^{{\scriptsize \rho }}f(y-rw)r^{d-1}dr
\bigg]\bigg\vert_{R}^{\infty }dw \\
&&\quad -\int_{R}^{\infty }\int_{|w|=1}\int_{R}^{{\scriptsize \rho }
}f(y-rw)r^{d-1}dr\bigl(\nabla v(\rho w),w\bigr)dwd\rho \\
&=&-\int_{R}^{\infty }\int_{|w|=1}\int_{R}^{{\scriptsize \rho }
}f(y-rw)r^{d-1}dr\bigl(\nabla v(\rho w),w\bigr)dwd\rho .
\end{eqnarray*}
Therefore, by H\"{o}lder's inequality,
\begin{eqnarray*}
|(f\ast v)(y)| &\leq &\int_{R}^{\infty }\bigg(\int_{R}^{{\scriptsize \rho }
}\int_{|w|=1}f^{2}(y-rw)r^{d-1}dwdr\bigg)^{\frac{1}{2}}\biggl(\int_{R}^{
{\scriptsize \rho }}r^{d-1}dr\biggr)^{\frac{1}{2}}\Phi (\rho )d\rho \\
&\leq &C\int_{R}^{\infty }\biggl(\int_{B_{{\scriptsize \rho }}(y)}f^{2}(z)dz
\biggr)^{\frac{1}{2}}\rho ^{\frac{d}{2}}\Phi (\rho )d\rho \\
&\leq &C\int_{R}^{\infty }\biggl(\int_{B_{R_{1}+{\scriptsize \rho }
}(x)}f^{2}(z)dz\biggr)^{\frac{1}{2}}\rho ^{\frac{d}{2}}\Phi (\rho )d\rho \\
&\leq &C\int_{R}^{\infty }(R_{1}+\rho )^{\frac{d}{2}}\rho ^{\frac{d}{2}}
\biggl(\sup_{{\scriptsize \rho >0}}(R_{1}+\rho )^{-d}\int_{B_{R_{1}+
{\scriptsize \rho }}(x)}f^{2}(z)dz\biggr)^{\frac{1}{2}}\Phi (\rho )d\rho \\
&\leq &C\bigl[\mathcal{M}f^{2}(x)\bigr]^{\frac{1}{2}}\int_{R}^{\infty
}(R_{1}+\rho )^{d}\Phi (\rho )d\rho .
\end{eqnarray*}
\end{proof}
\subsection{Proof of Proposition \protect\ref{main2}}
1$^{0}.$ Since $(I-\Delta )^{{\scriptsize \beta /2}}:H_{p}^{s}\rightarrow
H_{p}^{s-{\scriptsize \beta /2}},s\in \mathbf{R},$ is an isomorphism (see
\cite{Ste71}), it is enough to prove the first inequality for $\beta =0$.
Also, it is enough to consider $g\in C_{0}^{\infty }(\mathbf{R}^{d+1},V),$
the space of smooth $V$ -valued functions on $\mathbf{R}^{d+1}$ with compact
support.
Let us introduce the function
\begin{equation*}
\tilde{\psi}^{({\scriptsize \alpha )}}(t,\xi )=\psi ^{({\scriptsize \alpha )}
}\biggl(t,\frac{\xi }{|\xi |}\biggr),\quad \xi \in \mathbf{R}_{0}^{d}=\{\xi
\in \mathbf{R}^{d}:\xi \neq 0\}.
\end{equation*}
Obviously, if $\alpha \neq 1,$
\begin{equation}
\psi ^{({\scriptsize \alpha )}}(t,\xi )=|\xi |^{{\scriptsize \alpha }}\tilde{
\psi}^{({\scriptsize \alpha )}}(t,\xi ). \label{du}
\end{equation}
Since
\begin{eqnarray*}
(w,\xi )\ln |(w,\xi )| &=&|\xi |(w,\frac{\xi }{|\xi |})\ln [|(w,\frac{\xi }{
|\xi |})|\xi |] \\
&=&|\xi |(w,\frac{\xi }{|\xi |})\ln |(w,\frac{\xi }{|\xi |})+|\xi |(w,\frac{
\xi }{|\xi |})\ln |\xi |
\end{eqnarray*}
and $\int_{|w|=1}wm^{(1)}(t,w)dw=0$, the equality (\ref{du}) holds for $
\alpha =1$ as well. By Assumption \textbf{B},
\begin{equation*}
\mbox{Re}\,\tilde{\psi}^{({\scriptsize \alpha )}}(t,\xi )\leq -\mu <0,\quad
t\in \mathbf{R},\ \xi \in \mathbf{R}_{0}^{d}.
\end{equation*}
Let $p=2$ and $g\in H^0_2(E_{a,b},V)$. Then, by Parseval's equality,
\begin{eqnarray}
\vert \partial^{{\scriptsize \alpha /2}}\mathcal{I}g\vert^2_{H^0_2(\tilde
E_{a,b},V)} &=& \int_a^b\int_a^t\int \vert \partial^{{\scriptsize \alpha /2}}
\mathcal{I}g(s,t,x)\vert^2 _V dxdsdt \notag \\
&=&\int_a^b\int_a^t\int \big\vert \vert\xi\vert^{{\scriptsize \alpha /2}}
\text{e}^{\vert {\scriptsize \xi\vert^{\alpha}\int_s^t\tilde{\psi}^{(\alpha
)}(r,\xi)dr}} \mathcal{F}g(s,\xi) \big\vert^2_V d\xi dsdt \notag \\
&\leq&\int_a^b\int_a^t\int \vert\xi\vert^{{\scriptsize \alpha}} \text{e}^{-2
{\scriptsize \mu\vert \xi\vert^{\alpha}(t-s)}} \vert \mathcal{F}g(s,\xi)
\vert^2_V d\xi dsdt \notag \\
&=&\int\int_a^b\int_s^b \vert\xi\vert^{{\scriptsize \alpha}} \text{e}^{-2
{\scriptsize \mu\vert \xi\vert^{\alpha}(t-s)}} \vert \mathcal{F}g(s,\xi)
\vert^2_V dtdsd\xi \notag \\
&\leq&(2\mu)^{-1} \int_a^b\int \vert \mathcal{F}g(s,\xi) \vert^2_V d\xi ds
\notag \\
&=& (2\mu)^{-1} \vert g\vert^2_{H^0_2(E_{a,b},V)} . \label{eq23}
\end{eqnarray}
2$^{0}$. Let $p>2$. We extend the functions $g\in H_{p}^{0}(E_{a,b},V)$ by
zero outside the interval $[a,b]$ if necessary. Obviously, the extended
functions belong to $H_{p}^{0}(E,V)$, where $E=E_{-\infty ,\infty }=\mathbf{R
}^{d+1}$.
For $g\in H_{p}^{0}(E,V)$ we denote
\begin{eqnarray*}
Gg(s,y) &=&\left\{ \int_{-\infty }^{s}\left\vert \int \partial ^{
{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime })g(u,y^{\prime })dy^{\prime
}\right\vert _{V}^{2}du\right\} ^{1/2} \\
&=&\left\{ \int_{-\infty }^{s}\left\vert \int G_{u,s}(y-y^{\prime })\partial
^{{\scriptsize \alpha /2}}g(u,y^{\prime })dy^{\prime }\right\vert
_{V}^{2}du\right\} ^{1/2}.
\end{eqnarray*}
Note that by triangle inequality in $L_{2}((-\infty ,s],V)$ we have for $
g_{1},g_{2},\in H_{p}^{0}(E,V),$
\begin{eqnarray}
G(g_{1}+g_{2})(s,y) &\leq &Gg_{1}(s,y)+Gg_{2}(s,y), \label{maf2} \\
|G(g_{1}+g_{2})(s,y)-Gg_{1}(s,y)| &\leq &Gg_{2}(s,y). \notag
\end{eqnarray}
According to (\ref{eq19}) and (\ref{eq20}) it is enough to prove that there
is a constant $C$ such that for all $g\in H_{p}^{0}(E,V),\ (t,x)\in \mathbf{R
}^{d+1}$
\begin{equation}
\left( Gg\right) ^{\#}(t,x)\leq C(\mathcal{M}_{t}\mathcal{M}
_{x}|g|_{V}^{2}(t,x))^{1/2}, \label{2.5}
\end{equation}
where $\mathcal{M}_{t}$ and $\mathcal{M}_{x}$ denote the maximal functions
defined using the balls in $\mathbf{R}$ and $\mathbf{R}^{d}$ and
\begin{equation*}
\left( Gg\right) ^{\#}(t,x)=\sup_{B\in \mathbb{Q},(t,x)\in B}\frac{1}{\text{
mes(}B)}\int_{B}|Gg(s,y)-(Gg)_{B}|dsdy.
\end{equation*}
Since $B\in \mathbb{Q}$ is of the form
\begin{eqnarray*}
B &=&(s_{0}-\delta ^{\alpha },s_{0}+\delta ^{\alpha })\times (z_{1}-\delta
,z_{1}+\delta )\times \ldots \times (z_{d}-\delta ,z_{d}+\delta )\} \\
&=&(\tilde{s}_{0},z)+\tilde{B}(0,0;\delta ),
\end{eqnarray*}
with $\tilde{s}_{0}=s_{0}+\delta ^{\alpha }$,$\tilde{B}(0,0;\delta
)=(-2\delta ^{\alpha },0)\times (-\delta ,\delta )^{d},$ it is
straightforward to verify that
\begin{eqnarray*}
&&\frac{1}{\text{mes(}B)}\int_{B}|Gg(s,y)-(Gg)_{B}|dsdy \\
&=&\frac{1}{\text{mes(}Q_{0})}\int_{Q_{0}}|Gg(\tilde{s}_{0}+\delta ^{\alpha
}s,z+\delta y)-(Gg(\tilde{s}_{0}+\delta ^{\alpha }\cdot ,z+\delta \cdot
))_{Q_{0}}|dsdy,
\end{eqnarray*}
where $Q_{0}=\tilde{B}(0,0;1)$.
Changing the variable of integration, $u=\tilde{s}_{0}+\delta ^{\alpha
}s,y^{\prime }=z+\delta y$, we see that
\begin{eqnarray*}
&&Gg(\tilde{s}_{0}+\delta ^{\alpha }t,z+\delta x) \\
&=&\left\{ \int_{-\infty }^{\tilde{s}_{0}+\delta ^{\alpha }t}\left\vert \int
\partial ^{{\scriptsize \alpha /2}}G_{u,\tilde{s}_{0}+\delta ^{\alpha
}t}(z+\delta x-y^{\prime })g(u,y^{\prime })dy^{\prime }\right\vert
_{V}^{2}du\right\} ^{1/2} \\
&=&\delta ^{\frac{\alpha }{2}+d}\left\{ \int_{-\infty }^{t}\left\vert \int
\partial ^{{\scriptsize \alpha /2}}G_{\tilde{s}_{0}+\delta ^{\alpha }s,
\tilde{s}_{0}+\delta ^{\alpha }t}(\delta (x-y))g(\tilde{s}_{0}+\delta
^{\alpha }s,z+\delta y)dy\right\vert _{V}^{2}ds\right\} ^{1/2} \\
&=&\left\{ \int_{-\infty }^{t}\left\vert \int \partial ^{{\scriptsize \alpha
/2}}G_{s,t}^{\tilde{s}_{0},\delta }(x-y)g(\tilde{s}_{0}+\delta ^{\alpha
}s,z+\delta y)dy\right\vert _{V}^{2}ds\right\} ^{1/2},
\end{eqnarray*}
where
\begin{equation*}
G_{s,t}^{s_{0},\delta }(x)=\mathcal{F}^{-1}\left( \exp \left\{
\int_{s}^{t}\psi ^{({\scriptsize \alpha )}}(s_{0}+\delta ^{\alpha }r,\xi
)dr\right\} \right) (x)
\end{equation*}
with
\begin{eqnarray*}
{}\psi ^{({\scriptsize \alpha )}}(s_{0}+\delta ^{\alpha }t,\xi )
&=&i(b(s_{0}+\delta ^{\alpha }t),\xi )1_{{\scriptsize \alpha =1}
}-\sum_{i,j=1}^{d}B^{ij}(s_{0}+\delta ^{\alpha }t)\xi _{i}\xi _{j}1_{
{\scriptsize \alpha =2}} \\
&&-C\int_{S^{d-1}}|(w,\xi )|^{{\scriptsize \alpha }}\Big[1-i\Big(\tan \frac{
\alpha \pi }{2}\mbox{sgn}(w,\xi )1_{{\scriptsize \alpha \neq 1}} \\
&&-\frac{2}{\pi }\mbox{sgn}(w,\xi )\ln |(w,\xi )|1_{{\scriptsize \alpha =1}}
\Big)\Big]m^{(\alpha )}(s_{0}+\delta ^{\alpha }t,w)dw.
\end{eqnarray*}
Note that for every $\tilde{s}_{0}\in \mathbf{R}^{d},\delta >0,$ the
coefficients $b(s_{0}+\delta ^{\alpha }t),B^{ij}(s_{0}+\delta ^{\alpha
}t),m^{(\alpha )}(s_{0}+\delta ^{\alpha }t,w),t\in \mathbf{R,}w\in S^{d-1},$
satisfy the assumptions \textbf{A,B} with the same constants $K$ and $\mu $.
Therefore for (\ref{2.5}) it is enough to show the inequality
\begin{equation}
\left( Gg\right) _{Q_{0}}^{\#}\leq C\left( \mathcal{M}_{t}\mathcal{M}
_{x}|g(t,x\right) |_{V}^{2})^{1/2},(t,x)\in Q_{0}, \label{2.6}
\end{equation}
with
\begin{equation*}
Q_{0}=\tilde{B}(0,0;1)=\left\{ (t,x)\in \lbrack -2,0]\times \lbrack
-1,1]^{d}\right\}
\end{equation*}
We consider the following three cases: \vskip6pt
(1) $g(t,x)=0,(t\,,x)\notin \lbrack -12,12]\times B_{3\sqrt{d}}(0);$
\vskip3pt (2) $g(t,x)=0,(t,x)\notin \lbrack -12,12]\times \mathbf{R}^{d};$
\vskip3pt (3) $g(t,x)=0,t\geq -8,x\in \mathbf{R}^{d}\,$.
For the estimates of the derivatives of $G_{u,s}(x)$ the following
representation is helpful. For $u<s,x\in \mathbf{R}^{d},j,k=1,\ldots ,d,$
\begin{eqnarray}
\partial _{j}\partial _{k}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(x)
&=&(s-u)^{-\frac{d}{{\scriptsize \alpha }}-\frac{1}{2}-\frac{2}{{\scriptsize
\alpha }}}F_{u,s}^{j,k}\left( (s-u)^{-\frac{1}{{\scriptsize \alpha }}
}x\right) , \label{maf3} \\
\partial _{j}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(x) &=&(s-u)^{-\frac{d
}{{\scriptsize \alpha }}-\frac{1}{2}-\frac{1}{{\scriptsize \alpha }}
}F_{u,s}^{j}\left( (s-u)^{-\frac{1}{{\scriptsize \alpha }}}x\right) , \notag
\\
\partial _{s}\partial ^{\frac{{\scriptsize \alpha }}{2}}G_{u,s}(x)
&=&(s-u)^{-\frac{d}{{\scriptsize \alpha }}-\frac{3}{2}}\bar{F}_{u,s}((s-u)^{-
\frac{1}{{\scriptsize \alpha }}}x), \notag \\
\partial _{j}\partial _{s}\partial ^{\frac{{\scriptsize \alpha }}{2}
}G_{u,s}(x) &=&(s-u)^{-\frac{d}{{\scriptsize \alpha }}-\frac{3}{2}-\frac{1}{
{\scriptsize \alpha }}}\bar{F}_{u,s}^{j}((s-u)^{-\frac{1}{{\scriptsize
\alpha }}}x), \notag
\end{eqnarray}
with
\begin{eqnarray*}
F_{u,s}^{j,k} &=&\mathcal{F}^{-1}\left\{ -\xi _{j}\xi _{k}|\xi |^{\frac{
{\scriptsize \alpha }}{2}}\exp \left\{ -|\xi |^{{\scriptsize \alpha }}\frac{1
}{(s-u)}\int_{u}^{s}\tilde{\psi}^{({\scriptsize \alpha )}}(r,\xi )dr\right\}
\right\} , \\
F_{u,s}^{j} &=&\mathcal{F}^{-1}\left\{ i\xi _{k}|\xi |^{\frac{{\scriptsize
\alpha }}{2}}\exp \left\{ -|\xi |^{{\scriptsize \alpha }}\frac{1}{(s-u)}
\int_{u}^{s}\tilde{\psi}^{({\scriptsize \alpha )}}(r,\xi )dr\right\}
\right\} , \\
\bar{F}_{u,s} &=&\mathcal{F}^{-1}\left\{ -|\xi |^{\frac{3}{2}{\scriptsize
\alpha }}\tilde{\psi}^{({\scriptsize \alpha )}}(s,\xi )\exp \left\{ -|\xi |^{
{\scriptsize \alpha }}\frac{1}{(s-u)}\int_{u}^{s}\tilde{\psi}^{({\scriptsize
\alpha )}}(r,\xi )dr\right\} \right\} , \\
\bar{F}_{u,s}^{j} &=&\mathcal{F}^{-1}\left\{ -i\xi _{j}|\xi |^{\frac{3}{2}
{\scriptsize \alpha }}\tilde{\psi}^{({\scriptsize \alpha )}}(s,\xi )\exp
\left\{ -|\xi |^{{\scriptsize \alpha }}\frac{1}{(s-u)}\int_{u}^{s}\tilde{\psi
}^{({\scriptsize \alpha )}}(r,\xi )dr\right\} \right\} .
\end{eqnarray*}
By definition of the inverse Fourier transform, all functions $
F_{u,s}^{j,k},F_{u,s}^{j},\bar{F}_{u,s},\bar{F}_{u,s}^{j}$ are uniformly
bounded.
\vskip6pt 3$^{0}$. First, we prove that in the case (1)
\begin{equation}
\int_{Q_{0}}(Gg)(s,y)^{2}dsdy\leq C\mathcal{M}_{t}\mathcal{M}
_{x}|g(t,x)|_{V}^{2} \label{2.7}
\end{equation}
for all $(t,x)\in Q_{0}.$
Repeating the proof of (\ref{eq23}), we have
\begin{eqnarray*}
\int_{Q_{0}}(Gg)^{2}(s,y)dsdy &\leq &\int_{-\infty }^{\infty }\int_{-\infty
}^{s}\int |\xi |^{{\scriptsize \alpha }}e^{-2{\scriptsize \mu |\xi |^{\alpha
}(s-u)}}|\mathcal{F}g(u,\xi )|_{V}^{2}d\xi duds \\
&\leq &(2\mu )^{-1}\int_{-\infty }^{\infty }\int |g(u,y)|_{V}^{2}dudy \\
&=&(2\mu )^{-1}\int_{-12}^{12}\int_{B_{2\sqrt{d}}(0)}|g(u,y)|_{V}^{2}dydu.
\end{eqnarray*}
Now for every $(t,x)\in Q_{0},$
\begin{eqnarray*}
&&\int_{-12}^{12}\int_{B_{3\sqrt{d}}(0)}|g(u,y)|_{V}^{2}dydu \\
&&\quad \leq \text{mes}\,(B_{5\sqrt{d}}(0))\int_{-12}^{12}\frac{1}{\text{mes}
\,(B_{5\sqrt{d}}(x))}\int_{B_{5\sqrt{d}}(x)}|g(u,y)|_{V}^{2}dydu \\
&&\quad \leq \text{mes}\,B_{5\sqrt{d}}(0)\int_{-12}^{12}\mathcal{M}
_{x}|g(u,x)|_{V}^{2}du \\
&&\quad \leq C\mathcal{M}_{t}\mathcal{M}_{x}|g(t,x)|_{V}^{2}
\end{eqnarray*}
and (\ref{2.7}) is proven.
4$^{0}$. Now we prove that (\ref{2.7}) holds in the case (2) as well. Since (
\ref{2.7}) holds for $g(t,z)=0,(t\,,z)\notin \lbrack -12,12]\times B_{3\sqrt{
d}}(0)$, it is enough to consider $g(t,z)$ such that $g(t,z)=0$ if $|t|>12$
or $|z|\leq 2\sqrt{d}$. By Minkowski's inequality,
\begin{eqnarray*}
(Gg)^{2}(s,y) &=&\int_{-\infty }^{s}\left\vert \int \partial ^{{\scriptsize
\alpha /2}}G_{u,s}(y-y^{\prime })g(u,y^{\prime })dy^{\prime }\right\vert
_{V}^{2}du \\
&\leq &\int_{-12}^{s}\left( \int |\partial ^{{\scriptsize \alpha /2}
}G_{u,s}(y-y^{\prime })|~|g(u,y^{\prime })|_{V}dy^{\prime }\right) ^{2}du.
\end{eqnarray*}
According to Lemma \ref{r2} (in our case $R=\sqrt{d},R_{1}=2\sqrt{d}),$
\begin{eqnarray*}
&&\left( \int |\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})|~|g(u,y^{\prime })|_{V}dy^{\prime }\right) ^{2}\leq C\mathcal{M}
_{x}|g(u,x)|_{V}^{2}\times \\
&&\qquad \times \left( \int_{\sqrt{d}}^{\infty }(2\sqrt{d}+\rho )^{d}\left(
\int_{|w|=1}\sum_{j=1}^{d}|\partial _{j}\partial ^{{\scriptsize \alpha /2}
}G_{u,s}(\rho w)|^{2}dw\right) ^{1/2}d\rho \right) ^{2} \\
&&\quad \leq C\mathcal{M}_{x}|g(u,x)|_{V}^{2}\kappa (u,s),
\end{eqnarray*}
where
\begin{equation*}
\kappa (u,s)=\left( \int_{1}^{\infty }\rho ^{d}\left(
\int_{|w|=1}\sum_{j=1}^{d}|\partial _{j}\partial ^{{\scriptsize \alpha /2}
}G_{u,s}(\rho w)|^{2}dw\right) ^{1/2}d\rho \right) ^{2}.
\end{equation*}
By (\ref{maf3}),
\begin{equation*}
\kappa (u,s)=(s-u)^{-\frac{2d}{{\scriptsize \alpha }}-1-\frac{2}{
{\scriptsize \alpha }}}\left( \int_{1}^{\infty }\rho ^{d}\left(
\int_{|w|=1}\sum_{j=1}^{d}|F_{u,s}^{j}(\rho (s-u)^{-\frac{1}{{\scriptsize
\alpha }}}w)|^{2}dw\right) ^{1/2}d\rho \right) ^{2}.
\end{equation*}
Changing the variable of integration, $\rho (s-u)^{-\frac{1}{{\scriptsize
\alpha }}}=r,$ and using H\"{o}lder's inequality, we get
\begin{eqnarray*}
{}\kappa (u,s) &=&(s-u)^{-1}\left( \int_{(s-u)^{-\frac{1}{{\scriptsize
\alpha }}}}^{\infty }r^{d}\left(
\int_{|w|=1}\sum_{j=1}^{d}[F_{u,s}^{j}(rw)]^{2}dw\right) ^{1/2}dr\right) ^{2}
\\
&\leq &(s-u)^{-1}\int_{(s-u)^{-\frac{1}{{\scriptsize \alpha }}}}^{\infty
}r^{-1-\frac{{\scriptsize \alpha }}{2}}dr\int_{0}^{\infty }r^{2d+1+\frac{
{\scriptsize \alpha }}{2}}\int_{|w|=1}\sum_{j=1}^{d}[F_{u,s}^{j}(rw)]^{2}dwdr
\\
&\leq &C(s-u)^{-\frac{1}{2}}\int \sum_{j=1}^{d}\Bigl[|x|^{\frac{d}{2}+1+
\frac{{\scriptsize \alpha }}{4}}F_{u,s}^{j}(x)\Bigr]^{2}dx.
\end{eqnarray*}
Hence, by Parseval's equality,
\begin{equation*}
{}\kappa (s,u)\leq C(s-u)^{-\frac{1}{2}}\int \sum_{j=1}^{d}\big\vert\partial
^{\frac{d}{2}+1+\frac{{\scriptsize \alpha }}{4}}\mathcal{F}F_{u,s}^{j}(\xi )
\big\vert^{2}d\xi .
\end{equation*}
Due to our assumptions \textbf{A}, \textbf{B} and Lemma \ref{auxl2}, the
last integral is finite. Therefore
\begin{eqnarray}
\int_{Q_{0}}(Gg)^{2}dsdy &\leq &C\int_{-2}^{0}\int_{-12}^{s}(s-u)^{-\frac{1}{
2}}\mathcal{M}_{x}|g(u,x)|_{V}^{2}duds \notag \\
&=&C\left(
\int_{-12}^{-2}\int_{-2}^{0}l(s,u,x)~dsdu+\int_{-2}^{0}
\int_{u}^{0}l(s,u,x)~dsdu\right) \notag \\
&\leq &C\int_{-12}^{0}\mathcal{M}_{x}|g(u,x)|_{V}^{2}du\leq C\mathcal{M}_{t}
\mathcal{M}_{x}|g(t,x)|_{V}^{2} \label{eq25}
\end{eqnarray}
for all $(t,x)\in Q_{0}$ with
\begin{equation*}
l(s,u,x)=(s-u)^{-\frac{1}{2}}\mathcal{M}_{x}|g(u,x)|_{V}^{2}.
\end{equation*}
$.$
5$^{0}.$ We will show that in the case (3)
\begin{equation}
\int_{Q_{0}}|Gg(s,y)-Gg(t^{\prime },x^{\prime })|^{2}dsdy\leq C\mathcal{M}
_{t}\mathcal{M}_{x}|g|_{V}^{2}(t,x) \label{2.10}
\end{equation}
with all $(t,x),(t^{\prime },x^{\prime })\in Q_{0}$. We estimate the
Lipschitz constant of $Gg$ in $t$ and $x$. Obviously, for each $(s,y),\
(t^{\prime },x^{\prime })\in Q_{0}$
\begin{equation}
\left\vert Gg(s,y)-Gg(t^{\prime },x^{\prime })\right\vert \leq C\left(
\sup_{(s,y)\in Q_{0}}|\nabla Gg(s,y)|+\sup_{(s,y)\in Q_{0}}|\partial
_{s}Gg(s,y)|\right) . \label{maf4}
\end{equation}
First we estimate $|\nabla Gg(s,y)|$ in (\ref{maf4}). Let $\varphi \in
C_{0}^{\infty }(\mathbf{R}^{d}),0\leq \varphi \leq 1,\varphi (x)=1$ if $
|x|\leq 2\sqrt{d}$, $\varphi (x)=0$ if $|x|<3\sqrt{d}$, and
\begin{eqnarray*}
g_{2}(u,y^{\prime }) &=&g(u,y^{\prime })\varphi (y^{\prime }), \\
g_{1}(u,y^{\prime }) &=&g(u,y^{\prime })\left( 1-\varphi (y^{\prime
})\right) ,(u,y^{\prime })\in \mathbf{R}^{d+1}\text{.}
\end{eqnarray*}
Since $g(u,y^{\prime })=0$ if $u\geq -8,$ applying H\"{o}lder's and
Minkowski's inequalities, we derive for $s\in \lbrack -2,0],|y|\leq 1,$
\begin{eqnarray*}
|\nabla Gg(s,y)|^{2} &\leq &\int_{-\infty }^{-8}\left\vert \int \nabla
\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime })g(u,y^{\prime
})dy^{\prime }\right\vert _{V}^{2}du \\
&\leq &2\int_{-\infty }^{-8}\left( \int_{|y^{\prime }|>2\sqrt{d}}|\nabla
\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})|~|g_{1}(u,y^{\prime })|_{V}dy^{\prime }\right) ^{2}du \\
&&+2\int_{-\infty }^{-8}\left( \int_{|y^{\prime }|\leq 3\sqrt{d}}|\nabla
\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})|~|g_{2}(u,y^{\prime })|_{V}dy^{\prime }\right) ^{2}du \\
&=&2(A_{1}(s,y)+A_{2}(s,y)).
\end{eqnarray*}
For any $(t,x)\in Q_{0},$ according to (\ref{maf3}) and Lemma \ref{r1}
(applied for $d=1),$
\begin{eqnarray*}
A_{2}(s,y) &\leq &\int_{-\infty }^{-8}\sup_{|z|\leq 4\sqrt{d}}|\nabla
\partial ^{{\scriptsize \alpha /2}}G_{u,s}(z)|^{2}(\int_{|y^{\prime }|\leq 3
\sqrt{d}}|g(u,y^{\prime })|_{V}dy^{\prime })^{2}du \\
&\leq &C\int_{-\infty }^{-8}\sup_{|z|\leq 4\sqrt{d}}|\nabla \partial ^{
{\scriptsize \alpha /2}}G_{u,s}(z)|^{2}(\frac{1}{\text{mes~}B_{4\sqrt{d}}(x)}
\int_{|x-y^{\prime }|\leq 4\sqrt{d}}|g(u,y^{\prime })|_{V}^{2}dy^{\prime })
\\
&\leq &C\int_{-\infty }^{-8}(s-u)^{-\frac{2d}{{\scriptsize \alpha }}-1-\frac{
2}{{\scriptsize \alpha }}}\mathcal{M}_{x}(|g|_{V}^{2})(u,x)du\leq C\mathcal{M
}_{t}\mathcal{M}_{x}(|g|_{V}^{2})(t,x)
\end{eqnarray*}
According to Lemma \ref{r2} (in our case $R=\sqrt{d}$ and $R_{1}=2\sqrt{d}$),
\begin{eqnarray*}
&&\left( \int_{|y^{\prime }|\geq 2\sqrt{d}}|\nabla \partial ^{{\scriptsize
\alpha /2}}G_{u,s}(y-y^{\prime })|~|g_{1}(u,y^{\prime })|_{V}dy^{\prime
}\right) ^{2}\leq C\mathcal{M}_{x}|g(u,x)|_{V}^{2}\times \\
&&\qquad \times \bigg(\int_{\sqrt{d}}^{\infty }(2\sqrt{d}+\rho )^{d}\bigg(
\int_{|w|=1}\sum_{j=1}^{d}|\partial _{j}\nabla \partial ^{{\scriptsize
\alpha /2}}G_{u,s}(\rho w)|^{2}dw\bigg)^{1/2}d\rho \bigg )^{2} \\
&&\quad \leq C\mathcal{M}_{x}|g(u,x)|_{V}^{2}\tilde{\kappa}(u,s),
\end{eqnarray*}
where
\begin{equation*}
\tilde{\kappa}(u,s)=\bigg (\int_{1}^{\infty }\rho ^{d}\bigg (
\int_{|w|=1}\sum_{j=1}^{d}|\partial _{j}\nabla \partial ^{{\scriptsize
\alpha /2}}G_{u,s}(\rho w)|^{2}dw\bigg )^{1/2}d\rho \bigg )^{2}.
\end{equation*}
By (\ref{maf3}) and H\"{o}lder's inequality
\begin{eqnarray*}
\tilde{\kappa}(u,s) &=&(s-u)^{-p}\bigg(\int_{1}^{\infty }\rho ^{d}\bigg(
\int_{|w|=1}\sum_{j,k=1}^{d}\Big[F_{u,s}^{j,k}(\rho (s-u)^{-\frac{1}{
{\scriptsize \alpha }}}w)\Big]^{2}dw\bigg)^{1/2}d\rho \bigg)^{2} \\
&\leq &(s-u)^{-p}\int_{1}^{\infty }\rho ^{-2}d\rho \int_{1}^{\infty }\rho
^{2d+2}\int_{|w|=1}\sum_{j,k=1}^{d}\Bigl[F_{u,s}^{j,k}\big(\rho (s-u)^{-
\frac{1}{{\scriptsize \alpha }}}w\big)\Bigr]^{2}dwd\rho \\
&=&(s-u)^{-p}\int_{|x|\geq 1}|x|^{d+3}\sum_{j,k=1}^{d}\Bigl[F_{u,s}^{j,k}
\big((s-u)^{-\frac{1}{{\scriptsize \alpha }}}x\big)\Bigr]^{2}dx
\end{eqnarray*}
with $p=\frac{2d+4}{{\scriptsize \alpha }}+1$.
Changing the variable of integration, $y=(s-u)^{-\frac{1}{{\scriptsize
\alpha }}}x$, we get by Parseval's equality
\begin{eqnarray*}
\tilde{\kappa}(u,s) &\leq &(s-u)^{-1-\frac{1}{{\scriptsize \alpha }}}\int
|y|^{d+3}\sum_{j,k=1}^{d}\big[F_{u,s}^{j,k}(y)\big]^{2}dy \\
&=&(s-u)^{-1-\frac{1}{{\scriptsize \alpha }}}\int \sum_{j,k=1}^{d}\Big\vert
\partial ^{\frac{d+3}{2}}\mathcal{F}F_{u,s}^{j,k}(\xi )\Big\vert^{2}d\xi .
\end{eqnarray*}
Due to our assumptions and Lemma~\ref{auxl2}, the last integral is finite.
Hence,
\begin{equation*}
\tilde{\kappa}(u,s)\leq C(s-u)^{-1-\frac{1}{{\scriptsize \alpha }}}
\end{equation*}
and for $(s,y)\in Q_{0},$
\begin{equation*}
A_{1}(s,y)\leq \int_{-\infty }^{-8}\mathcal{M}_{x}|g|_{V}^{2}(u,x)\tilde{
\kappa}(u,s)du\leq C\int_{-\infty }^{-8}\mathcal{M}
_{x}|g|_{V}^{2}(u,x)(s-u)^{-1-\frac{1}{{\scriptsize \alpha }}}du.
\end{equation*}
Therefore by Lemma \ref{r1} (in the case $d=1),$ for $(s,y)\in
Q_{0},(t,x)\in Q_{0},$
\begin{equation}
|\nabla Gg(s,y)|^{2}\leq A_{1}(s,y)+A_{2}(s,y)\leq C\mathcal{M}_{t}\mathcal{M
}_{x}|g|_{V}^{2}(t,x). \label{28}
\end{equation}
Now we estimate $|\partial _{s}Gg(s,y)|$. Applying H\"{o}lder's and
Minkowski's inequalities, we get for $(s,y)\in Q_{0},$
\begin{eqnarray*}
\lbrack \partial _{s}G(s,y)]^{2} &\leq &\int_{-\infty }^{-8}\left\vert \int
\partial _{s}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})g(u,y^{\prime })dy^{\prime }\right\vert _{V}^{2}du \\
&\leq &2\int_{-\infty }^{-8}\bigg(\int_{|y^{\prime }|>2\sqrt{d}}|\partial
_{s}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})|\,|g_{1}(u,y^{\prime })|_{V}dy^{\prime }\bigg)^{2}du \\
&&+\int_{-\infty }^{-8}\bigg(\int_{|y^{\prime }|\leq 3\sqrt{d}}|\partial
_{s}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(y-y^{\prime
})|\,|g_{2}(u,y^{\prime })|_{V}dy^{\prime }\bigg)^{2}du \\
&=&2B_{1}(s,y)+2B_{2}(s,y).
\end{eqnarray*}
According to Lemma~\ref{r2},
\begin{eqnarray*}
&&\bigg(\int_{|y^{\prime }|>2\sqrt{d}}|\partial _{s}\partial ^{{\scriptsize
\alpha /2}}G_{u,s}(y-y^{\prime })|\,|g_{1}(u,y^{\prime })|_{V}dy^{\prime }
\bigg)^{2}\leq C\mathcal{M}_{x}|g|_{V}^{2}(u,x)\times \\
&&\qquad \times \biggl(\int_{\sqrt{d}}^{\infty }(2\sqrt{d}+\rho )^{d}\biggl(
\int_{|w|=1}\sum_{j=1}^{d}\bigl[\partial _{j}\partial _{s}\partial ^{
{\scriptsize \alpha /2}}G_{u,s}(\rho w)\bigr]^{2}dw\biggr)^{\frac{1}{2}
}d\rho \biggr)^{2} \\
&&\quad \leq C\bar{\kappa}(u,s)\mathcal{M}_{x}|g|_{V}^{2}(u,x),
\end{eqnarray*}
where
\begin{equation*}
\bar{\kappa}(u,s)=\bigg(\int_{1}^{\infty }\rho ^{d}\bigg(\int_{|w|=1}
\sum_{j=1}^{d}\big[\partial _{j}\partial _{s}\partial ^{{\scriptsize \alpha
/2}}G_{u,s}(\rho w)\big]^{2}dw\bigg)^{\frac{1}{2}}d\rho \bigg)^{2}.
\end{equation*}
According to (\ref{maf3}), we have by H\"{o}lder's inequality
\begin{eqnarray*}
\bar{\kappa}(u,s) &=&(s-u)^{-p}\bigg(\int_{1}^{\infty }\rho ^{d}\bigg(
\int_{|w|=1}\sum_{j=1}^{d}\big[\bar{F}_{u,s}^{j}(\rho (s-u)^{-\frac{1}{
{\scriptsize \alpha }}}w)\big]^{2}dw\bigg)^{\frac{1}{2}}d\rho \bigg)^{2} \\
&\leq &(s-u)^{-p}\int_{1}^{\infty }\rho ^{-1-{\scriptsize \alpha }}d\rho
\int_{1}^{\infty }\rho ^{2d+1+{\scriptsize \alpha }}\int_{|w|=1}
\sum_{j=1}^{d}\big[\bar{F}_{u,s}^{j}(\rho (s-u)^{-\frac{1}{{\scriptsize
\alpha }}}w)\big]^{2}dwd\rho \\
&\leq &C(s-u)^{-p}\int_{|x|\geq 1}|x|^{d+2+{\scriptsize \alpha }
}\sum_{j=1}^{d}\big[\bar{F}_{u,s}^{j}((s-u)^{-\frac{1}{{\scriptsize \alpha }}
}x)\big]^{2}dx,
\end{eqnarray*}
where $p=\frac{2d}{{\scriptsize \alpha }}+3+\frac{2}{{\scriptsize \alpha }}$
. Changing the variable of integration, $y=(s-u)^{-\frac{1}{{\scriptsize
\alpha }}}x$, we get by Parseval's equality
\begin{eqnarray*}
\bar{\kappa}(u,s) &\leq &C(s-u)^{-2}\int \sum_{j=1}^{d}\big[|y|^{\frac{d}{2}
+1+\frac{{\scriptsize \alpha }}{2}}\bar{F}_{u,s}^{j}(y)\big]^{2}dy \\
&\leq &C(s-u)^{-2}\int \sum_{j=1}^{d}\big\vert\partial ^{\frac{d}{2}+1+\frac{
{\scriptsize \alpha }}{2}}\mathcal{F}\bar{F}_{u,s}^{j}(\xi )\big\vert
^{2}d\xi .
\end{eqnarray*}
Due to our assumptions and Lemma~\ref{auxl2}, the last integral is finite.
Hence,
\begin{equation*}
\bar{\kappa}(u,s)\leq C(s-u)^{-2}
\end{equation*}
and, by Lemma \ref{r1} $(d=1)$ it follows for $(s,y),(t,x)\in Q_{0}$,
\begin{equation}
B_{1}(s,y)\leq C\int_{-\infty }^{-8}(s-u)^{-2}\mathcal{M}
_{x}|g|_{V}^{2}(u,x)du\leq C\mathcal{M}_{t}\mathcal{M}_{x}|g|_{V}^{2}(t,x).
\label{eq29}
\end{equation}
For any $(s,y),(t,x)\in Q_{0},$ according to (\ref{maf3}) and Lemma \ref{r1}
($d=1),$
\begin{eqnarray*}
B_{2}(s,y) &\leq &\int_{-\infty }^{-8}\sup_{|z|\leq 4\sqrt{d}}|\partial
_{s}\partial ^{{\scriptsize \alpha /2}}G_{u,s}(z)|^{2}(\int_{|y^{\prime
}|\leq 3\sqrt{d}}|g(u,y^{\prime })|_{V}dy^{\prime })^{2}du \\
&\leq &C\int_{-\infty }^{-8}\sup_{|z|\leq 4\sqrt{d}}|\partial _{s}\partial ^{
{\scriptsize \alpha /2}}G_{u,s}(z)|^{2}(\frac{1}{\text{mes~}B_{4\sqrt{d}}(x)}
\int_{|x-y^{\prime }|\leq 4\sqrt{d}}|g(u,y^{\prime })|_{V}^{2}dy^{\prime })
\\
&\leq &C\int_{-\infty }^{-8}(s-u)^{-\frac{2d}{{\scriptsize \alpha }}-3}
\mathcal{M}_{x}|g|_{V}^{2}(u,x)du\leq C\mathcal{M}_{t}\mathcal{M}
_{x}(|g|_{V}^{2})(t,x).
\end{eqnarray*}
Summarizing, we have for all $(s,y),(t,x)\in Q_{0},$
\begin{equation}
|\nabla G(s,y)|^{2}+[\partial _{s}G(s,y)]^{2}\leq C\mathcal{M}_{t}\mathcal{M}
_{x}(|g|_{V}^{2})(t,x)|_{V}^{2}. \notag
\end{equation}
Therefore (\ref{2.10}) follows and we showed that (\ref{2.7}) holds in the
first and second case.
6$^{0}$. Now we show that (\ref{2.7}) in the case (2)-(1) and (\ref{2.10})
in the case (3) imply (\ref{2.6}). Let $\varphi $ be a continuos function on
$\mathbf{R}$ with all bounded derivatives such that $0\leq \varphi \leq
1,\varphi (s)=0$ if $-8\leq s,\varphi (s)=1$ if $s\leq -9$. Let
\begin{eqnarray*}
g_{1}(s,y) &=&g(s,y)\varphi (s),(s,y)\in \mathbf{R}^{d+1}, \\
g_{2} &=&g-g_{1}\text{.}
\end{eqnarray*}
Then by (\ref{maf2}),
\begin{eqnarray*}
|Gg-(Gg)_{Q_{0}}| &\leq &|G(g_{1}+g_{2})-Gg_{1}|+|Gg_{1}-\left(
Gg_{1}\right) _{Q_{0}}| \\
+|\left( Gg_{1}\right) _{Q_{0}}-(Gg)_{Q_{0}}| &\leq &Gg_{2}+\left(
Gg_{2}\right) _{Q_{0}}+|Gg_{1}-\left( Gg_{1}\right) _{Q_{0}}|
\end{eqnarray*}
and
\begin{equation*}
\left( Gg\right) _{Q_{0}}^{\#}\leq (Gg_{1})_{Q_{0}}^{\#}+2(Gg_{2})_{Q_{0}}
\end{equation*}
Now, by (\ref{2.2}) and (\ref{2.3}), the required inequality (\ref{2.6})
follows from (\ref{2.7}), (\ref{eq25}) and (\ref{2.10}). The first assertion
of the proposition is proved.
7$^{0}.$ The estimates in Besov spaces follow immediately because
\begin{equation*}
\left( \partial ^{{\scriptsize \alpha /2}}\mathcal{I}g\right) _{j}=\partial
^{{\scriptsize \alpha /2}}\mathcal{I}g_{j}
\end{equation*}
and we have shown that
\begin{eqnarray*}
&&\int_{a}^{b}\int_{\mathbf{R}^{d}}\bigg(\int_{a}^{t}\big\vert\varphi
_{j}\ast f(s,t,x)\big\vert_{V}^{2}ds\bigg)^{p/2}dxdt \\
&&\qquad =\int_{a}^{b}\int_{\mathbf{R}^{d}}\bigg(\int_{a}^{t}\big\vert\big(
\partial ^{{\scriptsize \alpha /2}}\mathcal{I}g(s,t,x)\big)_{j}\big\vert
_{V}^{2}ds\bigg)^{p/2}dxdt \\
&&\qquad =\int_{a}^{b}\int_{\mathbf{R}^{d}}\bigg(\int_{a}^{t}\big\vert
\partial ^{{\scriptsize \alpha /2}}\mathcal{I}g_{j}(s,t,x)\big\vert_{V}^{2}ds
\bigg)^{p/2}dxdt \\
&&\qquad \leq C\int_{a}^{b}|g_{j}(s,\cdot )|_{V,p}^{p}ds.
\end{eqnarray*}
The proposition is proved.
| 200,475
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In keeping with Venice’s morbid reputation, some walking tours focus on spectacularly gruesome murders and intrigues. The finest might be the city-run evening tour through the area of Cannaregio in northwest Venice, about a 15-minute walk from the Rialto Bridge. Apart from telling you about the ghosts which haunt the alleyways—like the one who stalks Venice with blazing coals in his eyes—the walk gives you the chance to leave the tourists behind, and along the quiet lapping canals see how contemporary Venetians live. The city tour starts at 8pm from Campo San Bartolomeo. Tickets are available from the Venice tourist office, which goes by its Italian initials APT, off St. Mark’s Square.
Next The Ghetto
| 147,317
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Travel & Tourism Industry release:
PITTSBURGH, April 17, 2018 /PRNewswire/ — An inventor from Los Angeles, Calif., has developed the PORTABLE POTTY, a portable, hand-held urinal for males.
“I was caught in a traffic jam and needed relieve myself. Unfortunately, I was alone and could not leave my car. I developed my invention to help in these types of situations,” said the inventor. The PORTABLE POTTY enables a traveler to discreetly and safely urinate within a motor vehicle. It helps prevent physical discomfort and repeated stops at highway rest areas. By eliminating these unscheduled stops, travel delays will be prevented. Ultimately, this portable urinal will allow a driver to feel much more comfortable while traveling.
The original design was submitted to the Los Angeles office of InventHelp. It is currently available for licensing or sale to manufacturers or marketers. For more information, write Dept. 17-LAX-932, InventHelp, 217 Ninth Street, Pittsburgh, PA 15222, or call (412) 288-1300 ext. 1368. Learn more about InventHelp’s Invention Submission Services at.
View original content with multimedia:
SOURCE InventHelp
Related Links
Click here to read the full release with contact information on PR Newswire. Some releases are not in English.
To post and circulate your own press release on FIR and the eTN Network please click here
| 317,231
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Bеfоrе уоu apply fоr secured business loans check оut уоu borrowing capacity. Evеrу business іѕ dіffеrеnt whісh means thеrе wіll bе nо universal method tо knоw thе cost аnd thеrеfоrе budgeting іѕ important. Thіѕ enables уоu tо deduce whеthеr уоu саn afford secured business loan оr not.
Before You Apply For Small Business Loans
Tuck іn thоѕе documents! Generally а secured small business loan borrower wоuld require fеw оf thе documents fоr approval. Fіrѕt аnd foremost іѕ thе business profile discussing thе nature оf business, annual sales, length аnd time оf business ownership. In case оf nеw business уоu wоuld require tо project thе loan plan аnd hоw thе business wоuld bе successful еnоugh tо pay bасk thе loan. Fоr secured business loans thе loan application wіll аlѕо include а loan request. Thіѕ wіll include thе type оf secured business loan required, thе amount аnd thе purpose (how thе funds wіll bе used).
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Small Business Loans and Your Credit
A lender wіll inquire whеthеr уоur personal оr business credit іѕ good оr not. Gеt уоur latest credit report аnd mаkе ѕurе іt carries accurate information. Aѕ уоu hаvе applied fоr secured option іt means thаt уоu wоuld hаvе bеttеr choices. However, credit score wіll modify thе interest rates уоu gеt fоr secured business loans. Wіth bad credit score уоu wіll bе paying higher interest rates аѕ compared tо оnеѕ wіth perfect credit.
Whіlе preparing tо gеt money іt іѕ important tо ѕее уоur financing options. Thеrе іѕ bоth а financial аnd emotional component whіlе borrowing secured small business loans; уоur property іѕ аt stake. Mаkе ѕurе уоu аrе ready fоr it. Thеn dо ѕоmе market research. Tаkе іt аѕ а test whеrе уоu hаvе tо gеt thе bеѕt grade. Yоu wоuld рrоbаblу nоt start а business wіthоut researching thе market; mаkе ѕurе уоu follow thіѕ rule whіlе settling оn secured business loan. Yеt dо remember thаt time іѕ money. And don’t waste tоо muсh time іn settling оn thе option.
Evеrу big business starts оut small. Secured small business loans аrе іn fact thе fіrѕt thіng thаt соmеѕ tо thе mind аnd provide fundamental opportunity whеn оnе іѕ raising money fоr business аnd саn provide tо thоѕе whо аrе tо thоѕе whо аrе lооkіng fоr funds fоr business purposes. Secured small business loans аrе whаt уоu nееd whеn уоu аrе lооkіng fоr small business loans wіth security.
Copied wіth permission from:
Finance news you need to know today
ZURICH — The world’s biggest temporary staffing agency Adecco has reported a rise in third-quarter profits, which missed expectations as business took a hit in its main market, France, and slowed in Germany. Originally published as Finance news you …
Read more on Herald Sun and Small Business Loans
UK alternative finance market set to surge in 2015 – study
The report from innovation charity Nesta and the University of Cambridge said alternative Small Business Loans finance to businesses and consumers was set to hit 1.74 billion pounds in 2014, before jumping to 4.4 billion pounds in 2015. The market totalled some 666 million …
Read more on Reuters UK
People Who Work In Tech Are Investing Better Than Their Friends In Finance
By analyzing about 700 portfolios of Openfolio users working in the tech industry, about 500 portfolios of users working in finance, and about 160 from users working in advertising and media, the company found that finance users aren’t the most …
Read more on Business Insider
SMRT’s finance head quits after 8 months
SMRT said its CFO provides the overall leadership to the finance function of the group. The CFO is also expected to play a key role in developing, monitoring and evaluating overall corporate strategy with the CEO and leaders of business units, with …
Read more on AsiaOne
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WIXOM, Mich., Oct., November 4, 2010 at 8:30 a.m. ET to discuss its third quarter 2010 results. Robert Chioini, Chairman and CEO, and Thomas Klema, CFO, will be hosting the call to review Rockwell's results. Rockwell will be releasing the earnings results on the same date.
When calling in, investors should refer to the Rockwell Medical Investor Conference Call and provide the operator with their name and company affiliation.
Investors who prefer to participate via internet may use the following link:
Schedule this webcast into MS-Outlook calendar (click open when prompted):
About, a prevalence. Thus, actual results could be materially different. Rockwell expressly disclaims any obligation to update or alter statements whether as a result of new information, future events or otherwise, except as required by law.
“ ”
| 98,361
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TITLE: Overdetermined system of simple nonlinear equations
QUESTION [1 upvotes]: Let $I$, $J$ and $K$ denote index sets. There are two kinds of variables $x_{j,k} \in [0, 1]$ and $y_{i,k} \in [0, 8760]$ and two kinds of constants $a_{i,j,k} \geq 0$ and $b_j \geq 0$. I'm considering a system of $(|I| \cdot |K| + 1) \cdot |J|$ equations
\begin{align}
&0 = x_{j,k} \cdot b_j \cdot y_{i,k} - a_{i,j,k}\\[2mm]
&0 = \sum_{k \in K}{x_{j,k}} - 1
\end{align}
in $(|I| + |J|) \cdot |K|$ unknowns. I assume $|I|, |J| \geq 2$ such that the system is always overdetermined for all $|K|$. As the system is rather simple I was wondering whether
there exists a unique solution
the system is analytically tractable
REPLY [0 votes]: For the first part, for the case when $b_j =1, a_{i,j,k} = 0 \,\, \forall i,j,k$, we can have multiple solutions. Moreover, when $\exists i,j,k \,\,\,\, b_j = 0, a_{i,j,k} > 0$, there are no solutions.
As for tractability, we need to do a few cases
Case 1: If there exists a $j$ such that $b_j =0$ and $a_{i,k,j} > 0$ for any $i,k$ then there is no solution that satisfies all the constraints.
Case 2: In case $b_j =0$, we have $a_{i,j,k} = 0$ for all $i,k$.
In this case, neglect the equations with $b_j=0$.
If there is a $j$ such that $\exists i,k \,\, a_{i,j,k}/b_j > 8760$, there is no solution.
Let $c_{i,j,k} = a_{i,j,k}/b_j$
Hence, we have that $x_{j,k} = c_{i,j,k}/y_{i,k}$ from the first set of constraints.
Now, we can use the second set of constraints to solve for $y_{i,k}$. The problem can be converted to to solving set of linear equations with variables in the range $[1, \infty)$
| 204,381
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Update 22.12.2009
Great news has arrived just before Christmas. Adobe has released a new version of flash which finally gives us linux users what seems to be a proper improvement of the flash plugins capabilities in full screen mode. At least I am experiencing GREAT improvement when viewing flash videos in fullscreen. On my 3,5 year old laptop I can finally view HD videos on youtube in fullscreen without any problems.
The new version I am talking about is 10.0.42.34. The changelog can be viewed here. The 10.1 version is also in progress, but stay away from it if you are a linux user. Nothing new there for us, and I'm not even sure it has the latest linux tweaks available.
So, try to keep track that you always have the latest version of flash available, this you can easily do here. Downloading the latest version can be done here. Download the .tar.gz version. To install: Open up firefox/swiftfox and type about:plugins in the adressbar. Check where your current libflashplayer.so is located, and replace it. Make sure you do not have multiple installations of the plugin.
Important for INTEL graphics users:
Remember to have at least version 2.8.0 of xorg-x11-drv-intel, because:
* Fri Aug 07 2009 Kristian HÃgsberg - 2.8.0-4
- Add dri2-page-flip.patch to enable full screen pageflipping.
Fixes XKCD #619.
Old post with tweaks:
Last few days I've had a lot of problems regarding flash video in Linux. So I've spent a lot of time trying to tweak my browsers to work better. Here's my quickstep guide to getting flash working more or less "ok" in Linux. There is still a lot to be done for the flash developers making it work more flawless under Linux.
1.) Make sure you have the plugin for firefox, iceweasel og swiftfox. Check this by typing about:plugin in the address bar. If there are multiple instances of the same flash driver, remove them. Make sure you have the latest version, which at the moment (10th June 2009) shall be: Shockwave Flash 10.0 r22. If you have multiple drivers do a updatedb & locate libflashplayer.so in bash. Restart the browser when you are done.
2.) Cpu frequency stepping can be another problem. If you have powernowd or cpufreq installed make sure they are working proper. I had big problems with cpufreq only using 1ghz of my cpu although I had another 883mhz available. Check cpufreq-info and make sure:
A) current policy: frequency should be within 1000 MHz and 1.83 GHz.
Check that the upper frequency is correct.
B) The governor "ondemand" may decide which speed to use.
Make sure you are using the ondemand governor.
C) Now we are gonna tweak this so your computer will utilize your full cpu when viewing flash videos:
C.1) Stop cpufreq: /etc/init.d/cpufreqd stop
C.2) Edit your /sys/devices/system/cpu/cpu0/cpufreq/ondemand/up_threshold and set it to 30. This will make your computer utilize your full cpu once the cpu usage climbs above 30%. You can always tweak this one to another setting if you want.
C.3) Start cpufreq: /etc/init.d/cpufreqd start
3.) We need flash to use your gfx card for the rendering, also known as hardware rendering. Flash uses Opengl in order to do this. Open aptitude, and remove _all_ opengl related packages, and then reinstall the following packages with it's dependencies:
libgl1-mesa-glx
libgl1-mesa-dev
Now try again :)
Please post more tips if you got any. No registration required.
Hi There
I found that watching justin-tv (wich is flash based) is very slow /choppy on my old laptop. A way to clam down xorgs is to change the color depth to only thousnads instead of millions. It improved nicely and I can watch justintv with maximized window now.
Hope it help
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Dvsn Shares Trailer For ‘Morning After’ The upcoming sophomore album from dvsn receives a trailer! 00Share on Pinterest0Share with your friendsYour NameYour EmailRecipient EmailEnter a MessageI read this article and found it very interesting, thought it might be something for you. The article is called Dvsn Shares Trailer For ‘Morning After’ and is located at
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TITLE: What sort of language does the language of second-order arithmetic become if the 'numbers' are the finite ordinals?
QUESTION [0 upvotes]: The language of second-order arithmetic is defined as follows (the wording of this definition is due to Henry Towsner from a pdf file of "[Chapter 4], "Second Order Arithmetic and Reverse Mathematics" I found on the internet):
Definition 4.1. The language of second order arithmetic is a two-sorted language: there are two kinds of terms, numeric terms and set terms.
0 is a numeric term,
There are infinitely many numeric variables, $x_0$, $x_1$,..., each of which is a numeric term.
In $s$ is a numeric term, then $\mathbf S$$s$ is a numeric term,
If $s$, $t$ are numeric terms then $+$$s$$t$ and $\cdot$$s$$t$ are numeric terms (abbreviated $s$+$t$ and $s$$\cdot$$t$),
There are infinitely many set variables $X_0$, $X_1$,..., each of which is a set term,
If $t$ is a numeric term and $S$ then $\in$$t$$S$ is an atomic formula (abbreviated $t$$\in$$S$),Tf $s$,$t$ are numeric terms then $=$$s$$t$ and $\lt$$s$$t$ are atomic formulas (abbreviated $s$=$t$ and $s$$\lt$$t$).
The formulas are built from the atomic formulas in the usual way.
Since it is known that $ZF$-Infinity (that is, $ZF$ woth the axiom of infinity dropped--I will not add the alternate axiom $\lnot$Infinity since the fragment alone will do all I need) derives all of the theorems of $PA$, it might be interesting to see how to formulate second-order arithmetic in terms of that fragment.
Obviously, some adjustments in the language will have to be made since the 'numbers' will now be finite ordinals. For example, the phrase 'numeric term' would have to be replaced by 'set term' (as in "$\emptyset$ is a set term, 'If $s$ is a set term, then $\mathbf S$$s$ is a set term' (where $\mathbf S$$s$ will have to be defined as $s$$\cup${$s$}); '$+$', '$\cdot$' will have to be replaced by the set-theoretic analogues for finite sets and '$\lt$ will have to be replaced by $\subset$).
But now what to do with the phrase 'set term' in 'There are infinitely many set variables $X_0$, $X_1$,..., each of which is a set term', since for $ZF$ -infinity one already has '$\in$' (for example) defined for the finite sets, and Separation can be used to define properties of finite ordinals analogous to properties of natural numbers in $PA$?
Question 1: Can the axioms of $ZF$-Infinity be so adjusted that the language of second-order arithmetic need only one sort--the sort set?
There is, of course, a more natural way to adjust the language of second-order arithmetic so that it is still a two-sorted language:
Replace the term 'numeric term' with 'set term', and 'set term' with 'class term', where 'set' can now be defined as follows:
'$x$ is a set' $\leftrightarrow$ $\exists$$X$($x$$\in$$X$), where $X$ is a class variable.
Given this definition of set, it might be reasonable to allow sets to be classes that can be members of other classes and in this fashion reduce the number of sorts of the language of second-order arithmetic from two to one. From this also one can see the role of the Comprehension axioms in second-order arithmetic (the usualway of defining class--as extensions of formulas). As regards second-order induction ($\forall$$X$($\emptyset$$\in$$x$ $\land$ $\forall$$x$($x$$\in$$X$ $\rightarrow$ $x$$\cup${$x$}$\in$$X$) $\rightarrow$ $\forall$$x$$x$$\in$$X$)), it allows for the existence of infinite classes (which can be deemed sets if such classes can be members of other classes--this is the true significance of the axiom of infinity).
Question 2. Does the 'bi-sorted' language described above (with a 'set' sort and a 'class' sort, with a class being a set iff that class can be a member of some class) seem a better way of adjusting the language of second-order arithmetic if $PA$ is replaced by $ZF$-Infinity than the alternative in Question 1 ?
(Note: It might be that neither of the alternatives presented are suitable for the task; so in that case, how does one properly define the language of second-order arithmetic when $PA$ is replaced by the fragment $ZF$-Infinity? [Question 3]
REPLY [1 votes]: Your question $2$ has the right idea. The right version of "second-order finite set theory" is two-sorted, as is second-order arithmetic, with one sort being the "class sort." (Note that in a sense, hereditarily finite sets will show up both as "numbers" and as "sets," but this is no different from the fact in second-order arithmetic that a finite set can be coded by a single number; it's just a bit more striking here because the finite set of hereditarily finite sets literally is a hereditarily finite set.)
Now an important point here is that second-order arithmetic is much stronger than PA! For a really easy example of this, it implies the consistency of PA. So not every model of finite set theory can be expanded to a model of second-order finite set theory, identically to how not every model of PA can be expanded to a model of second-order arithmetic. That said, by restricting the comprehension axioms, we can get a second-order finite set theory which is conservative over the first-order finite set theory (identically to how, by restricting comprehension in second-order arithmetic, we can get ACA which is conservative over PA).
Note that this is exactly what we should expect: by the analogy "numbers $\sim$ hereditarily finite sets,", we expect "sets of numbers $\sim$ sets of hereditarily finite sets". So that's the language setup we should be looking for. Remember, the whole point of this analogy is that - once we're looking at sufficiently strong theories on each side - there is no real difference between finite set theory and arithmetic.
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\begin{document}
\def\le{\left}
\def\r{\right}
\def\cost{\mbox{const}}
\def\a{\alpha}
\def\d{\delta}
\def\ph{\varphi}
\def\e{\epsilon}
\def\la{\lambda}
\def\si{\sigma}
\def\La{\Lambda}
\def\B{{\cal B}}
\def\A{{\mathcal A}}
\def\L{{\mathcal L}}
\def\O{{\mathcal O}}
\def\bO{\bar{{\mathcal O}}}
\def\F{{\mathcal F}}
\def\K{{\mathcal K}}
\def\H{{\mathcal H}}
\def\D{{\mathcal D}}
\def\C{{\mathcal C}}
\def\M{{\mathcal M}}
\def\N{{\mathcal N}}
\def\G{{\mathcal G}}
\def\T{{\mathcal T}}
\def\R{{\mathcal R}}
\def\I{{\mathcal I}}
\def\bw{\bar{W}}
\def\phin{\|\varphi\|_{0}}
\def\s0t{\sup_{t \in [0,T]}}
\def\lt{\lim_{t\rightarrow 0}}
\def\iot{\int_{0}^{t}}
\def\ioi{\int_0^{+\infty}}
\def\ds{\displaystyle}
\def\pag{\vfill\eject}
\def\fine{\par\vfill\supereject\end}
\def\acapo{\hfill\break}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\barr{\begin{array}}
\def\earr{\end{array}}
\def\vs{\vspace{.1mm} \\}
\def\rd{\reals\,^{d}}
\def\rn{\reals\,^{n}}
\def\rr{\reals\,^{r}}
\def\bD{\bar{{\mathcal D}}}
\newcommand{\dimo}{\hfill \break {\bf Proof - }}
\newcommand{\nat}{\mathbb N}
\newcommand{\E}{\mathbb E}
\newcommand{\Pro}{\mathbb P}
\newcommand{\com}{{\scriptstyle \circ}}
\newcommand{\reals}{\mathbb R}
\title{Averaging principle for a class of stochastic reaction-diffusion equations \thanks{ {\em Key words and
phrases:} Stochastic reaction diffusion equations, invariant
measures and ergodicity, averaging principle, Kolmogorov equations
in Hilbert spaces.} }
\author{Sandra Cerrai\\
Dip. di Matematica per le Decisioni\\
Universit\`{a} di Firenze\\
Via C. Lombroso 6/17\\
I-50134 Firenze, Italy
\and
Mark Freidlin\\
Department of Mathematics\\
University of Maryland\\
College Park\\
Maryland, USA}
\date{}
\maketitle
\begin{abstract}
We consider the averaging principle for stochastic
reaction-diffusion equations. Under some assumptions providing
existence of a unique invariant measure of the fast motion with
the frozen slow component, we calculate limiting slow motion. The
study of solvability of Kolmogorov equations in Hilbert spaces
and the analysis of regularity properties of solutions, allow to
generalize the classical approach to finite-dimensional problems
of this type in the case of SPDE's.
\end{abstract}
\section{Introduction}
\label{intro}
Consider a Hamiltonian system with one degree of freedom. In the
area where the Hamiltonian has no critical points, one can
introduce action-angle coordinates $(I,\varphi)$, with $I \in
\mathbb{R}^{1}$ and $0\leq \varphi \leq 2\pi$, so that the system
has the form
\begin{equation}
\dot{I}_{t}=0, \hspace{0.4cm}
\dot{\varphi}_{t}=\omega(I_{t}).\label{ActionAngle1}
\end{equation}
Now, consider small perturbations of this system such that, after
an appropriate time rescaling, the perturbed system can be written
as follows
\begin{equation}
\dot{I}_{t}^{\epsilon}=\beta_{1}(I_{t}^{\epsilon},\varphi_{t}^{\epsilon}),
\hspace{0.4cm}
\dot{\varphi}_{t}^{\epsilon}=\frac{1}{\epsilon}\,\omega(I_{t}^{\epsilon})+\beta_{2}(I_{t}^{\epsilon},\varphi_{t}^{\epsilon}).\label{ActionAnglePerturbed}
\end{equation}
Here the perturbations $\beta_{1},\beta_{2}:\mathbb{R}^1\times
[0,2\pi]\to \mathbb{R}$ are assumed to be regular enough
functions, as well as $\omega:\mathbb{R}\to \mathbb{R}$, and
$0<\epsilon<<1$.
System (\ref{ActionAnglePerturbed}) has a fast component, which
is, roughly speaking, the motion along the non-perturbed
trajectories (\ref{ActionAngle1}), after the time change
$t\rightarrow t/\epsilon$, and the slow component which can be
described by the evolution of $I_{t}^{\epsilon}$. When $\epsilon$
goes to $0$, the slow component approaches the averaged motion
$\bar{I}_{t}$, defined by
\begin{equation}
\dot{\bar{I}}_{t}=\bar{\beta}_{1}(\bar{I}_{t}),\ \
\ \ \ \bar{I}_{0}=I_{0},\label{AveragedSlowMotion}
\end{equation}
where
\[\bar{\beta}_{1}(y)=\frac{1}{2\pi}\int_{0}^{2\pi}\beta_{1}(y,\varphi)d\varphi.\]
This is a classical manifestation of the averaging principle for
equation (\ref{ActionAnglePerturbed}).
To prove the convergence of $I^{\epsilon}_{t}$ to
$\bar{I}_{t}$, one can consider a $2\pi$-periodic in
$\varphi$ solution $u(I,\varphi)$ of an auxiliary equation
\begin{equation}
\mathcal{L}^Iu(I,\varphi):=\omega(I)\frac{\partial u}{\partial \varphi}=\beta_{1}(I,\varphi)-\bar{\beta}(I).
\label{AuxiliaryEquation}
\end{equation}
It is easy to see that such a solution exists and is unique up to
an additive function, depending just on $I$. Moreover, it can be
chosen in such a way that $u(I,\varphi)$ has continuous
derivatives in $I$ and $\varphi$. Actually, $u(I,\varphi)$ can be
written explicitly. It follows from (\ref{ActionAnglePerturbed})
and (\ref{AuxiliaryEquation}) that
\[\begin{array}{l}
\ds{u(I_{t}^{\epsilon},\varphi_{t}^{\epsilon})-u(I_{0}^{\epsilon},\varphi_{0}^{\epsilon})=
\frac{1}{\epsilon}\int_{0}^{t}\frac{\partial u}{\partial
\varphi}\,(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})\omega(I_{s}^{\epsilon})ds+ \int_{0}^{t}\frac{\partial u}{\partial
\varphi}\,(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})\beta_{2}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})ds}\\
\vs \ds{ +\int_{0}^{t}\frac{\partial
u}{\partial
I}\,(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})\beta_{1}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})ds=\frac
1\epsilon\,\int_{0}^{t}[\beta_{1}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})-\bar{\beta}(I_{s}^{\epsilon})]ds}\\
\vs \ds{ +
\int_{0}^{t}\frac{\partial u}{\partial
\varphi}\,(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})\beta_{2}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})ds+\int_{0}^{t}\frac{\partial u}{\partial
I}\,(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})\beta_{1}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})ds.}
\end{array}\]
Hence, by taking into account the boundedness of coefficients $\beta_1$ and $\beta_2$ and of function $u(I,\varphi)$
together with its first derivatives, one can conclude from the last
equality that for any $T>0$
\begin{equation}
\sup_{0\leq t\leq
T}|\int_{0}^{t}[\beta_{1}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})-\bar{\beta}(I_{s}^{\epsilon})]ds|\leq
c\,\epsilon,\label{convergence1}
\end{equation}
for some constant $c>0$. Now, from (\ref{ActionAnglePerturbed}) and
(\ref{AveragedSlowMotion}) it follows
\[
I^{\epsilon}_{t}-\bar{I}_{t}=\int_{0}^{t}[\beta_{1}(I_{s}^{\epsilon},\varphi_{s}^{\epsilon})-\bar{\beta}(I_{s}^{\epsilon})]ds+
\int_{0}^{t}[\bar{\beta}(I_{s}^{\epsilon})-\bar{\beta}(\bar{I}_{s})]ds,\]
so that, assuming that $\beta(I,\varphi)$ (and thus
$\bar{\beta}(I)$) is Lipschitz-continuous, thanks to
(\ref{convergence1}) and to Gronwall's lemma we get
\begin{displaymath}
\sup_{0\leq t\leq T}|I_{t}^{\epsilon}-\bar{I}_{t}|\leq
c\,\epsilon.
\end{displaymath}
On a first glance, one can think that consideration of the
auxiliary equation (\ref{AuxiliaryEquation}) for proving averaging
principle is an artificial trick. But, actually, this is not the
case; the use of equation (\ref{AuxiliaryEquation}) and its natural
generalizations helps to prove averaging principle in many cases.
For example, when deterministic perturbations of a completely
integrable system with many degrees of freedom (in a domain where
one can introduce action-angle coordinates) are considered, the
operator $\mathcal{L}$ is the generator of the corresponding flow
on a torus. Because of the existence of resonance tori, where
invariant measure of the flow is not unique, one has to consider
approximate solutions of the corresponding equation
(\ref{AuxiliaryEquation}). The price for this is that the
convergence of $\sup_{0\leq t\leq
T}|I_{t}^{\epsilon}-\bar{I}_{t}|$ to zero does not hold for
any fixed initial condition, but just in Lebesgue measure in the
phase space, given that the set of resonance tori is small enough
(see \cite{neistadt}). An approximate solution of the
corresponding analogue of equation (\ref{AuxiliaryEquation}) is
used in \cite{frven2} for averaging of stochastic perturbations.
In this case it is possible to prove weak convergence to the
averaged system in the space of continuous functions on the phase
space.
Moreover, concerning the use of the auxiliary equation \eqref{AuxiliaryEquation}, it is worthwhile mentioning that in \cite{papanicolaou} suitable {\em correction functions} arising as solutions of problems analogous to \eqref{AuxiliaryEquation} are introduced in order to prove some limit theorems for more general multi-scaling systems.
An analogue of equation (\ref{AuxiliaryEquation}) appears also in
the case when the fast motion is a stochastic process
\begin{displaymath}
\dot{I}_{t}^{\epsilon}=\beta_{1}(I_{t}^{\epsilon},\varphi_{t}^{\epsilon}),\hspace{0.2cm}
\dot{\varphi}_{t}^{\epsilon}=\frac{1}{\epsilon}\,\omega(I_{t}^{\epsilon},\varphi_{t}^{\epsilon})+
\frac{1}{\sqrt{\epsilon}}\,\sigma(I_{t}^{\epsilon},\varphi_{t}^{\epsilon})\dot{w}_{t}+
\beta_{2}(I_{t}^{\epsilon},\varphi_{t}^{\epsilon}).
\end{displaymath}
Here $I, \varphi:[0,+\infty)\to \reals^n$,
$\omega:\reals^n\times \reals^n\to \reals^n$,
$\sigma(I,\varphi)\sigma^{\ast}(I,\varphi)=\alpha(I,\varphi)$ is a
positive definite $n \times n-$matrix and $w_{t}$ is the
standard $n$-dimensional Wiener process. All functions are assumed
to be $2\pi-$periodic in the variables $\varphi_{i}$ and smooth enough. Under these conditions, for each $I\in
\mathbb{R}^{n}$ the diffusion process $\varphi_{t}^{I}$ on the
$n$-torus $T^{n}$ defined by the equation
\[\dot{\varphi}_{t}^{I}=\omega(I, \varphi_{t}^{I})+\sigma(I,
\varphi_{t}^{I})\dot{w}_{t},\]
has a unique invariant measure
with density $m_{I}(\varphi)$. Then equation
(\ref{AuxiliaryEquation}) should be replaced by
\begin{equation}
\label{AuxiliaryEquationReplaced}
\mathcal{L}^{I}u(I,\varphi)=\beta_{1}(I,\varphi)-\bar{\beta}_{1}(I),\end{equation}
where $\mathcal{L}^{I}$ is the generator of the process
$\varphi_{t}^{I}$ and for any $I\in \mathbb{R}^{n}$
\[\bar{\beta}_{1}(I):=\int_{T^{n}}\beta_{1}(I,\varphi)m_{I}(\varphi)d\varphi.\]
Taking into account the uniqueness of the invariant measure, one can
check that there exists a solution to problem
(\ref{AuxiliaryEquationReplaced}) which is smooth in $I$ and
$\varphi$. Applying It\^o's formula to
$u(I_{t}^{\epsilon},\varphi_{t}^{\epsilon})$, one can prove not
just weak convergence of $I_{t}^{\epsilon}$ to $\bar{I}_{t}$
on any finite time interval, but also convergence of
$(I_{t}^{\epsilon}-\bar{I}_{t})/\sqrt{\epsilon}$ to a
diffusion process.
Besides the situations described above, averaging principle both for deterministically and for randomly perturbed systems, having a finite number of degrees of freedom, has been studied by many authors, under different assumptions and with different methods. The first rigorous results are due to Bogoliubov
(see \cite{bogo}). Further developments were obtained by Volosov, Anosov and
Neishtadt (see \cite{neistadt} and
\cite{volosov}) and by Arnold et al. (see \cite{akn}). All these references are for the deterministic case. Concerning the stochastic case, it is worth quoting the paper by Khasminskii \cite{khas}, the works of Brin, Freidlin and Wentcell (see \cite{brfr},
\cite{freidlin}, \cite{frven}, \cite{frven2}), Veretennikov (see \cite{vere}) and
Kifer (see for example \cite{kif1}, \cite{kif2},
\cite{kif3}, \cite{kif4}).
\medskip
To the best of our knowledge, very few has been done as far as averaging for infinite dimensional systems is concerned. To this purpose we recall the papers \cite{massei} and \cite{seivr}, where the case of stochastic evolution equations in abstract Hilbert spaces is considered, and the paper \cite{kukpia}, where randomly perturbed KdV equation is studied.
In this paper we are dealing with a system of
reaction-diffusion equations with a stochastic fast component.
Namely, for each $0<\e<<1$, we consider the system of partial
differential equations
\begin{equation}
\label{sistema} \le\{
\begin{array}{l}
\ds{ \frac{\partial u^\e}{\partial t}(t,\xi)=\mathcal{A}
u^\e(t,\xi)+f(\xi,u^\e(t,\xi),v^\e(t,\xi)), \ \ \ \ \ t\geq 0,\ \ \ \xi \in\,[0,L],}\\
\vs \ds{\frac{\partial v^\e}{\partial t}(t,\xi)=\frac
1\e\,\le[\mathcal{B}
v^\e(t,\xi)+g(\xi,u^\e(t,\xi),v^\e(t,\xi))\r]+\frac
1{\sqrt{\e}}\,\frac{\partial w}{\partial t}(t,\xi), \ \ \ \ \
t\geq 0,\ \ \ \xi \in\,[0,L],}\\
\vs \ds{u^\e(0,\xi)=x(\xi),\ \ \ \ v^\e(0,\xi)=y(\xi),\ \ \ \ \
\xi
\in\,[0,L],}\\
\vs \ds{ \mathcal{N}_{1} u^\e\,(t,\xi)=\mathcal{N}_{2}
v^\e\,(t,\xi)=0,\ \ \ \ t\geq 0,\ \ \ \ \xi \in\,\{0,L\}.}
\end{array}\r.
\end{equation}
The present model describes a typical and relevant situation for reaction-diffusion systems in which the diffusion coefficients and the rates of reactions have
different order. In the case we are considering here, the noise is included only in the fast motion and it is of additive type. However, we would like to stress that the introduction of a noisy term of additive type in the slow equation would not lead to
any new effects, as it should be
included in the limiting slow motion without any substantial changes.
\medskip
The linear operators $\mathcal{A}$ and $\mathcal{B}$, appearing
respectively in the slow and in the fast equation, are
second order uniformly elliptic operators and $\mathcal{N}_1$ and
$\mathcal{N}_2$ are some operators acting on the boundary. The
operator $\mathcal{B}$, endowed with the boundary conditions
$\mathcal{N}_2$, is self-adjoint and strictly dissipative (see
Hypothesis \ref{H1}).
The reaction coefficients $f$ and $g$ are measurable mappings from
$[0,L]\times \mathbb{R}^2$ into $\mathbb{R}$ which satisfy
suitable regularity assumptions and for the reaction coefficient
$g$ in the fast motion equation some dissipativity assumption is
assumed (see Hypotheses \ref{H2} and \ref{H3}).
The noisy perturbation of the fast motion equation is given by
a space-time white noise $\partial w/\partial t(t,\xi)$, defined
on a complete stochastic basis $(\Omega, \mathcal{F},
\mathcal{F}_t, \mathbb{P})$.
\medskip
The corresponding fast motion $v^{x,y}(t)$, with frozen slow
component $x \in\,H:=L^2(0,L)$, (the counterpart of the process
$\varphi_{t}^{I}$ above in the case of a system with a finite
number of degrees of freedom) is a Markov process in a functional
space. Notice that the phase space of $v^{x,y}(t)$ is not just
infinitely dimensional but also not compact. Nevertheless, by
assuming that the system has certain dissipativity properties, for
any fixed $x \in\,H$ the process $v^{x,y}(t)$ has a unique
invariant measure $\mu^{x}$. If $\mathcal{L}^{x}$ is the generator
of this process, then the counterpart of equation
\eqref{AuxiliaryEquation} has the form
\begin{equation}
\label{counterpart}
c(\e)\,\Phi_h^{\e}(x,y)-\mathcal{L}^{x}\Phi_h^{\e}(x,y)=\le<F(x,y)-
\bar{F}(x),h\r>_{H},\ \ \ \ \ x,y,h \in\,H,
\end{equation}
where $c(\e)$ is a constant depending on $\e$ and vanishing at
$\e=0$, \[F(x,y)(\xi)=f(\xi, x(\xi),y(\xi)),\ \ \ \ \ \xi
\in\,[0,L],\] and
\[\bar{F}(x):=\int_{H} F(x,y)\,\mu^x(dy),\ \ \ \ x
\in\,H.\]
Notice that in \eqref{counterpart} we
cannot consider the Poisson equation ($c(\e)=0$), but we have to
add a zero order term $c(\e)\Phi^\e_h(x,y)$, in order to get bounds for $\Phi^\e_h(x,y)$ and its derivatives which are uniform with respect to $\e \in\,(0,1]$.
Due to the ergodicity of $\mu^x$, we prove that there exists some $\d>0$ such that for any
$\varphi:H\to \mathbb{R}$ and $x,y \in\,H$
\[\le|P^x_t\varphi(y)-\int_H \varphi(z)\,\mu^x(dz)\r|\leq
c\,\le(1+|x|_H+|y|_H\r)\,e^{-\d t}\,[\varphi]_{\text{\tiny
Lip}},\] where $P^x_t$ is the transition semigroup associated with
the fast motion $v^{x,y}(t)$, with frozen slow component $x$. This
implies that the solution $\Phi^\e_h(x,y)$ of equation
\eqref{counterpart} can be written explicitly as
\[\Phi^\e_h(x,y)=\int_0^t e^{-c(\e)t} P^x_t\le<F(x,\cdot)-
\bar{F}(x),h\r>_{H}(y)\,dt.\]
By using some techniques developed in \cite{cerrai}, we obtain bounds for
$\Phi^\e_h(x,y)$ and its derivatives, which in general are not
uniform in $x,y \in\,H$, as the reaction coefficients $f$ and $g$
are not assumed to be bounded. Moreover, we are able to apply an
infinite dimensional It\^o's formula to $\Phi^\e_h(u^\e,P_n
v^\e)$, where $P_n$ is the projection of $H$ onto the
$n$-dimensional space generated by the first $n$ modes of the operator $\cal{B}$, and $u^\e$
and $v^\e$ are the solutions of system \eqref{sistema}. In this
way, as in the case of a system with a finite number of degrees of
freedom, we are able to prove that
\begin{equation}
\label{re} \E\,\le|\int_0^t\le<F(u^\e(s),v^\e(s))-
\bar{F}(u^\e(s)),h\r>_{H}\,ds\r|\leq K_t(\e),\ \ \ \ \ t\geq 0,\ \
\e>0, \end{equation}
for some $K_t(\e)\downarrow 0$, as $\e$ goes to
zero. The proof of \eqref{re} is one of the major task of the
paper, as it requires several estimates for $\Phi^\e_h(x,y)$ and
its derivatives and uniform bounds with respect to $\e>0$, both for $u^\e$ and
for $v^\e$.
Once we have estimate \eqref{re}, we show that for any $T>0$ the
family $\{\mathcal{L}(u^\e)\}_{\e \in\,(0,1)}$ is tight in
$\mathcal{P}(C((0,T];H)\cap L^\infty(0,T;H))$ and then
we identify the weak limit of any subsequence
of $\{u^\e\}$ with the solution $\bar{u}$ of the averaged equation
\begin{equation}
\label{averagedbis} d\bar{u}(t)=A \bar{u}(t)+\bar{F}(\bar{u}(t)),\
\ \ \ \ \bar{u}(0)=x \in\,H. \end{equation}
Now, as a consequence of the dependence of
$\mu^x$ on $x \in\,H$, the nonlinear term $\bar{F}$ in
\eqref{averagedbis} is a functional of $\bar{u}$. Nevertheless,
one can prove that problem \eqref{averagedbis}, under certain
small assumptions, has a unique solution (see section \ref{sec5},
Proposition \ref{existence}).
Hence, by a uniqueness
argument, this allows us to conclude that the whole sequence $\{ u^\e\}_{\e>0}$ converges to $\bar{u}$ in probability, uniformly on any finite time interval $[0,T]$. That is
\begin{Theorem}
Under Hypotheses \ref{H1}, \ref{H2} and \ref{H3} (see Section 2), for any $T>0$, $x, y \in\,W^{\a,2}(0,L)$, with $\a>0$, and $\eta>0$ it holds
\[\lim_{\e\to 0}\,\Pro \le(\,\sup_{t \in\,[0,T]}|u^\e(t)-\bar{u}(t)|_{H}>\eta\,\r)=0.\]
\end{Theorem}
Notice that in the case of space dimension $d=1$ the fast equation with frozen slow component is a gradient system and then we have an explicit expression for the invariant measure $\mu^x$. This allows to prove that the mapping $\bar{F}$ is differentiable and to give an expression for its derivative. In such a way we can study dependence with respect to $x$ for the correction function $\Phi_h^\e$ and we can proceed with the use of It\^o's formula.
In the case of space dimension $d>1$ the fast equation is no more of gradient type. Nevertheless, in Section \ref{sec6} we show how it is possible to overcome this difficulty and how, by a suitable approximation procedure, it is still possible to prove averaging.
\medskip
Finally, we would like to recall that in a number of models, one can assume that the noise in the fast
motion is small. This results in replacement of $\e^{-1/2}$ by
$\d\,\e^{-1/2}$, with $0<\d<<1$ (to this purpose we refer to
\cite{freidlin}). Then, in generic situation, the invariant
measure of the fast motion with frozen slow component $x$ is
concentrated, as $\d$ goes to zero, near one point $y^\star(x)
\in\,H$. This is a result of large deviations bounds and
$y^\star(x)$ can be found as an extremal of a certain functional.
In particular, if the operator $\mathcal{B}$ in the fast equation
is self-adjoint and $g(\xi,\si,\rho)=h(\si) N(\rho)$, with, for
brevity, the antiderivative $H(\si)$ of $h(\si)$ having a unique
maximum point, then $y^\star(x)$ is a constant providing the
maximum of $H(\si)$. In this case we have that
$\bar{F}(x)(\xi)=f(\xi,x(\xi),y^\star(x)(\xi))$, $\xi
\in\,[0,\pi]$, and \eqref{averagedbis} is a classical
reaction-diffusion equation. We will address this problem
somewhere else.
\section{Assumptions and notations}
\label{sec2}
Let $H$ be the Hilbert $L^2(0,L)$, endowed with the usual scalar
product $\le<\cdot,\cdot\r>_H$ and the corresponding norm
$|\cdot|_H$. In what follows, we shall denote by $\mathcal{L}(H)$
the Banach space of bounded linear operators on $H$, endowed with
the usual sup-norm. $\mathcal{L}_1(H)$ denotes the Banach space of
{\em trace-class} operators, endowed with the norm
\[\|A\|_1:=\mbox{Tr}\,[\sqrt{A A^\star}], \]
and $\mathcal{L}_2(H)$ denotes the Hilbert space of {\em
Hilbert-Schmidt} operators on $H$, endowed with the scalar product
\[\le<A,B\r>_{2}=\mbox{Tr}\,[AB^\star]\]
and the corresponding norm $\|A\|_2=\sqrt{\mbox{Tr}\,[AA^\star]}$.
\medskip
The Banach space of bounded Borel functions $\varphi:H\to
\reals$, endowed with the sup-norm
\[\|\varphi\|_0:=\sup_{x \in\,H}|\varphi(x)|,\]
will be denoted by $B_b(H)$. $C_b(H)$ is the subspace of all
uniformly continuous mappings and $C^k_b(H)$ is the subspace of
all $k$-times differentiable mappings, having bounded and
uniformly continuous derivatives, up to the $k$-th order, for $k
\in\,\nat$. $C^k_b(H)$ is a Banach space endowed with the norm
\[\|\varphi\|_k:=\|\varphi\|_0+\sum_{i=1}^k\sup_{x
\in\,H}|D^i\varphi(x)|_{\mathcal{L}^i(H)}=:\|\varphi\|_0+\sum_{i=1}^k\,[\varphi]_i,\]
where $\mathcal{L}^1(H):=H$ and, by recurrence,
$\mathcal{L}^i(H):=\mathcal{L}(H,\mathcal{L}^{i-1}(H))$, for any
$i>1$.
In what follows we shall denote by $\text{Lip}(H)$ the set of all
Lipschitz-continuous mappings $\varphi:H\to \reals$ and we shall
set
\[[\varphi]_{\text{\tiny
Lip}}:=\sup_{x\neq y} \frac{|\varphi(x)-\varphi(y)|}{|x-y|}.\]
Moreover, for any $k\geq 1$ we shall denote by $\text{Lip}^k(H)$
the subset of all $k$-times differentiable mappings having
bounded and uniformly continuous derivatives, up to the $k$-th
order. Notice that for any $\varphi \in\,\text{Lip}(H)$
\begin{equation}
\label{fi} |\varphi(y)|\leq [\varphi]_{\text{\tiny
Lip}}|y|_H+|\varphi(0)|,\ \ \ \ \ y \in\,H.
\end{equation}
\medskip
The stochastic perturbation in the fast motion equation is given
by a space-time white noise $\partial w/\partial t(t,\xi)$, for $t \geq 0$ and $\xi
\in\,[0,L]$. Formally the cylindrical Wiener process $w(t,\xi)$ is
defined as the infinite sum
\[w(t,\xi)=\sum_{k=1}^\infty e_k(\xi)\,\beta_k(t),\ \ \ \ \ t\geq
0,\ \ \ \xi \in\,[0,L],\] where $\{e_k\}_{k \in\,\nat}$ is a
complete orthonormal basis in $H$ and $\{\beta_k(t)\}_{k
\in\,\nat}$ is a sequence of mutually independent standard
Brownian motions defined on the same complete stochastic basis
$(\Omega,\mathcal{F}, \mathcal{F}_t, \mathbb{P})$.
Now, for any $T>0$ and $p\geq 1$ we shall denote by
$\mathcal{H}_{T,p}$ the space of adapted processes in
$C((0,T];L^p(\Omega;H))\cap L^\infty(0,T;L^p(\Omega;H))$. $\mathcal{H}_{T,p}$ is a Banach space,
endowed with the norm
\[\|u\|_{\mathcal{H}_{T,p}}=\le(\,\sup_{t
\in\,[0,T]}\,\E\,|u(t)|_H^p\r)^{\frac 1p}.\] Moreover, we shall
denote by $\mathcal{C}_{T,p}$ the subspace of processes $u
\in\,L^p(\Omega;C((0,T];H)\cap L^\infty(0,T;H))$, endowed with the norm
\[\|u\|_{\mathcal{C}_{T,p}}=\le(\,\E\,\sup_{t
\in\,[0,T]}\,|u(t)|_H^p\r)^{\frac 1p}.\]
\bigskip
The linear operators $\mathcal{A}$ and $\mathcal{B}$, appearing
respectively in the slow and in the fast motion equation, are
second order uniformly elliptic operators, having continuous coefficients on $[0,L]$, and $\mathcal{N}_1$ and $\mathcal{N}_2$
can be either the identity operator (Dirichlet boundary
conditions) or first order operators of the following type
\[\beta(\xi) \frac{\partial}{\partial \xi}+\gamma(\xi),\ \
\ \ \xi \in\,\{0,L\},\] for some $\beta, \gamma \in\,C^1[0,L]$
such that $\beta(\xi)\neq 0$, for $\xi=0,L$.
As known, the realizations $A$ and $B$ in $H$ of the second order
operators $\mathcal{A}$ and $\mathcal{B}$, endowed respectively
with the boundary condition $\mathcal{N}_1$ and $\mathcal{N}_2$,
generate two analytic semigroups, which will be denoted by $e^{t
A}$ and $e^{t B}$, $t\geq 0$. Their domains $D(A)$ and $D(B)$ are
given by
\[W^{2,2}_{\mathcal{N}_i}(0,L):=\le\{\,x
\in\,W^{2,2}(0,L)\,:\,\mathcal{N}_i x(0)=\mathcal{N}_i
x(L)=0\,\r\},\ \ \ \ \ i=1,2.\] By interpolation we have that for
any $0\leq r\leq s\leq 1/2$ and $t>0$ the semigroups $e^{tA}$ and
$e^{t B}$ map $W^{r,2}(0,L)$ into $W^{s,2}(0,L)$\footnote{For any
$s>0$, $W^{s,2}(0,L)$ denotes the set of functions $x \in\,H$ such that
\[[x]_{s,2}:=\int_{[0,L]^2}\frac{|x(\xi)-x(\eta)|^2}{|\xi-\eta|^{2s+1}}\,d\xi\,d\eta<\infty.\]
$W^{s,2}(0,L)$ is endowed with the norm $|x|_{s,2}:=|x|_H+[x]_{s,2}.$} and
\begin{equation}
\label{wit21} |e^{tA} x|_{s,2}+|e^{tB} x|_{s,2}\leq c_{r,s}
(t\wedge 1)^{-\frac{s-r}2} e^{\gamma_{r,s} t}\,|x|_{r,2},
\end{equation}
for some constants $c_{r,s}\geq 1$ and $\gamma_{r,s} \in\,\reals$.
\medskip
In what follows, we shall assume that the operator $B$ arising in
the fast motion equation fulfills the following condition.
\begin{Hypothesis}
\label{H1} There exists a complete orthonormal basis $\{e_k\}_{k
\in\,\nat}$ in $H$ and a sequence $\{\a_k\}_{k \in\,\nat}$ such
that $B e_k=-\a_k e_k$ and
\begin{equation}
\label{av3bis} \la:=\inf_{k \in\,\nat}\,\a_k>0.
\end{equation}
\end{Hypothesis}
From \eqref{av3bis} it immediately follows
\begin{equation}
\label{av3} \|e^{t B}\|_{\mathcal{L}(H)}\leq e^{- \la t},\ \ \ \ \
t\geq 0.
\end{equation}
\begin{Lemma}
There exists $\gamma <1$ such that
\begin{equation}
\label{h11} \sum_{k=1}^\infty e^{-t\a_k} \leq c\,(t\wedge
1)^{-\gamma}\,e^{-\la t},\ \ \ \ t\geq 0. \end{equation}
In particular
\begin{equation}
\label{wit9} \|e^{t B}\|_2\leq c\,(t\wedge 1)^{-\frac \gamma
2}\,e^{-\la t},\ \ \ \ t\geq 0.
\end{equation}
\end{Lemma}
\begin{proof}
For any $\gamma>0$, there exists some $c_\gamma>0$ such that
\[\sum_{k=1}^\infty e^{-\a_k
t}\leq c_\gamma t^{-\gamma}\,\sum_{k=1}^\infty \a_k^{-\gamma}.\]
Now, for any second order uniformly elliptic operator on the interval $[0,L]$ having continuous coefficients, it holds $\a_k\sim k^2$. Hence, if we assume $\gamma
>1/2$ and take $t \in\,(0,1]$, we have that \eqref{h11} is satisfied. In the case $t>1$, thanks to
\eqref{av3} we have
\[\sum_{k=1}^\infty e^{-\a_k
t}=\sum_{k=1}^\infty |e^{(t-1)B} e^B e_k|_H\leq
c\,e^{-(t-1)\la}\sum_{k=1}^\infty | e^B e_k|_H\leq c\,e^{-\la
t},\] so that \eqref{h11} follows in the general case.
\end{proof}
According to \eqref{wit9}, there
exists some $\d>0$ such that
\begin{equation}
\label{h13} \int_0^t s^{-\d} \,\|e^{s B}\|_2^2\,ds<\infty,\ \ \ \
\ t\geq 0.
\end{equation}
As known (for a prof see e.g. \cite{dpz1}), this implies that the
so-called {\em stochastic convolution} \[w^B(t):=\int_0^t
e^{(t-s)B}\,dw(s),\ \ \ \ t\geq 0,\] is a $p$ integrable
$H$-valued process, for any $p\geq 1$, having continuous
trajectories. Moreover, as a consequence of the dissipativity
assumption \eqref{av3bis}, for any $p\geq 1$
\begin{equation}
\label{eq17} \sup_{t\geq 0}\,\E\,|w^B(t)|^p_{H}=:c_{p}<\infty.
\end{equation}
\medskip
Concerning the reaction coefficient $f$ in the slow motion
equation, we assume what follows.
\begin{Hypothesis}
\label{H2} The mapping $f:[0,L]\times \mathbb{R}^2\to \reals$ is
measurable and $f(\xi,\cdot):\reals^2\to \reals$ is continuously
differentiable, for almost all $\xi \in\,[0,L]$, with uniformly bounded derivatives.
\end{Hypothesis}
Concerning the reaction coefficient $g$ in the fast motion
equation, we assume the following conditions.
\begin{Hypothesis}
\label{H3}
\begin{enumerate}
\item The mapping $g:[0,L]\times
\mathbb{R}^2\to \reals$ is measurable.
\item For each fixed $\sigma_2 \in\,\reals$ and almost all $\xi \in\,[0,L]$, the
mapping $g(\xi,\cdot,\si_2):\reals\to \reals$ is of class $C^1$, with uniformly bounded derivatives.
\item For each fixed $\si_1 \in\,\reals$ and almost all $\xi \in\,[0,L]$, the
mapping $g(\xi,\si_1,\cdot):\reals\to \reals$ is of class $C^3$, with uniformly bounded derivatives.
Moreover,
\begin{equation}
\label{av4} \sup_{\substack{\xi \in\,[0,L]\\\si
\in\,\mathbb{R}^2}}|\frac{\partial g}{\partial
\si_2}(\xi,\si)|=:L_{g}<\la,
\end{equation}
where $\la$ is the positive constant introduced in \eqref{av3bis}.
\end{enumerate}
\end{Hypothesis}
\medskip
In what follows we shall denote by $F$ and $G$ the Nemytskii
operators associated respectively with $f$ and $g$, that is
\[F(x,y)(\xi)=f(\xi,x(\xi),y(\xi)),\ \ \ \ \
G(x,y)(\xi)=g(\xi,x(\xi),y(\xi)),\] for any $\xi \in\,[0,L]$ and
$x,y \in\,H$. Due to the boundedness assumptions on their derivatives, functions $f$ and $g$ are Lipschitz-continuous and hence the
mappings $F,G:H\times H\to H$ are Lipschitz-continuous.
Concerning their regularity properties, for any fixed $y \in\,H$
the mappings $F(\cdot,y)$ and $G(\cdot,y)$ are once G\^ateaux
differentiable in $H$ with
\[D_xF(x,y)z=\frac{\partial f}{\partial
\si_1}(\cdot,x,y)z,\ \ \ \ D_xG(x,y)z=\frac{\partial g}{\partial
\si_1}(\cdot,x,y)z.\] Moreover, for any fixed $x \in\,H$, the
mapping $F(x,\cdot):H\to H$ is once G\^ateaux
differentiable and the mapping $G(x,\cdot):H\to H$ is three times G\^ateaux
differentiable, with
\[D_yF(x,y)z=\frac{\partial f}{\partial
\si_2}(\cdot,x,y)z,\]
and
\[D_y^jG(x,y)(z_i,\ldots,z_j)=\frac{\partial g^j}{\partial
\si_2^j}(\cdot,x,y)z_1\cdots z_j,\ \ \ \ j=1,2,3.\] Notice that if
$h \in\, L^\infty(0,L)$, then for any fixed $x,y \in\,H$ the
mappings $\le<F(\cdot,y),h\r>_H$ and $\le<F(x,\cdot),h\r>_H$ are
both
Fr\'echet differentiable and
\begin{equation}
\label{stimaderf} \sup_{x, y
\in\,H}\,|D\le<F(\cdot,y),h\r>_H(x)|_H\leq L_f\,|h|_H,\ \ \ \ \
\sup_{x, y \in\,H}\,|D\le<F(x,\cdot),h\r>_H(y)|_H\leq L_f\,|h|_H,
\end{equation}
where $L_f$ is the Lipschitz constant of $f$.
\section{Preliminary results on the fast motion equation}
\label{sec3}
As \eqref{h13} holds and the mappings $F,G:H\times H\to H$ are both
Lipschitz-continuous,
for any $\e>0$ and any initial conditions $x, y \in\,H$ system
\eqref{sistema} admits a unique mild solution $(u^\e,v^\e)
\in\,\mathcal{C}_{T,p}\times \mathcal{C}_{T,p}$, with $p\geq 1$
and $T>0$ (for a proof see e.g. \cite[Theorem 7.6]{dpz1}). This
means that there exist two unique processes $u^\e$ and $v^\e$,
both in $\mathcal{C}_{T,p}$, such that for any $t \in\,[0,T]$
\begin{equation}
\label{mild} u^\e(t)=e^{t A}x+\int_0^t e^{(t-s)A}
F(u^\e(s),v^\e(s))\,ds\end{equation}
and
\[v^\e(t)=e^{t B/\e}y+\frac 1\e \int_0^t e^{(t-s)B/\e} G(u^\e(s),v^\e(s))\,ds+\frac 1{\sqrt{\e}}
\int_0^t e^{(t-s)B/\e}\,dw(s).\]
\subsection{The fast motion equation}
\label{3.0}
Now, for any fixed $x \in\,H$ we consider the problem
\begin{equation}
\label{av5} \le\{
\begin{array}{l}
\ds{\frac{\partial v}{\partial t}(t,\xi)=\mathcal{B}
v(t,\xi)+g(\xi,x(\xi),v(t,\xi))+\frac{\partial w}{\partial
t}(t,\xi), \ \ \ \ \ t\geq 0,\ \ \ \xi \in\,[0,L],}\\
\vs \ds{v(0,\xi)=y(\xi),\ \ \ \xi \in\,[0,L],\ \ \ \ \
\mathcal{N}_2v\,(t,\xi)=0,\ \ \ \ t\geq 0,\ \ \ \xi \in\,\{0,L\}.}
\end{array}\r.
\end{equation}
By arguing as above, for any fixed slow component $x \in\,H$ and any
initial datum $y \in\,H$, equation \eqref{av5} admits a unique
mild solution in $\mathcal{C}_{T,p}$, which will be denoted by
$v^{x,y}(t)$.
Moreover, as proved for example in \cite[Proposition 8.2.2]{cerrai}, there exists $\theta>0$ such that for any $t_0>0$
and $p\geq 1$
\begin{equation}
\label{suppo} \sup_{t\geq t_0}\E\,|v^{x,y}(t)|_{C^\theta([0,L])}^p<\infty.
\end{equation}
\begin{Lemma}
\label{L3.1}
Under Hypotheses \ref{H1} and \ref{H3}, for any
$p\geq 1$ and $x,y \in\,H$
\begin{equation}
\label{wit1bis} \E\,|v^{x,y}(t)|_H^p\leq c_{p}\,\le(e^{-\d p
t}|y|_H^p+|x|_H^p+1\r),\ \ \ \ t\geq 0,
\end{equation}
where $\d:=(\la-L_{g})/2$.
\end{Lemma}
\begin{proof}
If we set $\rho(t):=v^{x,y}(t)-w^B(t)$, thanks to \eqref{av3} and
\eqref{av4} and to the Lipschitz-continuity of $G$, we have
\[\begin{array}{l}
\ds{\frac 12 \frac{d}{dt}|\rho(t)|_H^2}\\
\vs \ds{=\le<B
\rho(t),\rho(t)\r>_H+\le<G(x,\rho(t)+w^B(t))-G(x,w^B(t)),\rho(t)\r>_H+\le<G(x,w^B(t)),\rho(t)\r>_H}\\
\vs \ds{\leq
-(\la-L_{g})\,|\rho(t)|_H^2+c\,\le(|w^B(t)|_H+|x|_H+1\r)\,|\rho(t)|_H}\\
\vs \ds{\leq -\frac{\la-L_{g}}2\,|\rho(t)|_H^2+
c\,\le(|w^B(t)|_H^2+|x|^2_H+1\r)}
\end{array}\]
and, by comparison, it easily follows
\begin{equation}
\label{wit1} |v^{x,y}(t)|_H\leq |\rho(t)|_H+|w^B(t)|_H\leq
c\,\le(e^{-\frac{\la-L_{g}}2\,t}\,|y|_H+\sup_{s\geq
0}\,|w^B(s)|_H+|x|_H+1\r).\end{equation} In particular, if we set
$\d:=(\la-L_{g})/2$, as a consequence of estimate \eqref{eq17}
we obtain \eqref{wit1bis}.
\end{proof}
Since we are assuming that for each fixed $\si_1
\in\,\reals$ and almost all $\xi \in\,[0,L]$ the mapping
$g(\xi,\si_1,\cdot):\reals\to \reals$ is of class $C^3$, with
uniformly bounded derivatives, for any $T>0$ and $p\geq 1$ and for
any fixed slow variable $x \in\,H$ the mapping
\begin{equation}
\label{av6}
y \in\,H\mapsto v^{x,y} \in\,\mathcal{H}_{T,p},
\end{equation}
is three times continuously differentiable (for a proof and all
details see e.g. \cite[Theorem 4.2.4]{cerrai}).
The first order derivative $D_y v^{x,y}(t)h$, at the point $y
\in\,H$ and along the direction $h \in\,H$, is the solution of the
first variation equation
\[\le\{
\begin{array}{l}
\ds{\frac{\partial z}{\partial
t}(t,\xi)=\mathcal{B}z(t,\xi)+\frac{\partial g}{\partial
\si_2}(\xi,x(\xi),y(\xi))z(t,\xi),}\\
\vs \ds{z(0)=h,\ \ \ \ \ \mathcal{N}_2 z\,(t,\xi)=0,\ \ \ \ \xi
\in\,\{0,L\}.}
\end{array}
\r.\] Hence, thanks to \eqref{av3} and \eqref{av4}, it is
immediate to check that for any $t\geq 0$
\begin{equation}
\label{av7} \sup_{x,y \in\,H}\,|D_y v^{x,y}(t)h|_H\leq e^{-\d
t}\,|h|_H,\ \ \ \ \ \mathbb{P}-\mbox{a.s.}
\end{equation}
where, as in the previous lemma, $\d=(\la-L_{g})/2$. Moreover, as
shown in \cite[Lemma 4.2.2]{cerrai}, for any $1\leq r\leq p\leq
\infty$ and $h \in\,L^r(0,L)$ we have that $D_y v^{x,y}(t)h
\in\,L^p(0,L)$, $\mathbb{P}$-a.s. for $t>0$, and
\[\sup_{y
\in\,H}|D_y v^{x,y}(t)h|_{L^p}\leq \mu_{r,p}(t)\,
t^{-\frac{p-r}{2rp}}\,|h|_{L^r},\ \ \ \ \ \mathbb{P}-\mbox{a.s.}
\]
for a continuous increasing function $\mu_{r,p}$ which is
independent of $x \in\,H$.
Concerning the second and the third order derivatives, they are
respectively solutions of the second and of the third variation
equations. As proved in \cite[Proposition 4.2.6]{cerrai}, for any
$h_1,h_2,h_3 \in\,H$ and $p\geq 1$ both $D_y^2
v^{x,y}(t)(h_1,h_2)$ and $D_y^3 v^{x,y}(t)(h_1,h_2,h_3)$ belong to
$L^p(0,L)$, $\mathbb{P}$-a.s. for any $t\geq 0$, and
\begin{equation}
\label{av8} \sup_{y \in\,H}|D_y^j
v^{x,y}(t)(h_1,\ldots,h_j)|_{L^p}\leq
\nu^j_{r,p}(t)\prod_{i=1}^j|h_i|_H,\ \ \ \ \
\mathbb{P}-\mbox{a.s.}, \end{equation} for $j=2,3$. It is
important to notice that, as for $\mu_{r,p}$, due to the
boundedness assumption on the derivatives of the reaction term
$g$, all $\nu^j_{r,p}$ are
continuous increasing functions independent of $x\in\,H$.
\medskip
We conclude this subsection by proving the smooth dependence of
the solution $v^{x,y}(t)$ of equation \eqref{av5} on the frozen
slow component $x \in\,H$. In the space $\mathcal{H}_{T,2}$ we
introduce the equivalent norm
\[\||u\||:=\sup_{t \in\,[0,T]}\,e^{-\a t}\,\E\,|u(t)|_H^2,\]
for some $\a>0$. Moreover, for any $x \in\,H$ and $v
\in\,\mathcal{H}_{T,2}$ we define
\[
\mathcal{F}(x,v)(t):= e^{t B}y+\int_0^t e^{(t-s)
B}G(x,v(s))\,ds+w^B(t), \ \ \ t \in\,[0,T].\] If $\a$ is chosen
large enough, the mapping $\mathcal{F}(x,\cdot)$ is a contraction in
the space $\mathcal{H}_{T,2}$, endowed with the norm defined
above.
It is easy to show that for all $v \in\,\mathcal{H}_{T,2}$, the
mapping $x \in\,H\mapsto \mathcal{F}(x,v) \in\,\mathcal{H}_{T,2}$
is Fr\'echet differentiable and the derivative is continuous.
Furthermore, for all $x \in\,H$ the mapping $v
\in\,\mathcal{H}_{T,2}\mapsto \mathcal{F}(x,v)
\in\,\mathcal{H}_{T,2}$ is G\^ateaux differentiable and the
derivative is continuous. Hence, by using the generalized theorem
on contractions depending on a parameter given in
\cite[Proposition C.0.3]{cerrai}, we have that the solution
$v^{x,y}$ of equation \eqref{av5}, which is the fixed point of
the mapping $\mathcal{F}(x,\cdot)$, is differentiable with respect
to $x \in\,H$ and the derivative along the direction $h \in\,H$
satisfies the following equation
\[\frac{d\rho}{dt}(t)=[B +G_y(x,v^{x,y}(t))]\,\rho(t)+G_x(x,v^{x,y}(t))h,\ \ \ \ \ \
\rho(0)=0.\] According to Hypothesis \ref{H3}, we have
\[\begin{array}{l}
\ds{\frac 12 \frac{d}{dt}|\rho(t)|_H^2=\le<[B
+G_y(x,v^{x,y}(t))]\rho(t),\rho(t)\r>_H+\le<G_x(x,v^{x,y}(t))h,\rho(t)\r>_H}\\
\vs \ds{\leq
-\frac{\la-L_{g}}2\,|\rho(t)|_H^2+|G_x(x,v^{x,y}(t))|^2_{\mathcal{L}(H)}|h|_H^2,}
\end{array}\]
so that, due to the boundedness of $G_x$,
\begin{equation}
\label{finis} \sup_{x, y \in\,H}\,|D_x v^{x,y}(t)h|_H\leq c\,
e^{-\frac{\la-L_{g}}2\,t}|h|_H,\ \ \ \ \ \mathbb{P}-\text{a.s.}
\end{equation}
\subsection{The fast transition semigroup}
\label{3.1}
For any fixed $x \in\,H$, we denote by $P^x_t$, $t\geq 0$, the
transition semigroup
associated with the fast equation \eqref{av5} with frozen slow component $x$. For any $\varphi
\in\,B_b(H)$ and $t\geq0$, it is defined by
\[P_t^x \varphi(y)=\E\,\varphi(v^{x,y}(t)),\ \ \ \ \ y \in\,H.\]
As the mapping introduced in
\eqref{av6} is differentiable and \eqref{av7} holds, it is
immediate to check that $P^x_t$ is a Feller contraction semigroup and maps $C_b(H)$ into itself.
Thanks to estimate \eqref{wit1bis} and to \eqref{fi}, the
semigroup $P^x_t$ is well defined on $\text{Lip}(H)$ and for any
$\varphi \in\,\text{Lip}(H)$, $x,y \in\,H$ and $t\geq 0$
\begin{equation}
\label{lippt} |P^x_t \varphi(y)|\leq [\varphi]_{\text{\tiny
Lip}}\,\E\,|v^{x,y}(t)|_H+|\varphi(0)|\leq
c\,[\varphi]_{\text{\tiny Lip}}\,(1+|x|_H+|y|_H)+|\varphi(0)|.
\end{equation}
Furthermore, $P^x_t$ maps $\text{Lip}(H)$ into itself and
according to \eqref{av7}
\begin{equation}
\label{lipptbis}
[P^x_t\varphi]_{\text{\tiny Lip}}\leq e^{-\d t}[\varphi]_{\text{\tiny
Lip}},\ \ \ \ \ t\geq 0.
\end{equation}
As known, the semigroup $P^x_t$ is not strongly continuous on
$C_b(H)$, in general. Nevertheless, it is {\em weakly continuous}
on $C_b(H)$ (for a definition and all details we refer to
\cite[Appendix B]{cerrai}). For any $\la>0$ and $\varphi
\in\,C_b(H)$, we set
\[F^x(\la)\varphi(y):=\int_0^\infty e^{-\la t}P^x_t\varphi(y)\,dt,\ \ \
\ \ x,y \in\,H.\] As proved in \cite[Proposition B.1.3 and
Proposition B.1.4]{cerrai}, since $P^x_t$ is a weakly continuous
semigroup, for any $\la>0$ and $x \in\,H$ the linear operator
$F^x(\la)$ is bounded from $C_b(H)$ into itself and there exists a
unique closed linear operator $L^x:D(L)\subseteq C_b(H)\to C_b(H)$
such that
\[F^x(\la)=R(\la,L^x)\ \ \ \ \ \ \la>0.\]
Such an operator is, by definition, the {\em infinitesimal weak
generator} of $P^x_t$.
It is important to stress that, thanks to \eqref{lippt} and
\eqref{lipptbis}, the operator $F^x(\la)$ is also well defined
from $\text{Lip}(H)$ into itself.
\medskip
Concerning the regularity properties of $P^x_t$,
as the mapping \eqref{av6} is three times
continuously differentiable, by differentiating under the sign of
expectation, for any $t\geq 0$ and $k\leq 3$ we get
\[\varphi \in\,\text{Lip}^k(H)\Longrightarrow P^x_t\varphi
\in\,\text{Lip}^k(H),\] and thanks to estimates \eqref{av7}, for
$k=1$, and \eqref{av8}, for $k=2,3$,
\[\sup_{x \in\,H}[P^x_t\varphi]_k\leq c_k(t)\,\sum_{1\leq h\leq k}[\varphi]_h,\ \ \ \ \ t \geq 0,\]
where $c_k(t)$ is some continuous increasing function. Moreover,
the semigroup $P^x_t$ has a smoothing effect. Actually, as proved
in \cite[Theorem 4.4.5]{cerrai}, for any $t>0$
\[\varphi \in\,B_b(H)\Longrightarrow P^x_t\varphi \in\,C^3_b(H),\]
and for any $0\leq i\leq j\leq 3$
\begin{equation}
\label{eq13}
\sup_{x \in\,H}\|D^j (P^x_t\varphi)\|_0\leq
c\,(t\wedge 1)^{-\frac{j-i}2}\,\|\varphi\|_i,\ \ \ \ \ t>0.
\end{equation}
By adapting the arguments used in \cite[Theorem 4.4.5]{cerrai}, it
is possible to prove that if $\varphi \in\,\text{Lip}(H)$, then
$P^x_t\varphi$ is three times continuously differentiable, for any
$t>0$. Moreover, the following estimates for the derivatives of
$P^x_t\varphi$ hold
\begin{equation}
\label{deriprima} [P^x_t\varphi]_1=\sup_{y \in\,H}\,|D
P^x_t\varphi(y)|_H\leq e^{-\d t}\,[\varphi]_{\text{\tiny Lip}},
\end{equation}
and for $j=2,3$
\begin{equation}
\label{derisucc} |D^j P^x_t\varphi(y)|_{\mathcal{L}^j(H)}\leq
c\,(t\wedge 1)^{-\frac{j-1}2}\,\le([\varphi]_{\text{\tiny
Lip}}(1+|x|_H+|y|_H)+|\varphi(0)|\r).
\end{equation}
Moreover, by adapting the proof of \cite[Theorem 5.2.4]{cerrai}, which is
given for bounded functions, to the case of general Lipschitz-continuous functions, it is possible to prove the following
crucial fact.
\begin{Theorem}
\label{av11} Under Hypotheses \ref{H1} and \ref{H3}, the operator
$D^2(P^x_t \varphi)(y)$ belongs to $\mathcal{L}_1(H)$, for any
fixed $x,y \in\,H$, $t>0$ and $\varphi \in\,\text{{\em Lip}}(H)$.
Besides, the mapping
\[(t,y) \in\,(0,\infty)\times H\mapsto
\mbox{{\em Tr}}\,[D^2(P^x_t\varphi)(y)] \in\,\reals,\] is
continuous and
\begin{equation}
\label{eq14} \le|\mbox{{\em Tr}}\,[D^2(P^x_t\varphi)(y)]\r|\leq
c_\gamma\,(t\wedge
1)^{-\frac{1+\gamma}2}\le(\,[\varphi]_{\text{{\em \tiny
Lip}}}(1+|x|_H+|y|_H)+|\varphi(0)|\r),
\end{equation}
where $\gamma$ is the constant introduced in \eqref{h11}.
\end{Theorem}
\begin{Remark}
{\em Even if the semigroup $P^x_t$ has a smoothing effect, the
proof of the validity of the trace-class property for the operator $D^2(P^x_t
\varphi)(y)$ is far from being trivial. Actually, it is based on
the two following facts. First (see \cite[Lemma 5.2.1]{cerrai}),
if $\{e_k\}_{k \in\,\nat}$ is the orthonormal basis introduced in
Hypothesis \ref{H1} and if $\gamma <1$ is the constant
introduced in \eqref{h11}, then it holds
\[\sup_{y \in\,H}\,\sum_{k=1}^\infty \int_0^t|D_y
v^{x,y}(s)e_k|_H^2\,ds\leq c(t)\,t^{1-\gamma},\ \ \ \
\mathbb{P}-\mbox{a.s.},\] for some continuous increasing function
$c(t)$ independent of $x \in\,H$. Secondly (see \cite[Lemma
5.2.2]{cerrai}), there exists some continuous increasing function
$c(t)$ such that for any $N \in\,\mathcal{L}(H)$ and $x \in\,H$
\[\sup_{y \in\,H}\sum_{k=1}^\infty \int_0^t|D^2_y
v^{x,y}(s)(e_k,N e_k)|_H^2\,ds\leq c(t)\,\|N\|,\ \ \ \
\mathbb{P}-\mbox{a.s.}\]
It is important to stress that both the estimate for the first
derivative and the estimate for the second derivative are a
consequence of \eqref{wit9} and \eqref{h11}.
}
\end{Remark}
\subsection{The asymptotic behavior of the fast equation}
\label{3.2}
We describe here the asymptotic behavior of the semigroup $P^x_t$.
Namely, we show that, for any fixed $x \in\,H$, it admits a
unique invariant measure $\mu^x$ which is explicitly given and we
describe the convergence of $P^x_t$ to equilibrium. Most of these
results are basically known in the literature, but we shortly
recall them for the reader convenience.
\medskip
According to \eqref{wit9}, the self-adjoint operator
\[\int_0^\infty e^{2 s B}\,ds=\frac 12\,(-B)^{-1}\]
is well defined in $\mathcal{L}_1(H)$, so that the Gaussian
measure $\mathcal{N}(0,(-B)^{-1}/2)$ of zero mean and covariance
operator $(-B)^{-1}/2$ is well defined on $(H,\mathcal{B}(H))$.
For any $x, y \in\,H$ we define
\[U(x,y):=\int_0^1 \le<G(x,\theta y),y\r>_H\,d\theta=
\int_0^L\int_0^{y(\xi)} g(\xi,x(\xi),s)\,ds\,d\xi.\] Due to the
Lipschitz-continuity of $g(\xi,\cdot)$ (see Hypothesis \ref{H3}),
for any $x, y \in\,H$ we have
\begin{equation}
\label{G} |G(x,y)|_H\leq
L_{g}\,|y|_H+c\,|x|_H+|G(0,0)|_H,\end{equation}
so that
for any $\eta>0$ we can fix a constant $c_\eta\geq 0$ such that
\[|U(x,y)|\leq
\frac{L_{g}+\eta}2\,|y|_H^2+c_\eta\,(1+|x|_H^2).\] As $\eta>0$ can be chosen as small as we wish, thanks to
\eqref{av4} this implies that for any fixed $x \in\,H$ the
mapping
\[y \in\,H\mapsto \exp 2 U(x,y) \in\,\reals\]
is integrable with respect to the Gaussian measure
$\mathcal{N}(0,(-B)^{-1}/2)$ and
\[Z(x):=\int_H \exp 2 U(x,y)\,\mathcal{N}(0,(-B)^{-1}/2)\,dy
\in\,(0,\infty),\ \ \ \ \ x \in\,H.\] This means that for each
fixed $x \in\,H$ the measure
\begin{equation}
\label{3.17}
\mu^x(dy):=\frac 1{Z(x)}\,\exp 2
U(x,y)\,\mathcal{N}(0,(-B)^{-1}/2)\,(dy)
\end{equation}
is well defined on $(H,\mathcal{B}(H))$.
Now, it is immediate to check that the mapping $U(x,\cdot):H\to
\reals$ is differentiable and
\begin{equation}
\label{dery} U_y(x,y)=G(x,y),\ \ \ \ x,y \in\,H.
\end{equation}
Therefore, as well known from the existing literature, the
measure $\mu^x$ defined in \eqref{3.17} is invariant for equation \eqref{av5}.
Because of the way the measure $\mu^x$ has been constructed, we
immediately have that it has all moments finite. In particular,
for any $x \in\,H$ we have
\begin{equation}
\label{lip} \text{Lip}(H)\subset L^p(H,\mu^x),\ \ \ \ p\geq
1.\end{equation} In the next lemma we show how the moments of
$\mu^x$ can be estimated in terms of the slow variable $x$.
\begin{Lemma}
\label{tight} Under Hypotheses \ref{H1} and \ref{H3}, for any $x
\in\,H$ and $p\geq 1$
\begin{equation}
\label{wit22} \int_H |z|^p_H\,\mu^x(dz)\leq c\,\le(1+|x|^p_H\r).
\end{equation}
\end{Lemma}
\begin{proof}
By using the invariance of $\mu^x$, thanks to estimate
\eqref{wit1bis} for any $p\geq 1$ and $t\geq 0$ we have
\[\begin{array}{l}
\ds{\int_H|z|_H^p\,\mu^x(dz)=\int_H
P^x_t|z|_H^p\,\mu^x(dz)=\int_H\E\,|v^{x,z}(t)|_H^p\,\mu^x(dz)}\\
\vs \ds{\leq c\,e^{-\d p
t}\int_H|z|_H^p\,\mu^x(dz)+c\,(1+|x|_H^p)}
\end{array}\]
Then, if we take $t=t_0$ such that $c\,e^{-\d p t_0}<1$, we have
\eqref{wit22}.
\end{proof}
Once we have the explicit invariant measure $\mu^x$, we show that
it is unique and we describe its convergence to equilibrium.
\begin{Theorem}
\label{ergo} Under Hypotheses \ref{H1} and \ref{H3}, for any
fixed $x \in\,H$ equation \eqref{av5} admits a unique ergodic
invariant measure $\mu^x$, which is strongly mixing and such that
for any $\varphi \in\,B_b(H)$ and $x, y \in\,H$
\begin{equation}
\label{mis5} \le|P^x_t\varphi(y)-\int_H
\varphi(z)\,\mu^x(dz)\r|\leq c\,\le(1+|x|_H+|y|_H\r)\,e^{-\d
t}(t\wedge 1)^{-\frac 12}\,\|\varphi\|_0, \end{equation} where
$\d:=(\la-L_{g})/2$.
\end{Theorem}
\begin{proof}
We fix $y,z \in\,H$ and set $\rho(t):=v^{x,y}(t)-v^{x,z}(t)$. We
have
\[\frac 12\frac{d}{dt}|\rho(t)|_H^2=\le<B \rho(t),\rho(t)\r>_H+\le<G(x,v^{x,y}(t))-G(x,v^{x,z}(t)),
\rho(t)\r>_H,\] and then, according to \eqref{av3} and
\eqref{av4}, we easily get
\[|v^{x,y}(t)-v^{x,z}(t)|_H^2=|\rho(t)|_H^2\leq e^{-2(\la-L_{g})t}|y-z|_H^2,\ \ \ \ \mathbb{P}-\text{a.s}.\]
This means that for any $\varphi \in\,\text{Lip}(H)$
\[ |P^x_t\varphi(y)-P^x_t\varphi(z)|\leq [\varphi]_{\text{\tiny Lip}}
\,\E\,|v^{x,y}(t)-v^{x,z}(t)|_H\leq [\varphi]_{\text{\tiny Lip}}
\,e^{-(\la-L_{g})t}|y-z|_H,\ \ \ \ t\geq 0.
\]
Hence, if $\varphi \in\,B_b(H)$, due to the
semigroup law and to estimate \eqref{eq13} (with $j=1$ and $i=0$),
for any $t>0$ we obtain
\begin{equation}
\label{wit8}
\begin{array}{l} \ds{|P^x_t\varphi(y)-P^x_t\varphi(z)|
\leq
[P^x_{t/2}\varphi]_1\,e^{-\d\,t}\,|y-z|_H
\leq c\,\|\varphi\|_0 \,(t\wedge 1)^{-\frac 12}\,e^{-\d\,
t}\,|y-z|_H,}
\end{array}\end{equation}
where $\d:=(\la-L_{g})/2$. In particular,
\[\lim_{t\to \infty} P^x_t\varphi(y)-P^x_t\varphi(z)=0,\]
so that the invariant measure $\mu^x$ is unique and strongly
mixing.
Now, due to the invariance of $\mu^x$, if $\varphi \in\,B_b(H)$
from \eqref{wit8} we have
\[\begin{array}{l}
\ds{\le|P^x_t\varphi(y)-\int_H
\varphi(z)\,\mu^x(dz)\r|=\le|\int_H\le[P^x_t\varphi(y)-P^x_t\varphi(z)\r]\,\mu^x(dz)\r|}\\
\vs \ds{\leq c\,\|\varphi\|_0 \,e^{-\d\, t}\,(t\wedge
1)^{-\frac 12}\int_H|y-z|_H\,\mu^x(dz)}\\
\vs \ds{\leq c\,\|\varphi\|_0\,e^{-\d\, t}\,(t\wedge 1)^{-\frac
12}\le(|y|_H+\int_H|z|_H\,\mu^x(dz)\r).}
\end{array}\]
and then, thanks to \eqref{wit22} (with $p=1$), we obtain
\eqref{mis5}.
\end{proof}
\begin{Remark}
{\em From the proof of estimate \eqref{mis5}, we immediately see
that if $\varphi \in\,\text{Lip}(H)$, then for any $x,y \in\,H$
\begin{equation}
\label{mis5bis} \le|P^x_t\varphi(y)-\int_H
\varphi(z)\,\mu^x(dz)\r|\leq c\,\le(1+|x|_H+|y|_H\r)\,e^{-\d
t}\,[\varphi]_{\text{\tiny Lip}} ,
\end{equation} where $\d:=(\la-L_{g})/2$.}
\end{Remark}
\subsection{The Kolmogorov equation associated with the fast
equation} \label{3.3}
For any frozen slow component $x \in\,H$, the Kolmogorov operator
associated with equation \eqref{av5} is given by the following
second order differential operator
\[\mathcal{L}^x \varphi(y)=\frac12 \mbox{Tr}\,[D^2
\varphi(y)]+\le<By+G(x,y),D\varphi(y)\r>_H,\ \ \ \ \ y
\in\,D(B).\] The operator $\mathcal{L}^x$ is defined for functions
$\varphi:H\to \reals$ which are twice continuously
differentiable, such that the operator $D^2 \varphi(y)$ is in
$\mathcal{L}_1(H)$, for all $y \in\,H$, and the mapping \[y
\in\,H\mapsto \mbox{Tr}\,D^2 \varphi(y) \in\,\reals,\] is
continuous. In what follows it will be important to study the
solvability of the elliptic equation associated with the infinite
dimensional operator $\mathcal{L}^x$
\begin{equation}
\label{av12} \la \varphi(y)-\mathcal{L}^x \varphi(y)=\psi(y),\ \ \
\ \ y \in\,D(B),
\end{equation}
for any fixed $x \in\,H$, $\la>0$ and $\psi:H\to \reals$ regular
enough. To this purpose we recall the notion of {\em strict}
solution for the elliptic problem \eqref{av12}.
\begin{Definition}
\label{def3} A function $\varphi$ is a strict solution of problem
\eqref{av12} if
\begin{enumerate}
\item $\varphi$ belongs to $D(\mathcal{L}^x)$, that is $\varphi:H\to \reals$
is twice continuously differentiable, the operator $D^2\varphi(y)
\in\,\mathcal{L}_1(H)$, for any $y \in\,H$, and the mapping
$y\mapsto \mbox{{\em Tr}}\,D^2\varphi(y)$ is continuous on $H$
with values in $\reals$;
\item $\varphi(y)$ satisfies \eqref{av12}, for any $y \in\,D(B)$.
\end{enumerate}
\end{Definition}
In the next theorem we see how it is possible to get the
existence of a strict solution of problem \eqref{av12} in terms of
the Laplace transform of the semigroup $P^x_t$ (see subsection
\ref{3.1} for the definition and \cite[Theorem 5.4.3]{cerrai} for
the proof).
\begin{Theorem}
\label{av14} Fix any $x \in\,H$ and $\la>0$. Then under Hypotheses
\ref{H1} and \ref{H3}, for any $\psi \in\,\text{{\em Lip}}(H)$ the
function
\[y \in\,H\mapsto \varphi(x,y):=\int_0^\infty e^{-\la
t}P^x_t\psi(y)\,dt \in\,\reals,\] is a strict solution of problem
\eqref{av12}.
\end{Theorem}
\begin{Remark}
{\em A detailed proof of the theorem above can be found in
\cite[Theorem 5.4.3]{cerrai} in the case $\psi \in\,C^1_b(H)$. The case of $\psi \in\,\text{Lip}(H)$ is analogous:
we have to
prove that for any $\psi \in\,\text{{ Lip}} (H)$ the function
$R(\la,L^x)\psi$ is a strict solution. To this purpose, by using
\eqref{deriprima} and \eqref{derisucc}, we have that
$R(\la,L^x)\psi$ is twice continuously differentiable and then,
thanks to Theorem \ref{av11} and estimate \eqref{eq14}, we have
that $D^2[R(\la,L^x)\psi] \in\,\mathcal{L}_1(H)$ and
continuity for the trace holds. Notice that in all these results
it is crucial that $\psi \in\,\text{{ Lip}} (H)$, because in this
case all singularities arising at $t=0$ are integrable. }
\end{Remark}
\section{A priori bounds for the solution of the system}
\label{sec4}
With the notations introduced in section \ref{sec2}, system
\eqref{sistema} can be written as \begin{equation}
\label{astratta} \le\{
\begin{array}{l}
\ds{ \frac{d u^\e}{dt}(t)=A u^\e(t)+F(u^\e(t),v^\e(t)),\ \ \ \
u^\e(0)=x,}\\
\vs \ds{ dv^\e(t)=\frac 1\e\,\le[B
v^\e(t)+G(u^\e(t),v^\e(t))\r]\,dt+\frac 1{\sqrt{\e}}\,dw(t),\ \ \
\ v^\e(0)=y.}
\end{array}\r.
\end{equation}
Our aim here is proving uniform bounds with respect to $\e>0$ for
the solutions $u^\e$ and $v^\e$ of system \eqref{astratta}.
\begin{Lemma}
\label{stime1} Under Hypotheses \ref{H1}, \ref{H2} and \ref{H3},
for any $x, y \in\,H$ and $T>0$ we have
\begin{equation}
\label{stima11} \sup_{\e>0}\,\E \sup_{t \in\,[0,T]}|u^\e(t)|_H^2\leq
c_T\le(1+|x|_H^2+|y|^2_H\r),
\end{equation}
and
\begin{equation}
\label{stima12} \sup_{\e>0}\,\sup_{t
\in\,[0,T]}\E\,|v^\e(t)|_H^2\leq c_T\le(1+|x|_H^2+|y|^2_H\r),
\end{equation}
for some constant $c_T>0$.
\end{Lemma}
\begin{proof}
We have
\[\begin{array}{l}
\ds{\frac 12 \frac{d}{dt}|u^\e(t)|_H^2=\le<A u^\e(t),
u^\e(t)\r>_H+\le<F(u^\e(t),v^\e(t))-F(0,v^\e(t)),u^\e(t)\r>_H}\\
\vs \ds{+\le<F(0,v^\e(t)),u^\e(t)\r>_H\leq
c\,|u^\e(t)|_H^2+c\,\le(1+|v^\e(t)|_H^2\r),}
\end{array}\]
so that
\begin{equation}
\label{wit10} |u^\e(t)|_H^2\leq e^{ct}\,|x|_H^2+c\,\int_0^t
e^{c(t-s)}\,\le(1+|v^\e(s)|_H^2\r)\,ds.
\end{equation}
Now, for any $\e>0$ we denote by $w^{\e,B}(t)$ the solution of the
problem
\[dz(t)=\frac 1\e Bz(t)\,dt+\frac 1{\sqrt{\e}}\, dw(t),\ \ \ \ \ \
z(0)=0.\] We have \[w^{\e,B}(t)=\frac 1{\sqrt{\e}} \int_0^t
e^{(t-s)B/\e}\,dw(s),\] and, due to \eqref{wit9}, with a simple
change of variables we get
\[\begin{array}{l}
\ds{\E\,|w^{\e, B}(t)|_H^2=\frac
1\e\,\int_0^{t}\|e^{(t-s)B/\e}\|_2^2\,ds=\int_0^{t/\e}\|e^{\rho
B}\|_2^2\,d\rho\leq c\int_0^\infty (\rho\wedge
1)^{-\gamma}e^{-2\la \rho}\,d\rho<\infty.}
\end{array}\]
This means that
\begin{equation}
\label{wit20} \sup_{\e>0}\,\sup_{t\geq 0}\,\E\,|w^{\e,
B}(t)|_H^2<\infty.
\end{equation}
Notice that the same uniform bound is true for moments of any
order of the $H$-norm of $w^{\e, B}(t)$.
If we set $\rho^\e(t):=v^\e(t)-w^{\e, B}(t)$, by proceeding as in
the proof of Lemma \ref{L3.1} we have
\[\frac 12 \frac d{dt}\,|\rho^\e(t)|_H^2\leq
-\frac{\la-L_{g}}{2 \e}\,|\rho^\e(t)|_H^2+\frac c
\e\,\le(1+|u^\e(t)|_H^2+|w^{\e, B}(t)|_H^2\r).\] Hence, by
comparison
\begin{equation}
\label{dea1} |\rho^\e(t)|_H^2\leq
e^{-\frac{\la-L_{g}}{\e}\,t}\,|y|_H^2+\frac c \e\,\int_0^t
e^{-\frac{\la-L_{g}}{\e}\,(t-s)}\le(1+|u^\e(s)|_H^2+|w^{\e,
B}(s)|_H^2\r)\,ds.
\end{equation}
According to \eqref{wit10}, this
implies
\[\begin{array}{l}
\ds{|v^\e(t)|_H^2\leq 2\,|w^{\e,
B}(t)|_H^2+2\,|\rho^\e(t)|_H^2\leq 2\,|w^{\e,
B}(t)|_H^2+c_T\,\le(1+|x|_H^2+|y|_H^2\r)}\\
\vs
\ds{+\frac{c_T}\e\int_0^te^{-\frac{\la-L_{g}}{\e}\,(t-s)}\int_0^s
|v^\e(r)|_H^2\,dr\,ds+\frac
c\e\int_0^te^{-\frac{\la-L_{g}}{\e}\,(t-s)}|w^{\e,
B}(s)|_H^2\,ds}
\end{array}\]
and by taking expectation, thanks to \eqref{wit20} we have
\[\begin{array}{l}
\ds{\E\,|v^\e(t)|_H^2\leq
c_T\,\le(1+|x|_H^2+|y|_H^2\r)}\\
\vs
\ds{+\frac{c_T}\e\int_0^te^{-\frac{\la-L_{g}}{\e}\,(t-s)}\int_0^s
\E\,|v^\e(r)|_H^2\,dr\,ds+\frac
c\e\int_0^te^{-\frac{\la-L_{g}}{\e}\,s}\,ds.}
\end{array}\]
With a change of variables, this yields
\[\begin{array}{l}
\ds{\E\,|v^\e(t)|_H^2\leq
c_T\,\le(1+|x|_H^2+|y|_H^2\r)}\\
\vs \ds{+c_T\int_0^t\le[\int_0^{\frac{t-r}\e}
e^{-(\la-L_{g})\,\si}\,d\si\r]\E\,|v^\e(r)|_H^2\,dr+c\int_0^{\frac t \e}
e^{-(\la-L_{g})\,s}\,ds}\\
\vs \ds{\leq
c_T\,\le(1+|x|_H^2+|y|_H^2\r)+c_T\int_0^t\E\,|v^\e(r)|_H^2\,dr,}
\end{array}\]
so that
\[\sup_{t \in\,[0,T]}\,\E\,|v^\e(t)|_H^2\leq c_T\,\le(1+|x|_H^2+|y|_H^2\r), \]
which gives \eqref{stima12}. By replacing the estimate above in
\eqref{wit10}, we immediately obtain \eqref{stima11}.
\end{proof}
\begin{Remark}
{\em In the previous lemma we have proved uniform bounds, with
respect to $\e>0$, for $\sup_{t \in\,[0,T]}\,\E\,|v^\e(t)|_H$ and
not for $\E\,\sup_{t \in\,[0,T]}\,|v^\e(t)|_H$. This is a
consequence of the fact that we can only prove the following
estimate for the second moment of the $C([0,T];H)$-norm of the stochastic
convolution $w^{\e,B}$
\[\E\,\sup_{t\in\,[0,T]}|w^{\e, B}(t)|^2_H\leq c_{T,\d} \,\e^{\d-1},\ \ \ \ \
t \in\,[0,T],\] for any $0<\d<1/2$. Then, due to the previous
estimate, we are only able to prove that
\begin{equation}
\label{dea} \E\,\sup_{t \in\,[0,T]}\,|v^\e(t)|_H^2\leq
c_{T,\d}\le(1+|x|^2_H+|y|^2_H+\e^{\,\d-1}\r),\ \ \ \ \ \e>0.
\end{equation}}
\end{Remark}
\begin{Theorem}
\label{tightness}
Assume that $x \in\,W^{\a,2}(0,L)$, for some
$\a>0$. Then, under Hypotheses \ref{H1}, \ref{H2} and \ref{H3},
for any $T>0$ the family of probability measures
$\{\,\mathcal{L}(u^\e)\,\}_{\e
>0}$ is tight in $C((0,T];H)\cap L^\infty(0,T;H)$.
\end{Theorem}
\begin{proof}
As known, if $\d\leq 1/4$
\begin{equation}
\label{analytic}
W^{2\d,2}(0,L)=(H,W^{2,2}_{\mathcal{N}_i}(0,L))_{\d,\infty}=\le\{\,x
\in\,H\,:\,\sup_{t \in\,(0,1]}t^{-\d}\,|e^{t
A}x-x|_H<\infty\,\r\},
\end{equation}
with equivalence of norms.
Moreover, if $f \in\,L^2(0,T;H)$, it is possible to prove that for any $t>s$ and $\d<1/2$
\[\le|\int_0^t e^{(t-r)A}f(r)\,dr-\int_0^s e^{(s-r)A}
f(r)\,dr\r|_H\leq c_{T,\d}\,(t-s)^\d\|f\|_{L^2(0,T;H)}.\] Then, if $x
\in\,W^{\a,2}(0,L)$, for any $t>s$ and $\theta\leq 1/4\wedge \a/2$ we
have
\[ \begin{array}{l} \ds{|u^\e(t)-u^\e(s)|_H}\\
\vs \ds{\leq \le|e^{s A}(e^{(t-s)A}x-x)\r|_H+\le|\int_0^t
e^{(t-r)A}F(u^\e(r),v^\e(r))\,dr-\int_0^s e^{(s-r)A}
F(u^\e(r),v^\e(r))\,dr\r|_H}\\
\vs \ds{ \leq
c_{T,\theta}\,(t-s)^{\theta}\,|x|_{2\theta,2}+c_{T,\theta}\,(t-s)^\theta\|F(u^\e,v^\e)\|_{L^2(0,T;H)}.}
\end{array}
\] This implies that for any $\theta\leq 1/4\wedge \a/2$
\begin{equation}
\label{holder} [u^\e]_{C^\theta([0,T];H)}=\sup_{s\neq
t}\,\frac{|u^\e(t)-u^\e(s)|}{|t-s|^{\theta}}\leq
c_T\,\le(|x|_{2\theta,2}+\|F(u^\e,v^\e)\|_{L^2(0,T;H)}\r).\end{equation}
Now, according to estimates \eqref{stima11} and \eqref{stima12},
we have
\begin{equation}
\label{wit28} \E\,\|F(u^\e,v^\e)\|_{L^2(0,T;H)}\leq
c\le(1+\E\,|u^\e|_{L^2([0,T];H)}+\E\,|v^\e|_{L^2([0,T];H)}\r)\leq
c_T\le(1+|x|_H+|y|_H\r),
\end{equation}
and hence
\begin{equation}
\label{wit27} \sup_{\e>0}\,\E\,|u^\e|_{C^{\theta}([0,T];H)}\leq
c_T\le(1+|x|_{2\theta,2}+|y|_H\r).
\end{equation}
Next, if $\theta<1/2\wedge \a$, thanks to \eqref{wit21} for any $t
\in\,[0,T]$ we have
\[\begin{array}{l}
\ds{|u^\e(t)|_{\theta,2}\leq c_T\,|x|_{\theta,2}+c_T\int_0^t
(t-s)^{-\frac \theta 2}\,|F(u^\e(s),v^\e(s))|_H\,ds}\\
\vs \ds{\leq
c_T\,|x|_{\theta,2}+c_{T,\theta}\,\|F(u^\e,v^\e)\|_{L^2(0,T;H)}.}
\end{array}\] Then, by using again \eqref{wit28}, we have
\begin{equation}
\label{wit29} \sup_{\e>0}\,\E\sup_{t
\in\,[0,T]}|u^\e(t)|_{\theta,2}\leq c_{T,\theta}\le(1+|x|_{\theta,2}+|y|_H\r).
\end{equation}
Combining together \eqref{wit27} and \eqref{wit29}, we conclude
that for any $\eta>0$ there exists $R(\eta)>0$ such that
\[\mathbb{P}\le(u^\e \in\,\mathcal{K}_{R(\eta)}\r)\geq 1-\eta,\ \ \ \ \
\e>0,\] where, by the Ascoli-Arzel\`a theorem, $\mathcal{K}_{R(\eta)}$ is
the compact subset of $C((0,T];H)\cap L^\infty(0,T;H)$ defined by
\[\mathcal{K}_{R(\eta)}:=\le\{\,u \,:\,\sup_{t
\in\,[0,T]}|u(t)|_{\theta,2}\leq R(\eta),\ |u|_{C^{\theta}([0,T];H)}\leq
R(\eta)\,\r\},\] for some $\theta<1/4\wedge \a/2$. This implies the tightness of
the family $\{\mathcal{L}(u^\e)\}_{\e>0}$ in $C((0,T];H)\cap L^\infty(0,T;H)$.
\end{proof}
We conclude the present section by proving that if $x$ and $y$ are taken
in $W^{\a,2}(0,L)$, for some $\a>0$, then $u^\e(t) \in\,D(A)$, for
$t>0$. Moreover, we provide an estimate for the momentum of the norm of
$A u^\e(t)$, which is uniform with respect to $\e \in\,(0,1]$.
\begin{Lemma}
\label{lemma4.4}
Assume that $x, y \in\,W^{\a,2}(0,L)$, for some
$\a \in\,(0,2]$. Then, under Hypotheses \ref{H1}, \ref{H2} and \ref{H3},
we have that $u^\e(t) \in\,D(A)$, $\mathbb{P}$-{\em a.s.}, for any
$t>0$ and $\e>0$. Moreover, for any $T>0$ and $\e \in\,(0,1]$ it holds
\begin{equation}
\label{cosimo} \E\,|A u^\e(t)|_H\leq c\,t^{\frac \a
2-1}\,|x|_{\a,2}+c_T\,(1+\e^{-\frac {\a\vee (1-\gamma)}
2})\,(1+|x|_{\a,2}+|y|_{\a,2}),\ \ \ \ t \in\,(0,T],
\end{equation}
where $\gamma$ is the constant introduced in \eqref{h11}.
\end{Lemma}
\begin{proof}
We decompose $u^\e(t)$ as
\[\begin{array}{l}
\ds{u^\e(t)=u^\e_1(t)+u^\e_2(t):=\le[e^{t A} x+\int_0^t e^{(t-s)A}\,F(u^\e(t),v^{\e}(t))\,ds\r]}\\
\vs \ds{+\int_0^t
e^{(t-s)A}\,\le[F(u^\e(s),v^{\e}(s))-F(u^\e(t),v^{\e}(t))\r]\,ds.}
\end{array}\]
According to \eqref{wit21} we have
\[\begin{array}{l}
\ds{|A u^\e_1(t)|_H\leq |A e^{t A} x|_H +|(e^{t
A}-I)F(u^\e(t),v^{\e}(t))|_H}\\
\vs \ds{\leq c_T\,t^{\frac \a
2-1}\,|x|_{\a,2}+c_T\le(1+|u^\e(t)|_H+|v^\e(t)|_H\r),}
\end{array}\]
so that, thanks to \eqref{stima11} and \eqref{stima12}
\begin{equation}
\label{u1} \E\,|A u^\e_1(t)|_H\leq c_T\,t^{\frac \a
2-1}\,|x|_{\a,2}+c_T\le(1+|x|_H+|y|_H\r),\ \ \ \ t \in\,(0,T].
\end{equation}
Concerning $u^\e_2(t)$, we have
\[\begin{array}{l}
\ds{|A u^\e_2(t)|_H\leq c_T\int_0^t
(t-s)^{-1}\le(\,|u^\e(t)-u^\e(s)|_H+|v^\e(t)-v^\e(s)|_H\r)\,ds}\\
\vs \ds{\leq c_T\int_0^t (t-s)^{\frac \a
2-1}\,ds\,[u^\e]_{C^{\frac \a
2}(0,T;H)}+c\int_0^t(t-s)^{-1}|v^\e(t)-v^\e(s)|_H\,ds,}
\end{array}\]
and then, due to \eqref{wit27}, by taking
expectation we have
\begin{equation}
\label{u2} \E\,|A
u^\e_2(t)|_H\leq
c_T\,(1+|x|_{\a,2}+|y|_H)+c\int_0^t(t-s)^{-1}\E\,|v^\e(t)-v^\e(s)|_H\,ds.
\end{equation}
This means that, in order to conclude the proof, we have to
estimate $\E\,|v^\e(t)-v^\e(s)|_H$, for any $0\leq s<t\leq T$.
It holds
\[\begin{array}{l}
\ds{v^\e(t)-v^\e(s)=\le[e^{t\frac B\e} y-e^{s\frac B \e} y\r]+\frac 1\e
\int_s^t e^{(t-\si)\frac{B}\e}G(u^\e(\si),v^{\e}(\si))\,d\si}\\
\vs \ds{+\frac 1\e \int_0^s
\le[e^{(t-\si)\frac B\e}-e^{(s-\si)\frac B \e}\r]G(u^\e(\si),v^{\e}(\si))\,d\si+\le[w^{\e,B}(t)-w^{\e,B}(s)\r]=:
\sum_{k=1}^4 I^\e_k(t,s).}
\end{array}\]
Proceeding as in the proof of Theorem \ref{tightness}, we have
\begin{equation}
\label{i1} |I^\e_1(t,s)|_H\leq c\,\e^{-\frac \a 2}\,(t-s)^{\frac
\a 2}\,|y|_{\a,2}.
\end{equation}
Concerning $I^\e_2(t,s)$, we have
\[|I^\e_2(t,s)|_H\leq
\frac c\e\int_s^te^{-\la\frac
{(t-\si)}\e}\le(1+|u^\e(\si)|_H+|v^\e(\si)|_H\r)\,d\si,\] and
then, with a change of variables, according to \eqref{stima11} and
\eqref{stima12} we get
\begin{equation}
\label{i2} \E\,|I^\e_2(t,s)|_H\leq c_T\int_0^{\frac{t-s}\e}
e^{-\la \si}\,d\si (1+|x|_H+|y|_H)\leq c_T \e^{-\frac \a
2}\,(t-s)^{\frac \a 2} (1+|x|_H+|y|_H).
\end{equation}
By proceeding with analogous arguments we prove that
\begin{equation}
\label{i3} \E\,|I^\e_3(t,s)|_H\leq c_T\, \e^{-\frac \a
2}\,(t-s)^{\frac \a 2} (1+|x|_H+|y|_H).
\end{equation}
Therefore, it remains to estimate $\E\,|I^\e_4(t,s)|_H$. By
straightforward computations, we have
\[\begin{array}{l}
\ds{\E\,|I^\e_4(t,s)|_H^2=\E\,|w^{\e,B}(t)-w^{\e,B}(s)|_H^2}\\
\vs \ds{=\frac 1\e \int_s^t\|e^{(t-\si)B/\e}\|_2^2\,d\si+\frac 1\e
\int_0^s\|e^{(t-\si)B/\e}-e^{(s-\si)B/\e}\|_2^2\,d\si=:J^\e_1+J^\e_2.}
\end{array}\]
According to \eqref{wit9}, with a change of
variables we have
\[J^\e_1\leq c\int_0^{\frac{t-s}\e} e^{-2\la \si}(\si\wedge
1)^{-\gamma}\,d\si\leq c\, \e^{-(1-\gamma)} (t-s)^{1-\gamma}.\]
Concerning $J^\e_2$, due to \eqref{wit21}
and \eqref{analytic} for any $\eta \in\,[0,1/2]$ and $s,t>0$
\[\|(e^{t B}-I)e^{sB}\|_{\mathcal{L}(H)}\leq c_\eta\,(t\wedge
1)^{\eta}(s\wedge 1)^{-\eta}.\]
Hence, thanks to \eqref{wit9}, if $0<\eta<1-\gamma$, by
proceeding with the same change of variables
\[\begin{array}{l}
\ds{J^\e_2=\frac 1\e \int_0^s
\le\|[e^{(t-s)B/\e}-I]e^{(s-\si)B/2\e}e^{(s-\si)B/2\e}\r\|_2^2\,d\si}\\
\vs \ds{\leq \frac c\e \le(\frac {t-s}\e\wedge1\r)^{\eta}
\int_0^s\le(\frac {s-\si}{2\e}\wedge1\r)^{-(\eta+\gamma)}
e^{-\la \frac {s-\sigma}{2\e}}\,d\si\leq c\, \e^{-\eta}
(t-s)^{\eta},}
\end{array}
\]
so that
\begin{equation}
\label{i4} \E\,|I^\e_4(t,s)|_H\leq
\le(\E\,|I^\e_4(t,s)|_H^2\r)^{\frac 12}\leq c\, \e^{-(1-\gamma)}
\le[(t-s)^{1-\gamma}+(t-s)^\eta\r].
\end{equation}
Collecting together \eqref{i1}, \eqref{i2}, \eqref{i3} and
\eqref{i4}, we obtain
\[\E\,|v^\e(t)-v^\e(s)|_H\leq c_T\,\e^{-\frac {\a\vee (1-\gamma)} 2}
\le(1+|x|_H+|y|_{\a,2}\r)\,\le[(t-s)^{\frac \a 2}+(t-s)^{\eta}+(t-s)^{1-\gamma}\r], \]
so that, from \eqref{u2},
\[\begin{array}{l}
\ds{\E\,|A u^\e_2(t)|_H
\leq c_T\,(1+|x|_{\a,2}+|y|_{\a,2})(1+\e^{-\frac {\a\vee (1-\gamma)} 2}).}
\end{array}\]
Together with \eqref{u1}, this yields \eqref{cosimo}.
\end{proof}
\section{The averaging result}
\label{sec5}
Our aim here is proving the main result of the present paper.
Namely, we are going to prove that for any fixed $T>0$ the
sequence $\{u^\e\}_{\e>0}\subset C((0,T];H)\cap L^\infty(0,T;H)$ converges in
probability to the solution $\bar{u}$ of the averaged equation
\begin{equation}
\label{averaged} du(t)=A u(t)+\bar{F}(u(t)),\ \ \ \ u(0)=x.
\end{equation}
The non-linear coefficient $\bar{F}$ in the equation above is
obtained by averaging the reaction coefficient $F$
appearing in the slow motion equation, with respect to the unique
invariant measure $\mu^x$ of the fast motion equation \eqref{av5}, with frozen slow component $x$. More precisely,
\begin{equation}
\label{wit31} \bar{F}(x):=\int_H F(x,y)\,\mu^x(dy),\ \ \ \ \ x
\in\,H.
\end{equation}
Notice that, as the mapping $y \in\,H\mapsto F(x,y) \in\,H$ is
Lipschitz-continuous, due to \eqref{lip} the integral above is
well defined. Moreover, as $\mu^x$ is ergodic, for any $h \in\,H$
we have \begin{equation} \label{mixing}
\le<\bar{F}(x),h\r>_H=\lim_{t\to \infty}\frac
1t\int_0^t\le<F(x,v^{x,y}(s)),h\r>_H\,ds,\ \ \ \
\mathbb{P}-\text{a.s.}
\end{equation}
This implies that $\bar{F}$
is Lipschitz-continuous. Actually, as $F:H\times H\to H$ is
Lipschitz-continuous (with Lipschitz-constant $L_f$) and
$v^{x,y}(t)$ is differentiable with respect to $x \in\,H$, with
its derivative fulfilling \eqref{finis}, for any $x_1,x_2 \in\,H$
and $t>0$ we have
\[\begin{array}{l}
\ds{\frac 1t\le|\int_0^t
\le<F(x_1,v^{x_1,y}(s))-F(x_2,v^{x_2,y}(s)),h\r>_H\,ds\r|}\\
\vs \ds{\leq \frac{L_f}t\int_0^t
(|x_1-x_2|_H+|v^{x_1,y}(s)-v^{x_2,y}(s)|_H)\,ds|h|_H}\\
\vs \ds{\leq L_f\,|h|_H\,(|x_1-x_2|_H+\sup_{\substack{x,y
\in\,H\\t\geq 0}}|D_x v^{x,y}(t)|_{\mathcal{L}(H)}|x_1-x_2|_H)\leq c\,(L_f+1)\,|h|_H\,|x_1-x_2|_H.}
\end{array}\]
Therefore, as \eqref{mixing} holds, we can conclude that
$\bar{F}$ is Lipschitz-continuous, with
\begin{equation}
\label{lipfbar} [\bar{F}]_{\text{\tiny Lip}}\leq
c\,(L_f+1).\end{equation} In particular, we have the following
existence and uniqueness result for the averaged equation.
\begin{Proposition}
\label{existence}
Under Hypotheses \ref{H1}, \ref{H2} and
\ref{H3}, equation \eqref{averaged} admits a unique mild solution
$\bar{u} \in\,C((0,T];H)\cap L^\infty(0,T;H)$, for any $T>0$ and $p\geq 1$
and for any initial datum $x \in\,H$.
\end{Proposition}
As far as the differentiability of $\bar{F}$ is concerned, we have
the following result.
\begin{Lemma}
\label{5.1}
For any $h \in\,L^\infty(0,L)$, the mapping
$\le<\bar{F}(\cdot),h\r>_H:H\to \reals$ is Fr\'echet
differentiable and for any $k \in\,H$
\[\begin{array}{l}
\ds{\le<D\bar{F}(x)k,h\r>_H=\int_H\le<D_x
F(x,y)k,h\r>_H\,\mu^x(dy)+2\int_H
\le<U_x(x,y),k\r>_H\le<F(x,y),h\r>_H\,\mu^x(dy)}\\
\vs \ds{-2\int_H\le<U_x(x,y),k\r>_H\,\mu^x(dy)\,
\int_H\le<F(x,y),h\r>_H\,\mu^x(dy),}
\end{array}\]
where $U_x(x,y)$ is the Fr\'echet derivative of the mapping
$U(\cdot,y):H\to \reals$ introduced in Subsection \ref{3.2}, for $y \in\,L^\infty(0,L)$ fixed.
\end{Lemma}
\begin{proof}
It is immediate to check that for any $y \in\,L^\infty(0,L)$ the
mapping
\[x \in\,H\mapsto U(x,y)=\int_0^1\le<G(x,\theta
y),y\r>_H\,d\theta \in\,\mathbb{R},\] is Fr\'echet differentiable
and for any $k \in\,H$
\[\le<U_x(x,y),k\r>_H=\int_0^1\le<G_x(x,\theta
y)k,y\r>_H\,d\theta,\] where $G_x(x,y)$ is the G\^ateaux
derivative of $G(\cdot,y)$ introduced in Section \ref{sec2}.
Then, if we define
\[V(x,y):=\frac 1{Z(x)}\, \exp\,2 U(x,y),\ \ \ \ \ x,y \in\,H,\]
by straightforward computations, for any $y \in\,L^\infty(0,L)$
the mapping $V(\cdot,y):H\to H$ is differentiable and we have
\begin{equation}
\label{mis50} D_xV(x,y)=2\,V(x,y)\le[U_x(x,y)-\int_H U_x(x,z)
\mu^x(dz)\r]=:2\,V(x,y) H(x,y). \end{equation} Notice that, as we
are assuming $\partial g/\partial \si_1(\xi,\si)$ to be uniformly
bounded, we have
\[|U_x(x,y)|_H\leq c\,|y|_H,\ \ \ \ x,y \in \,H,\]
so that thanks to \eqref{wit22} $D_xV(x,y)$ is well defined.
Now, according to \eqref{suppo}, the measure $\mu^x$
is supported on $C([0,L])$, so that
\[\le<\bar{F}(x),h\r>_H=\int_{C([0,L])}\le<F(x,y),h\r>_H\,\mu^x(dy).\]
Hence, if we set $\mu:=\mathcal{N}(0,(-B)^{-1}/2)$, by
differentiating under the sign of integral from \eqref{mis50} we
have
\[\begin{array}{l}
\ds{\le<D\bar{F}(x)k,h\r>_H=\le<D\int_{C([0,L])}\le<F(x,y),h\r>_H\,V(x,y)\,\mu(dy),k\r>_H}\\
\vs
\ds{=\int_{C([0,L])}\le<D_xF(x,y)k,h\r>_H\mu^x(dy)+2\int_{C([0,L])}\le<F(x,y),h\r>_H\,H(x,y)\,\mu^x(dy)}\\
\vs
\ds{=\int_H\le<D_xF(x,y)k,h\r>_H\mu^x(dy)+2\int_H\le<F(x,y),h\r>_H\,H(x,y)\,\mu^x(dy),}
\end{array}\]
and recalling how $H(x,y)$ is defined, we can conclude the proof
of the lemma.
\end{proof}
\bigskip
Now, as $u^\e$ is a mild solution of the slow motion equation (in fact
it is a classical solution, as $u^\e(t) \in\,D(A)$ for any $t>0$,
and estimate \eqref{cosimo} holds), for any $h \in\,D(A^\star)$
\[\le<u^\e(t),h\r>_H=\le<x,h\r>_H+\int_0^t \le<u^\e(s),A^\star
h\r>_H\,ds+\int_0^t\le<F(u^\e(s),v^\e(s)),h\r>_H\,ds,\ \ \ \ t\geq
0.\] Hence, we have
\begin{equation}
\label{byparts}
\le<u^\e(t),h\r>_H=\le<x,h\r>_H+\int_0^t
\le<u^\e(s),A^\star
h\r>_H\,ds+\int_0^t\le<\bar{F}(u^\e(s)),h\r>_H\,ds+R_h^\e(t),\ \ \
\ t\geq 0
\end{equation}
where the remainder $R_h^\e(t)$ is clearly given by
\begin{equation}
\label{remainder}
R_h^\e(t):=\int_0^t\le<F(u^\e(s),v^\e(s))-\bar{F}(u^\e(s)),h\r>_H\,ds,\
\ \ \ \ t\geq 0.
\end{equation}
Our purpose is proving that the remainder $R_h^\e(t)$ converges to
zero, as $\e$ goes to zero. We will see that, thanks to Theorem
\ref{tightness}, this will imply the averaging result.
\begin{Lemma}
\label{resto} Assume Hypotheses \ref{H1}, \ref{H2} and \ref{H3}
and fix any $\a>0$. Then, for any $T>0$, $x, y \in\,W^{\a,2}(0,L)$
and any $h \in\,H$
\begin{equation} \label{restobis}
\lim_{\e\to 0}\,\sup_{t \in\,[0,T]}\,\E\,|R_h^\e(t)|=0.
\end{equation}
\end{Lemma}
\begin{proof}
Fix $h \in\,L^\infty(0,L)$. For any $x,y \in\,H$ and $\e>0$ we
define
\begin{equation}
\label{phieps} \Phi_h^{\e}(x,y):=\int_0^\infty e^{-c(\e)\,
t}\,P^{x}_t\le[\le<F(x,\cdot),h\r>_H-\le<\bar{F}(x),h\r>_H\r](y)\,dt,
\end{equation}
where $c(\e)$ is some positive constant, depending on $\e>0$, to
be chosen later on. As for any $y,z \in\,H$
\[|\le<F(x,y),h\r>_H-\le<F(x,z),h\r>_H|\leq c\,|y-z|_H\,|h|_H,\]
for some constant $c$ independent of $x \in\,H$, we have that the
mapping
\[y \in\,H\to \le<F(x,y),h\r>_H-\le<\bar{F}(x),h\r>_H \in\,\reals\]
is Lipschitz-continuous and
\begin{equation}
\label{gore}
[\le<F(x,\cdot),h\r>_H-\le<\bar{F}(x),h\r>_H]_{\text{{\tiny
Lip}}}\leq c\,|h|_H. \end{equation}
According to Theorem
\ref{av14}, this means that the function $\Phi^{\e}_h(x,\cdot)$ is
a strict solution of the problem
\begin{equation}
\label{kolmo}
c(\e)\,\Phi^{\e}_h(x,y)-\mathcal{L}^{x}\Phi^{\e}_h(x,y)=\le<F(x,y),h\r>_H-\le<\bar{F}(x),h\r>_H,\
\ \ \ \ y \in\,D(B). \end{equation}
\medskip
Now, we prove uniform bounds in $\e>0$ for $\Phi^{\e}_h(x,y)$, for
its first derivatives with respect to $y$ and $x$ and for
$\text{Tr}\,[D^2_y\Phi^{\e}_h(x,y)]$. Due to \eqref{mis5bis} and \eqref{gore}, we
have
\[\le|P^{x}_t\le<F(x,\cdot),h\r>_H(y)-\le<\bar{F}(x),h\r>_H\r|\leq
c\,\le(1+|x|_H+|y|_H\r) e^{-\d t}\,|h|_H,\] and then
\begin{equation}
\label{stimauni} |\Phi_h^{\e}(x,y)|\leq c\int_0^\infty
e^{-c(\e)\,t} e^{-\d t}\,dt \le(1+|x|_H+|y|_H\r)\,|h|_H\leq
\frac{c}{\d}\,\le(1+|x|_H+|y|_H\r)\,|h|_H,
\end{equation}
for some constant $c$ independent of $\e>0$.
For the first derivative with respect to $y$, from
\eqref{deriprima} and \eqref{gore} we get
\[\le[P^{x}_t\le<F(x,\cdot),h\r>_H-\le<\bar{F}(x),h\r>_H\r]_1\leq
c\,e^{-\d t}\,|h|_H,\] and then
\begin{equation}
\label{der1}
\begin{array}{l} \ds{|D_y
\Phi^{\e}_h(x,y)|_H=\le|\int_0^\infty e^{-c(\e) t}\,D_y
\le[P^{x}_t\le<F(x,\cdot),h\r>_H(y)-\le<\bar{F}(x),h\r>_H\r]\,dt\r|_H}\\
\vs \ds{\leq c\int_0^\infty e^{-c(\e) t}\,e^{-\d t}\,dt\,|h|_H\leq
\frac c\d \,|h|_H,}
\end{array}
\end{equation}
for a constant $c$ independent of $\e>0$.
For the trace of $D^2_y \Phi_h^\e(x,y)$, according to \eqref{eq14}
we have
\[\le|\text{Tr}\,\le[D^2_y
P^x_t\,\le<F(x,\cdot),h\r>_H(y)\r]\r| \leq c\,(t\wedge
1)^{-\rho}\le(1+|x|_H+|y|_H\r)\,|h|_H,\] for some $\rho
<1$, and hence if $c(\e)\leq 1$
\begin{equation}
\label{traccia}
\begin{array}{l} \ds{\le|\text{Tr}\,\le[D^2_y
\Phi^\e(x,y)\r]\r|\leq \int_0^\infty e^{-c(\e) t}
\le|\text{Tr}\,\le[D^2_y\le(
P^x_t\,\le<F(x,\cdot),h\r>_H(y)-\le<\bar{F}(x),h\r>_H\r)\r]\r|\,dt
}\\
\vs \ds{\leq \int_0^\infty e^{-c(\e) t}\,(t\wedge
1)^{-\rho}\,dt\,\le(1+|x|_H+|y|_H\r)\,|h|_H\leq \frac
c{c(\e)}\,\le(1+|x|_H+|y|_H\r)\,|h|_H.}
\end{array}\end{equation}
Next, concerning the regularity of $\Phi^{\e}_h$ with
respect to $x \in\,H$, we first compute the derivative of the
mapping
\[x \in\,H\mapsto
P^{x}_t\le<F(x,\cdot),h\r>_H(y)=\E\,\le<F(x,v^{x,y}(t)),h\r>_H
\in\,\reals.\] As we are assuming that $h \in\,L^\infty$, we have
that the mappings $\le<F(x,\cdot),h\r>_H$ and
$\le<F(\cdot,y),h\r>_H$ are both Fr\'echet differentiable (see
Section \ref{sec2}). Beside, as shown at the end of subsection
\ref{3.0}, the process $v^{x,y}$ is differentiable with respect to
$x$. Then, by differentiating above under the sign of integral,
for any $k \in\,H$ we obtain
\[\begin{array}{l}
\ds{\le<D_x\le[ P^{x}_t\le<F(x,\cdot),h\r>_H(y)\r],k\r>_H}\\
\vs \ds{ =\E\,\le<D_x F(x,v^{x,y}(t))k,h\r>_H
+\E\,\le<D_y F(x,v^{x,y}(t))D_x v^{x,y}(t)k,h\r>_H}\\
\vs \ds{=P^{x}_t\le<D_x F(x,\cdot)k,h\r>_H(y)+\E\,\le<D_y
F(x,v^{x,y}(t))D_x v^{x,y}(t)k,h\r>_H,}
\end{array}\]
so that, thanks to \eqref{stimaderf} and \eqref{finis}
\[|D_x\le[ P^{x}_t\le<F(x,\cdot),h\r>_H(y)\r]|_H\leq c L_f\,|h|_H,\
\ \ \ \ t\geq 0.\] Moreover, as shown in Lemma \ref{5.1}, the
mapping $x\in\,H\mapsto \le<\bar{F}(x),h\r>_H \in\,\reals$ is
Fr\'echet differentiable and, due to estimate \eqref{lipfbar}, we
have
\[[\le<\bar{F}(x),h\r>_H]_1=[\le<\bar{F}(x),h\r>_H]_{\text{\tiny
Lip}}\leq c (L_f+1)\,|h|_H.\] Therefore
\begin{equation}
\label{der1x}
\begin{array}{l} \ds{|D_x
\Phi^{\e}_h(x,y)|_H=\le|\int_0^\infty e^{-c(\e) t}\,D_x
\le[P^{x}_t\le<F(x,\cdot),h\r>_H(y)-\le<\bar{F}(x),h\r>_H\r]\,dt\r|_H}\\
\vs \ds{\leq c\int_0^\infty e^{-c(\e) t}\,dt\,(L_f+1)\,|h|_H=
\frac {c (L_f+1)}{c(\e)}\,|h|_H.}
\end{array}
\end{equation}
\medskip
Next, for any $n \in\,\nat$, we define $v^\e_n:=P_n v^\e$, where
$P_n$ the projection of $H$ onto $\le<e_1,\ldots,e_n\r>$ and
$\{e_k\}_{k \in\,\nat}$ is the complete orthonormal system,
introduced in Hypothesis \ref{H2}, which diagonalizes $B$. It is
immediate to check that $v_n^\e$ is a strong solution of equation
\[dv_n^\e(t)=\frac 1\e\,\le[B
v_n^\e(t)+P_n\,G(u^\e(t),v^\e(t))\r]\,dt+\frac
1{\sqrt{\e}}\,P_n\,dw(t),\ \ \ \ \ v^\e_n(0)=P_n y.\] Moreover,
according to Lemma \ref{lemma4.4}, $u^\e$ is a strong solution of
the slow motion equation.
Therefore, we can apply
It\^o's formula to
$\Phi^{\e}_h(u^\e(t),v^\e_n(t))$ and we get
\[\begin{array}{l}
\ds{\Phi^{\e}_h(u^\e(t),v^\e_n(t))=\Phi^{\e}_h(x,P_ny)
+\int_0^t\le<D_x
\Phi^{\e}_h (u^\e(s),v^\e_n(s)),A u^\e(s)+F(u^\e(s),v^\e(s))\r>_H\,ds}\\
\vs \ds{+\frac 1\e \int_0^t \le<D_y\Phi^{\e}_h
(u^\e(s),v^\e_n(s)),B v^\e_n(s)+P_n G(u^\e(s),v^\e(s))\r>_H\,ds}\\
\vs \ds{+\frac 1{2\e}\int_0^t \text{Tr}\,[P_n\,D^2_y\Phi^{\e}_h
(u^\e(s),v^\e_n(s))]\,ds +\frac 1{\sqrt{\e}}\int_0^t\le<D_y
\Phi^{\e}_h (u^\e(s),v^\e_n(s)),P_n dw(s)\r>_H,}
\end{array}\]
and hence
\begin{equation}
\label{striscia}
\begin{array}{l}
\ds{\Phi^{\e}_h(u^\e(t),v^\e_n(t))=\Phi^{\e}_h(x,P_ny)
+\int_0^t\le<D_x \Phi^{\e}_h
(u^\e(s),v^\e_n(s)),A u^\e(s)+F(u^\e(s),v^\e(s))\r>_H\,ds}\\
\vs \ds{+\frac 1\e \int_0^t \mathcal{L}^{u^\e(s)}\Phi^{\e}_h
(u^\e(s),v^\e_n(s))\,ds+\frac 1{\sqrt{\e}}\int_0^t\le<D_y
\Phi^{\e}_h (u^\e(s),v^\e_n(s)),P_n dw(s)\r>_H}\\
\vs \ds{+\frac 1\e \int_0^t \le<D_y\Phi^{\e}_h
(u^\e(s),v^\e_n(s)),[P_n
G(u^\e(s),v^\e(s))-G(u^\e(s),v^\e_n(s))]\r>_H\,ds}\\
\vs \ds{+\frac 1{2\e}\int_0^t
\text{Tr}\,[(P_n-I)\,D^2_y\Phi^{\e}_h (u^\e(s),v^\e_n(s))]\,ds.}
\end{array}
\end{equation}
Recalling that $\Phi^{\e}_h(x,\cdot)$ is a strict solution of the
elliptic equation \eqref{kolmo}, for any $s\geq 0$ we have
\[\begin{array}{l}
\ds{\mathcal{L}^{u^\e(s)}\Phi^{\e}_h(u^\e(s),v^\e_n(s))=c(\e)\,\Phi^{\e}_h(u^\e(s),v^\e_n(s))-
\le(\le<F(u^\e(s),v^\e_n(s)),h\r>_H-\le<\bar{F}(u^\e(s)),h\r>_H\r).}
\end{array}\]
Then, multiplying both sides of
\eqref{striscia} by $\e$,
\[\begin{array}{l}
\ds{R_\e(t)=\int_0^t
\le[\le<F(u^\e(s),v^\e(s)),h\r>_H-\le<\bar{F}(u^\e(s)),h\r>_H\r]\,ds=c(\e)
\int_0^t \Phi^{\e}_h
(u^\e(s),v^\e_n(s))\,ds}\\
\vs \ds{+\sqrt{\e}\int_0^t\le<D_y
\Phi^{\e}_h (u^\e(s),v^\e_n(s)),P_n\,dw(s)\r>_H-\e\,\le[\Phi^{\e}_h(u^\e(t),v^\e_n(t))-\Phi^{\e}_h(x,y)\r]}\\
\vs \ds{+\e\,\int_0^t\le<D_x \Phi^{\e}_h (u^\e(s),v^\e_n(s)),A
u^\e(s)+F(u^\e(s),v^\e(s))\r>_H\,ds+H^{n,\e}(t),}
\end{array}\]
where
\[\begin{array}{l}
\ds{H^{n,\e}(t):=\int_0^t \le<D_y\Phi^{\e}_h
(u^\e(s),v^\e_n(s)),[P_n
G(u^\e(s),v^\e(s))-G(u^\e(s),v^\e_n(s))]\r>_H\,ds}\\
\vs \ds{+\frac 1{2}\int_0^t\!\!
\text{Tr}\,[(P_n-I)\,D^2_y\Phi^{\e}_h
(u^\e(s),v^\e_n(s))]\,ds+\!\!\int_0^t\!\!
\le<F(u^\e(s),v^\e(s))-F(u^\e(s),v_n^\e(s)),h\r>_H\,ds.}
\end{array}\]
Thanks to \eqref{stimauni}, \eqref{der1} and \eqref{der1x}, this yields
\[\begin{array}{l}
\ds{|R^{\e}_h(t)|\leq c \le(\frac{\e}{c(\e)}+c(\e)
\r)\,\int_0^t\le(1+|u^\e(s)|_H+|v^\e(s)|_H+|v^\e_n(s)|_H\r)\,ds
|h|_H}\\
\vs \ds{+c\,\frac {\e}{c(\e)}\int_0^t |A
u^\e(s)|_H\,ds\,|h|_H+\sqrt{\e}\le|\int_0^t\le<D_y \Phi^{\e}_h
(u^\e(s),v^\e_n(s)),P_n\,dw(s)\r>_H\r|}\\
\vs
\ds{+\e\,\le(1+|u^\e(t)|_H+|v^\e_n(t)|_H+|x|_H+|y|_H\r)\,|h|_H+|H^{n,\e}|,}
\end{array}
\]
and hence, by taking expectation, due to \eqref{stima11},
\eqref{stima12}, \eqref{cosimo} and \eqref{der1}, for any $n
\in\,\nat$
\[\begin{array}{l}
\ds{\E\,|R^{\e}_h(t)|\leq c_T \le(\frac{\e}{c(\e)}+c(\e)+\e
\r)\,\le(1+|x|_H+|y|_H\r)\,
|h|_H}\\
\vs \ds{+c_T\,\frac {\e}{c(\e)}(1+\e^{-\frac {\a\vee (1-\gamma)}
2})(1+|x|_{\a,2}+|y|_{\a,2})|h|_H+c_T\,\sqrt{\e}\,|h|_H+\E\,|H^{n,\e}|.}
\end{array}
\]
Now, thanks to estimates \eqref{stima11}, \eqref{stima12},
\eqref{der1} and \eqref{traccia}, by using the dominated
convergence theorem, for any $\e>0$ we have
\[\lim_{n\to \infty}\,\E\,|H^{n,\e}|=0.\]
This means that if we take $c(\e)=\e^\d$, with $0<\d<1-[\a\vee (1-\gamma)]/2$, and
$n_\e \in\,\nat$ such that $\E\,|H^{n_\e,\e}|\leq \e$, it follows
\[\begin{array}{l}
\ds{\sup_{t \in\,[0,T]}\,\E\,|R^{\e}_h(t)|\leq c_T
\le(\frac{\e}{c(\e)}+c(\e)+\e
\r)\,\le(1+|x|_H+|y|_H\r)\, |h|_H}\\
\vs \ds{+c_T\,\frac
{\e}{c(\e)}(1+\e^{-\frac {\a\vee (1-\gamma)} 2})(1+|x|_{\a,2}+|y|_{\a,2})|h|_H+c_T\,\sqrt{\e}\,|h|_H+\E\,|H^{n_\e,\e}|}\\
\vs \ds{ \leq c_T\,e^{\rho}\,(1+|x|_{\a,2}+|y|_{\a,2})|h|_H,}
\end{array}\] for some $\rho>0$. This immediately yields
\eqref{restobis} for $h \in\,L^\infty(0,L)$.
Now, if $h \in\,H$ we fix a sequence $\{h_n\}_{n \in\,\nat}\subset
L^\infty(0,L)$ converging to $h$ in $H$ and such that $|h_h|_H\leq
|h|_H$. As
\[\sup_{t \in\,[0,T]}\,\E\,|R_{h_n}^\e(t)|\leq
c_T\,\e^{\rho}\,(1+|x|_{\a,2}+|y|_{\a,2})|h_n|_H,\] we obtain
\eqref{restobis} also in the general case.
\end{proof}
Once we have proved the key Lemma \ref{resto}, we can prove the
main result of the paper, the convergence of the solution of the
slow motion equation to the solution of the {\em averaged}
equation.
\begin{Theorem}
\label{averaging} Assume that $x, y \in\,W^{\a,2}(0,L)$, for some
$\a>0$. Then, under Hypotheses \ref{H1}, \ref{H2} and \ref{H3},
for any $T>0$ and $\eta>0$ we have
\begin{equation}
\label{wit30} \lim_{\e\to
0}\mathbb{P}\,\le(\sup_{t \in\,[0,T]}|u^\e(t)-\bar{u}(t)|_{H}>\eta\,\r)=0,
\end{equation}
where $\bar{u}$ is the solution of the averaged equation
\eqref{averaged}.
\end{Theorem}
\begin{proof}
Due to Theorem \ref{tightness}, the sequence
$\{\mathcal{L}(u^\e)\}_{\e>0}$ is tight in $C((0,T];H)\cap L^\infty(0,T;H)$, and then
as a consequence of the Skorokhod theorem, for any two sequences
$\{\e_n\}_{n \in\,\nat}$ and $\{\e_m\}_{m \in\,\nat}$ converging
to zero, there exist subsequences $\{\e_{n(k)}\}_{k \in\,\nat}$
and $\{\e_{m(k)}\}_{k \in\,\nat}$ and a sequence of random
elements
\[\{\rho_k\}_{k \in\,\nat}:=\le\{(u_1^k,u_2^k)\r\}_{k \in\,\nat},\]
in $\mathcal{C}:=\le[C((0,T];H)\cap L^\infty(0,T;H)\r]^2$, defined on some probability
space $(\hat{\Omega},\hat{\F},\hat{\Pro})$, such that the law of
$\rho_k$ coincides with the law of
$(u^{\e_{n(k)}},u^{\e_{m(k)}})$, for each $k \in\,\nat$, and
$\rho_k$ converges $\hat{\Pro}$-a.s. to some random element
$\rho:=(u_1,u_2) \in\,\mathcal{C}$. By a well known
argument due to Gy\"ongy and Krylov (see \cite{gk}), if we show
that $u_1=u_2$, then we can conclude that there exists some $u
\in\,C((0,T];H)\cap L^\infty(0,T;H)$ such that the whole sequence $\{u^\e\}_{\e>0}$
converges to $u$ in probability.
For any $k \in\,\nat$ and $i=1,2$ we define
\begin{equation}
\label{fine?} R_i^k(t):=\langle u^k_i(t),h\rangle_H-\langle
x,h\rangle_H-\int_0^t\langle u^k_1(s),A^\star h\rangle_H\,ds
-\int_0^t\langle\bar{F}(u^k_i(s)),h\rangle_H\,ds.\end{equation}
As
$\mathcal{L}(u^k_1)=\mathcal{L}(u^{\e_{n(k)}})$ and
$\mathcal{L}(u^k_1)=\mathcal{L}(u^{\e_{m(k)}})$, according to
\eqref{restobis} we have
\[\lim_{k\to \infty }\,\sup_{t \in\,[0,T]}\hat{\E}\,|R^k_i(t)|=0,\]
so that, as the sequences $\{u^k_1\}_{k \in\,\nat}$ and
$\{u^k_2\}_{k \in\,\nat}$ converge $\hat{\mathbb{P}}$-a.s. in
$C((0,T];H)\cap L^\infty(0,T;H)$ respectively to $u_1$ and $u_2$, by taking the limit
for some $\{k_i(n)\}\subseteq \{k\}$ going to infinity in
\eqref{fine?}, we have that both $u_1$ and $u_2$ fulfill the
equation
\[
\langle u(t),h\rangle_H=\langle x,h\rangle_H+\int_0^t \langle
u(s),A^\star h\rangle_H\,ds+\int_0^t\langle
\bar{F}(u(s)),h\rangle_H\,ds,\] for any $h \in\,D(A^\star)$,
and then they coincide with the unique solution of
the {\em averaged} equation \eqref{averaged}.
As we have recalled before, this implies that the sequence
$\{u^\e\}_{\e>0}$ converges in probability to some $u
\in\,C([0,L];H)$, and, by using again a uniqueness argument, such $u$ has to coincide with the solution $\bar{u}$ of equation \eqref{averaged}.
\end{proof}
\section{Some remarks on the case of space dimension $d>1$}
\label{sec6}
In the case of space dimension $d=1$, the fast equation \eqref{av5} with frozen slow component $x \in\,H$ is a gradient system and hence its unique invariant measure $\mu^x$ admits a density $V(x,y)$ with respect to the Gaussian measure $\cal{N}(0,(-B)^{-1}/2)$. This allows to prove in Lemma \ref{5.1} that for any $h \in\,L^\infty(0,L)$ the mapping
\begin{equation}
\label{fbar}
x \in\,H\mapsto \le<\bar{F}(x),h\r>_H \in\,\reals
\end{equation}
is Fr\'echet differentiable and also allows to compute its derivative.
In space dimension $d>1$, in order to have function-valued solutions to system \eqref{sistema} we have to take a noise colored in space, and hence the fast equation is no more a gradient system. For this reason we cannot say anything about the differentiability of mapping \eqref{fbar} and hence we cannot say anything about the differentiability with respect to $x \in\,H$ of the mapping $\Phi^\e_h(x,y)$ introduced in \eqref{phieps}.
Nevertheless, under suitable assumptions on the noise in the fast equation, it is possible to prove a result analogous to that proved in Lemma \ref{resto} and hence to get averaging.
Instead of working in the interval $(0,L)$, now we work in a bounded open set $D\subset \reals^d$, with $d>1$, having a regular boundary. In the fast motion equation we take a noise of the following form
\[w^Q(t,\xi)=\sum_{k=1}^\infty Q e_k(\xi)\beta_k(t),\ \ \ \ t\geq 0,\ \ \ \ \xi \in\,D,\]
and we assume that the operators $B$ and $Q$ satisfy the following conditions.
\begin{Hypothesis}
\label{H1bis}
\begin{enumerate}
\item There exists a complete orthonormal system $\{e_k\}_{k
\in\,\nat}$ in $H$ and two positive sequences $\{\a_k\}_{k
\in\,\nat}$ and $\{\la_k\}_{k \in\,\nat}$ such that
$B e_k=-\a_k e_k$ and $Q e_k=\la_k e_k$
and, for some
$\gamma<1$,
\[
\sum_{k=1}^\infty \frac
{\la_k^2}{\a_k^{1-\gamma}}<\infty.
\]
\item There exists $\la>0$ such that
$\a_k\geq \la$, for any $k \in\,\nat$.
\item There exists $\eta<1/2$ such that
\[\inf_{k \in\,\nat}\la_k \a_k^\eta>0.\]
\end{enumerate}
\end{Hypothesis}
Notice that, as $\a_k\sim k^{2/d}$, the conditions above imply that we have to work with $d\leq 3$.
Under Hypothesis \ref{H1bis} and Hypotheses \ref{H2} and \ref{H3} (with obvious changes due to the passage from $d=1$ to $d\geq 1$) system \eqref{sistema} admits a unique mild solution $(u^\e,v^\e) \in\,\cal{C}_{T,p}\times \cal{C}_{T,p}$, for any $\e>0$, $p\geq 1$ and $T>0,$ and for any fixed slow component $x \in\,H$ the fast equation \eqref{av5} admits a unique mild solution $v^{x,y} \in\,\cal{C}_{T,p}$, fulfilling \eqref{suppo} and \eqref{wit1bis}. As in the one dimensional case, the process $v^{x,y}$ is three times differentiable with respect to $y \in\,H$ and once with respect to $x \in\,H$ and estimates analogous to
\eqref{av7}, \eqref{av8} and \eqref{finis} hold (for all details see \cite{cerrai}).
The fast transition semigroup $P^x_t$ maps $C_b(H)$ into itself and $\text{Lip}(H)$ into itself and \eqref{lipptbis} holds. Moreover, it has a smoothing effect and maps $B_b(H)$ into $C_b^3(H)$ and estimates \eqref{eq13}, \eqref{deriprima} and \eqref{derisucc} are still true, with the singularity $(t\wedge 1)^{(j-i)/2}$ replaced by $(t\wedge 1)^{(j-i)(\eta+1/2)}$.
As far as the asymptotic behavior of the fast semigroup is concerned, it admits a unique invariant measure $\mu^x$ which is strongly mixing and fulfills \eqref{wit22}, \eqref{mis5} (with the singularity $(t\wedge 1)^{1/2}$ replaced by the singularity $(t\wedge 1)^{-(\eta+1/2)}$) and \eqref{mis5bis}. But, as we have said before, as equation \eqref{av5} is not of gradient type, we do not have any explicit expression for the measure $\mu^x$.
All uniform bounds for $u^\e$ and $v^\e$ proved in Section \ref{sec4} are still valid, so that the family of probability measures $\{\cal{L}(u^\e)\}_{\e \in\,(0,1)}$ is tight in $C((0,T];H)\cap L^\infty(0,T;H)$. This means that in order to have averaging in this multidimensional case it suffices to prove Lemma \ref{resto}. The proof in this case follows the same lines as in the one dimensional case, but it requires some extra approximation arguments.
Actually, one has to introduce
the approximating problems
\begin{equation}
\label{appne} dv^\e_n(t)=\frac 1\e \le[B_n
v^\e_n+G_n(u^\e(t),v^\e_n(t))\r]\,dt+\frac 1{\sqrt{\e}}\,Q_n
\,dw(t),\ \ \ \ v^\e_n(0)=P_n y,
\end{equation}
and
\begin{equation}
\label{appn} dv_n^{x,y}(t)=\le[B_n
v_n^{x,y}+G_n(x,v^{x,y}_n(t))\r]\,dt+Q_n \,dw(t),\ \ \ \
v^{x,y}_n(0)=P_n y,
\end{equation}
where $B_n x:=BP_n x$, $Q_n x:=Q P_n x$ and $G_n(x,y):=P_n G(x,P_nx)$, for any $n \in\,\nat$ and $x,y \in\,H$.
As the operators $B_n$ and $Q_n$ fulfill Hypothesis \ref{H1bis} and
$G_n$ has the same regularity properties of $G$, all properties
satisfied by $v^\e$, $v^{x,y}$ and $P^x_t$ are still valid
for $v_n^\e$, $v_n^{x,y}$ and for the transition semigroup
$P^{n,x}_t$ associated with \eqref{appn}.
Moreover, all estimates for $v^{x,y}_n$ and
$P^{n,x}_t$ are uniform with respect to $n \in\,\nat$, and for
each fixed $\e>0$ and $x,y \in\,H$
\begin{equation}
\label{limye} \lim_{n\to\infty}\E\,\sup_{t\geq
0}|v^\e_n(t)-v^\e(t)|_H^2=0,
\end{equation}
and
\begin{equation}
\label{limyn} \lim_{n\to\infty}\E\,\sup_{t\geq
0}|v^{x,y}_n(t)-v^{x,y}(t)|_H^2=0.
\end{equation}
Clearly, equation \eqref{appn} shows the same long-time behavior
as equation \eqref{av5}. Then for any $n \in\,\nat$ there exists a
unique invariant measure $\mu^{n,x}$ for the semigroup
$P^{n,x}_t$, which fulfills all properties described for $\mu^x$, with all estimates uniform with respect to
$n \in\,\nat$.
Next, we define
\[\bar{F}_n(x):=\int_HF(x,y)\,\mu^{n,x}(dy),\ \ \ \ \ x \in\,H.\]
As for $\bar{F}$, we obtain that all $\bar{F}_n:H\to H$ are Lipschitz-continuous and
\begin{equation}
\label{bis30} \sup_{n \in\,\nat}\,[\bar{F}_n]_{\text{Lip}}\leq
c\,L_f.
\end{equation}
Moreover, for any $x \in\,H$
\begin{equation}
\label{limfbar} \lim_{n\to
\infty}\le|\bar{F}_n(x)-\bar{F}(x)\r|_H=0.
\end{equation}
For any $n \in\,\nat$ we define
\[ H_n(x):=\int_{\mathbb{R}^n}\bar{F}_n(P_n
x-\sum_{k=1}^n \xi_k e_k)\rho_n(\xi)\,d\xi,\ \ \ \ \ \ x
\in\,H,\]
where
$\rho_n:\mathbb{R}^n\to\reals$ is a $C^1$ mapping having support
in $B_{\mathbb{R}^n}(0,1/n)$ and having total mass equal $1$. All
mappings $H_n$ are in $C^1(H;H)$ and
\begin{equation}
\label{effen} \lim_{n\to \infty}\,|\bar{F}_n(x)-H_n(x)|_H=0,\ \ \
\ \ x \in\,H.
\end{equation}
Moreover, due to \eqref{bis30}, we have
\begin{equation}
\label{bis20} |\bar{F}_n(x)-H_n(x)|_H\leq c\,\le(1+|x|_H\r),\ \ \
\ \ \sup_{n \in\,\nat}\,[H_n]_{\text{Lip}(H)}<\infty.
\end{equation}
Then, in the proof of Lemma \ref{resto} we introduce the following correction function
\[\Phi_n^{\e}(x,y):=\int_0^\infty e^{-c(\e)\,
t}\,P^{n,x}_t\le[\le<F(x,\cdot),h\r>_H-\le<H_n(x),h\r>_H\r](y)\,dt,
\]
As in the one dimensional case, we have that the function $\Phi^{\e}_n(x,\cdot)$ is
a strict solution of the problem
\[c(\e)\,\Phi^{\e}_n(x,y)-\mathcal{L}^{n,x}\Phi^{\e}_n(x,y)=\le<F(x,y),h\r>_H-\le<H_n(x),h\r>_H,\
\ \ \ \ y \in\,H, \]
where $\mathcal{L}^{n,x}$ is the Kolmogorov operator associated with the approximating fast equation \eqref{appn}.
Concerning the regularity of $\Phi^{\e}_n$ with respect to $y$, we proceed as in the proof of Lemma \ref{resto} and all estimates are uniform with respect to $n \in\,\nat$. As far as regularity in $x$ is concerned, we also proceed as in the one dimensional case, by noticing that the mapping $x\in\,H\mapsto
\le<H_n(x),h\r>_H \in\,\reals$ is Fr\'echet differentiable and,
due to estimate \eqref{effen}, the $C^1$-norm is uniformly bounded in $n \in\,\nat$, that is
\[\sup_{n \in\,\nat}\,[\le<H_n(\cdot),h\r>_H]_1=\sup_{n \in\,\nat}\,[\le<H_n(\cdot),h\r>_H]_{\text{\tiny
Lip}}=c\,|h|_H<\infty.\]
This implies an estimate for $D_x\Phi_n^\e$, which is uniform with respect to $n \in\,\nat$.
Next, as in the proof of Lemma \ref{resto} we apply It\^o's formula to
$\Phi^{\e}_n(u^\e(t),v^\e_n(t))$ and, by some estimates not different from those already used, by taking $c(\e)=\e^\delta$, for some $\delta>0$ we obtain
\[\begin{array}{l}
\ds{\E\,\le|\int_0^t
\le[\le<F(u^\e(s),v_n^\e(s)),h\r>_H-\le<H_n(u^\e(s)),h\r>_H\r]\,ds\r|}\\
\vs \ds{\leq c_T \e^{\d^\prime}
\le(1+|x|_{\a,2}+|y|_{\a,2}\r)\,|h|_H+\int_0^T\E\,\le|\bar{F}_n(u^\e(s))-H_n(u^\e(s))\r|_H\,ds\,|h|_H,}
\end{array}\]
for some $\d^\prime>0$. Due to \eqref{limfbar} and \eqref{effen}, this allow to conclude that \eqref{restobis} holds.
| 26,562
|
Your Powder Puff (1934) More at IMDbPro »
Overview
User Rating:
MOVIEmeter: Up 201% in popularity this week. See rank & trends on IMDbPro.
Director:Friz Freleng
Plot Keywords:
Additional Details
Parents Guide:Add content advisory for parents
Runtime:USA:6 min
Color:Black and White
Company:Leon Schlesinger Studios more
Fun Stuff
Movie Connections:Edited into Bugs vs. Daffy: Battle of the Music Video Stars (1988) (TV) more
FAQWhich series is this from: Merrie Melodies or Looney Tunes?
more
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Message BoardsDiscuss this movie with other users on IMDb message board for Shake Your Powder Puff (1934)
Recommendations
If you enjoyed this title, our database also recommends: Show more recommendations
This Merrie Melodies short proves nothing except that animators in the 1930s were on drugs.
It's inconceivable to me that children who watched these cartoons wouldn't be absolutely terrified. In this one, we're witness to a sort of vaudeville show with an all-animal cast, including a myopic turtle and a pig very proud of his ability to play the flute. There's a troupe of dancing "girls" who sing about "shaking your powder puff," and a dog (or wolf, or fox or something) keeps getting thrown out by the management.
This is an interesting curio, but it's totally whacked out. If you want to see it for yourself, you can find it as a special feature on the DVD release of the 1934 Astaire/Rogers film, "The Gay Divorcée."
| 111,861
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Kratt Bros: Be The Creature - Expedition Meerkats
Kratt Bros: Be The Creature, Episode 17 - Expedition Meerkats
Series Synopsis
Naturalists and documentary filmmakers Chris and Martin Kratt explore the biology, physiology, and natural behaviour of a unique creature. Travelling all over the world they search for what it means to 'Be The Creature'.
Episode Synopsis
Episode 17 - Expedition Meerkats
Martin and Chris Kratt take their passion for creature adventuring to the desert of the Kalahari in Southern Africa for a round-the-clock expedition to discover two very different but equally fascinating creatures.
Previous Episode Synopsis
Episode 16 - Expedition Komodo Dragon
Martin and Chris Kratt journey across the world and back in time to enter the realm of the Komodo dragon on the secluded islands of the Indonesian Archipelago.
| 153,444
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Proactive or Reactive Maintenance – Does it Really Matter?
Companies seem to revisit this issue with each change of management, whether it is the plant, engineering, operations, or maintenance manager. Based on his or her background, each manager will bring a different perspective on this question. Most managers, even those who may have progressed through the plant hierarchy, will fail to take into consideration the true cost of reactive maintenance. The investment it takes to properly maintain the plant equipment should be financially balanced, comparing proactive costs versus reactive costs. On one side, there is the cost of proactive maintenance. In most organizations, this is usually easy to calculate. It is the cost of the maintenance labour, materials, tools, equipment, contractors, etc., performed in a proactive (budgeted) mode. This information is tracked in most accounting and budgeting systems. On the other side is the cost of reactive maintenance (decreased maintenance efficiency and effectiveness) including the unavailability of the asset or equipment. This information is not always known or as easy to calculate. What is the cost of unavailability or downtime? It begins with the cost for the equipment to sit idle (operational and maintenance labour, utilities, depreciation, etc.) all of which is lumped into a budget line item called “cost to produce” — but these costs are only the tip of the iceberg. Additional costs would also factor in the cost of a product not produced on schedule. This impacts the delivery schedule. If the company is going to minimize the impact on their delivery chain, they may have to notify their customers to expect a late delivery, which in a competitive marketplace could result in a dissatisfied or lost customer. The company may choose to work additional shifts to make up the lost product, which impacts future maintenance and operations schedules. This also incurs additional costs since it requires additional operational and maintenance labor, usually in the form of overtime. It also requires additional energy to operate the equipment when it was originally not scheduled to run. If this energy is required during a penalty period for unscheduled energy consumption, then the cost can skyrocket. The last three paragraphs assumed that there is unused capacity for the production or process equipment. However, what occurs when the equipment is already at capacity and it fails? How does the company recover the lost production? In some cases, such as the utility industry, they can purchase additional energy from the grid; this is usually at a much higher cost than the company would have produced it. However, in most cases, the lost production cannot be made up and the company must notify its customer that a delivery will be late. Consider also, the start of the supply chain. If the production process is down in a just in time or lean manufacturing environment, then the suppliers will have to be notified to reschedule their delivery or the company will be forced to accept the delivery and stockpile the material at the start of their manufacturing process. After considering this newsletter, it must be recognized that proactive maintenance is a more effective business model than reactive maintenance. How much more effective? Wasted reactive maintenance resources may average 30 percent or more than the required resources in a proactive work model. Production downtime losses will average at least four times the wasted maintenance resources. How much will your company be willing to spend to move from a reactive to proactive maintenance business model? These questions will be topics for discussion in the next newsletters. Stay tuned. Terry Wireman is senior vice-president of Vesta Partners LLC. You can reach him at This e-mail address is being protected from spambots. You need JavaScript enabled to view it . Click here to subscribe to his Wireman's Wire enewsletter.
| 64,333
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David Wharton directed this film for a poem by UK performance poet Tony Walsh, A.K.A. Longfella. Videopoetry critic Erica Goss writes,
Actor James Foster delivers an emotional punch you won’t forget: This is one of the few video poems I’ve seen that features the talents of a professional actor, and the results are striking. Foster tells the story of the devastation of divorce with his facial expressions and body language, increasing the tension with repetitions of the word, “Sometimes:” “When I’m eating it cold from a tin in the kitchen / and sometimes, when I’ve stood in a line to collect my prescription.” Watching him break apart is at once humbling and terrifying.
| 338,209
|
TITLE: Why is the variation of position in the boundary of configuration space not reversible?
QUESTION [1 upvotes]: [...] we have investigated the question of an extremum under the condition that we are inside the boundaries of configuration space. A function which does not assume any extremum inside a certain region may well assume it on the boundary of that region. On the boundary of the configuration space the displacements are no longer reversible and hence our argument that the first variation must vanish$-$because otherwise it can be made positive as well as negative$-$no longer holds. For non-reversible displacements a function may well assume an extremum without having a stationary value at that point. In that case an extremum exists without the vanishing of the first variation.
The above is excerpted from Stationary value versus extreme value from Cornelius Lanczos' The Variational Principles of Mechanics; here the author explains
The extremum of a function requires a stationary value only for reversible displacements. On the boundary of the configuration space, where the variation of position is not reversible, an extremum is possible without a stationary value.
I couldn't comprehend what the author meant by non-reversible variation of position; for the very time in the book he did use the term reversible displacement and its antonym.
But what are actually they?
What does make a "variation of position" reversible or non-reversible?
Why is the displacement not reversible in the boundary of a configuration space?
Could anyone shed light on these terms?
REPLY [3 votes]: This is strange wording, but it's trying to explain the fact that at the boundary of a function's domain of definition, it can be extremal without being stationary - because the very idea of "stationary" doesn't make sense at the boundary.
By definition, a "stationary" point is one where the first derivative (or "variation", which is just a disguised functional derivative) vanishes. But standard differentiability is only defined on open sets (see this math.SE question) - although one might try to extend the derivative to the boundary by continuity, it doesn't make sense in general to speak of the derivative at the boundary.
That the variation is "not reversible" means that you can vary in one direction (inwards) at the boundary, but not outwards, since that variation would carry the function out of its domain of definition. Supposing that Lanzcos' argument for why the first variation has to vanish at a extremum is a variant of the standard argument, it's now clear why "non-reversibility" of the variation destroys the argument: The variation would acquire one sign under variations inwards, and the opposite sign under variations outwards, but the latter don't exist else we wouldn't be at the boundary, so we simply can't derive the vanishing of the variation/derivative.
| 142,310
|
TITLE: Let $f(x)=(x+1)(x+2)..(x+100)$ and $g(x)=f(x)f’’(x)-(f’(x))^2$. Find number of roots $n$ of $g(x)=0$
QUESTION [2 upvotes]: After relevant simplification ie. Log on both sides and then differentiating, the expression received is
$$\frac{f’(x)}{f(x)} =\sum_{k=1}^{100} \frac{1}{x+k}$$
Differentiating again will give
$$\frac{g(x)}{(f(x))^2}=-\sum_{k=1}^{100} \frac{1}{(x+k)^2}$$
So in turn, the roots of $g(x)=0$ are simply the roots of $f(x)=0$, so ans should be 100, but that’s not correct.
Where am I going wrong?
REPLY [2 votes]: $$\frac{f’(x)}{f(x)} =\sum_{k=1}^{100} \frac{1}{x+k}$$
D.w.r.t. $x$ both sides
$$\frac{f(x)f''(x)-f'^2(x)}{f^2(x)}=-\sum_{k=1}^{100} \frac{1}{(x+k)^2}<0 \ne 0.$$
So $$g(x)=f(x)f''(x)-f'^2(x)=\sum_{k=1}^{n}\frac{f^2(x)}{(x+k)^2}$$
Then $$-g(x)=(x+2)^2(x+3)^2(x+4)^2....(x+n)^2+ (x+1)^2(x+3)^2(x+4)^2....(x+n)^2+......>0 \ne 0,$$ has no real root.
| 181,381
|
TITLE: Definition of a quotient group in Dummit-Foote
QUESTION [2 upvotes]: I'm reading the section on quotient groups in Dummit and Foote, and they give somewhat non-standard definition of a quotient group. I was wondering whether there is an easy way to see right away for someone who is familiar with the standard definition of a quotient group that DF's definition is equivalent to the standard one?
What I can see. The standard definition defines $G/K$ (as a set) as the set of left cosets of $K$ in $G$. Every such coset is an equivalence class under the equivalence relation on $G$ given by $g_1\sim g_2\iff g_1=g_2k$ for some $k\in K$. The fibers of $\phi:G\to H$ are the equivalence classes on $G$ given by $g_1\sim g_2\iff \phi(g_1)=\phi(g_2)$. So there are two partitions of $G$, and I think I need to see (1) why they are the same, (2) why multiplication is "the same".
REPLY [2 votes]: Note that if $g_1 = g_2k$, then $\varphi(g_1) = \varphi(g_2)$ since $\varphi(k) = e$. Thus elements in the same coset are in the same fiber of $\varphi$.
If $g_1$ and $g_2$ are in the same fiber of $\varphi$, then $\varphi(g_1) = \varphi(g_2)$, so that $g_1g_2^{-1}\in K$, or $g_1 = g_2k$ for some $k\in K$.
| 65,645
|
Black Spirituality Religion : Forgiving others, improving yourself
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XenPorta 2 PRO © Jason Axelrod from 8WAYRUN.COM
| 184,804
|
\begin{document}
\author {A. Borichev}
\address{Centre de Math\'ematiques et Informatique, Universit\'e d'Aix-Marseille I, 39, rue Joliot Curie, 13453, Marseille Cedex
13, France}
\email{borichev@cmi.univ-mrs.fr}
\author{K. Gr\"ochenig}
\address{Faculty of Mathematics \\
University of Vienna \\
Nordbergstrasse 15 \\
A-1090 Vienna, Austria}
\email{karlheinz.groechenig@univie.ac.at}
\author{Yu. Lyubarskii}
\address{Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway}
\email{yura@math.ntnu.no}
\title[ Frame constants near the critical density]
{ Frame Constants of Gabor Frames near the Critical Density}
\thanks{A.B. was partially supported by the ANR projects
ANR-07-BLAN-0249 and ANR-09-BLAN-0058-01;
K.G. was supported by the Marie-Curie Excellence Grant
MEXT-CT-2004-517154; Yu.L. was partly supported by the Research
Council of Norway, grants 160192/V30 and 177355/V30.}
\begin{abstract}
We consider Gabor frames generated by a Gaussian function and
describe the behavior of the frame constants as the density of the
lattice approaches the critical value.
\end{abstract}
\subjclass[2000]{ Primary 30H05; Secondary 42C15, 33C90, 94A12.}
\keywords{Gabor frame, frame bounds, sampling inequality, Balian-Low
theorem, Fock space, atomization technique}
\maketitle
\section{Introduction}
In this article we study the stability problem for the expansions of functions
on the real line with respect to a discrete set of phase-space shifts of a Gaussian,
precisely
\begin{equation}
\label{eq:1}
f(x) = \sum _{k,l\in \bZ } c_{kl} e^{2\pi i lax} e^{-\pi (x-bk)^2} \, .
\end{equation}
Expansions of such form (with $a=1$, $b=1$) were introduced by D.~Gabor in his
classical article \cite{Gabor}. Now expansions of type~\eqref{eq:1}, so-called Gabor
expansions, appear in signal processing, quantum mechanics,
time-frequency analysis, the theory of pseudodifferential operators, and other
applications.
During the last decades an extensive theory of expansions \eqref{eq:1} as well as more general Gabor expansions has been developed (see, for instance \cite{Charlybook, Christensen} and the references therein). However, not much is known about numerical stability property of such expansions.
In modern language, the existence of Gabor expansions is derived from
frame theory. To fix terminology and notation, take some $g\in \lr$, it will be called a
window function, and let $\Lambda = M \bZ ^2 \subset \bR ^2$ be a lattice
in $\bR ^2$, where $M $ is a $2\times 2$ invertible real-valued
matrix.
Given a point $\lambda=(x,\xi )$ in phase-space $\bR ^2$, the
corresponding \tfs\ is
$$
\pi_\lambda f(t) = e^{2\pi i \xi t } f(t-x), \qquad t\in \bR
\, .
$$
The set of functions
$\cG (g,\Lambda ) = \{\pi_\lambda g : \lambda \in \Lambda\}$ is
called the \emph{Gabor system} generated by $g$ and $\Lambda$.
We say that such a system is a {\em Gabor frame}
or \emph{Weyl-Heisenberg frame}, whenever
there exist constants $A,B >0$ such that,
for all $f\in \lr$,
\begin{equation}
\label{Eq:2.8}
A\|f\|^2_{\lr} \leq \sum_{\lambda \in
\Lambda}|\langle f, \pi_\lambda g\rangle_{\lr}|^2 \leq
B\|f\|^2_{\lr}\, .
\end{equation}
The (best possible) constants $A=A(\Lambda,g)$ and
$B=B(\Lambda,g)$ in~\eqref{Eq:2.8} are called the
{\em lower and upper frame bounds} for the frame $\cG (g,\Lambda )$.
There is a standard procedure for constructing expansions of type \eqref{eq:1}
for each Gabor frame. Namely there exists a
\emph{dual window} $\gamma \in L^2(\bR )$, such that every $f\in
L^2(\bR )$ can be
expanded as a Gabor series
\begin{equation}
\label{eq:2}
f= \sum _{\lambda \in \Lambda } \langle f, \pi_\lambda \gamma
\rangle \pi_\lambda g.
\end{equation}
Such dual window is, in general, non-unique.
We refer the reader to e.g. ~\cite{Charlybook} for an exposition of Gabor analysis and related matters.
The property of the system $\cG (g,\Lambda )$ to form a frame in $\lr$ depends (among other factors) on geometrical characteristics of $\Lambda$. We say that the area of its fundamental
domain $s(\Lambda ) = \mbox{Area}(M[0,1)^2)= |\det M|$ is the \emph{size} of $\Lambda$.
By the {\em density} of $\Lambda$ we mean $d(\Lambda)=s(\Lambda)^{-1}$; for the lattice case this definition of density coincides with numerous
standard density definitions (see e.g. \cite{seip}).
We refer to~\cite{heil} for a comprehensive account of the density
theorems for Gabor frames. In fact for \emph{any} window function $g$
the condition $s(\Lambda )\leq 1$ is necessary
for $\cG (g, \Lambda )$ to be a frame in $\lr$.
For ``nice'' windows $g$ (in the Schwartz class, say) a
fascinating form of the uncertainty principle, the so-called
Balian-Low Theorem (BLT), requires even that $s(\Lambda )<1$ for $\cG
(g, \Lambda )$ to be a frame~\cite{BHW95}.
The results of \cite{sw,lyu} yield in particular that in the case of
the
Gaussian window
$$
\vf (t) = e^{-\pi t^2},\qquad t\in \bR .
$$
the condition $s(\Lambda ) <1$ is also sufficient:
\medskip
\noindent{\bf Theorem A.} {\em
The set $\cG (\vf , \Lambda )$ is a frame for $\lr $ \fif\
$s(\Lambda )<1$ .}
\medskip
Together with BLT this implies that the lower frame bound
$A=A(\Lambda )$ must tend to $0$, as the size of the lattice
$s(\Lambda ) $ approaches one. Thus the original Gabor series
\eqref{eq:1} with $a=1$, $b=1$ corresponds to the critical case $s(\Lambda )=1$
and does not provide an $L^2$-stable expansion.
\footnote{In his article \cite{Gabor} Gabor considered expansions of functions $f$
which possess additional decay in time and frequency. As we now know (see e.g. \cite{LS}), for such functions the series \eqref{eq:1} converges in $\lr$. }
In this case, there exist $L^2$-functions with polynomially growing
coefficients. See~\cite{janssen, LS} for the convergence properties
of~\eqref{eq:1}.
In this article we are concerned exclusively with Gabor frames for the
Gaussian window $\cG (\vf , \Lambda )$ for the square lattice
$\Lambda(a)=a\ZZ\times a\ZZ$ and study the behavior of its frame constants
$A(a)=A(\Lambda(a))$ and $B(a)=B(\Lambda(a))$ near the critical density $d(\Lambda)=1$. The main result of the article reads as follows.
\begin{th1}
\label{th:1}
There exist constants $0<c<C<\infty$ such that for each $a\in (1/2,1)$ the frame bounds $A(a)$,
$B(a)$ for the frame $\cG (\vf , \Lambda(a) )$ satisfy
\begin{equation}
\label{eq:3}
c (1-a ^2)\leq A(a ) \leq C (1-a ^2)
\end{equation}
and
\begin{equation}
\label{xlst}
c< B(a) < C.
\end{equation}
\end{th1}
\begin{rem}
A similar statement holds for arbitrary rectangular lattices.
The values of $c, C$ in this theorem then depend upon the shape of
the lattice. Nevertheless, one can prove
that given a number $K>0$ there exist
constants $c$ and $C$ valid
for all matrices $M$ such that the diameter
of the fundamental domain $M (0,1]^2$ does not exceed $K$.
\end{rem}
The ratio $B(\Lambda)/A(\Lambda)$ plays the role of the condition number for the frame
$\cG (\vf , \Lambda)$. Thus Theorem \ref{th:1} says how fast does the frame
$\cG (\vf , \Lambda)$ "numerically degenerate" as its density approaches the critical value.
The asymptotical behavior $A(a ) \asymp (1-a ^2) $ has been first observed
numerically by Thomas Strohmer~\cite{str09} and by Peter Sondergaard~\cite{So}.
Moreover, the numerical simulation in \cite{str09} allowed us to guess the construction which
gives
the second i
nequality in \eqref{eq:3}. This construction is described
in Section 4 below.
Next let $g_1(t) = (\mathrm{cosh}\, \pi \gamma t)\inv$, $\gamma >0$,
be the hyperbolic cosine function. Janssen and Strohmer~\cite{JS02}
have shown that $\cG (g_1, a\bZ \times b\bZ)$ is a frame, if and only
if $ab<1$. To do this, they showed that the frame bounds for $\cG (g_1, a\bZ \times b\bZ)$
are equivalent to those of $\cG (g_0, a\bZ \times b\bZ)$ with the
Gaussian $g_0$ and applied Theorem~A. Therefore we obtain the same
asymptotic estimates for the frame bounds for the hyperbolic cosine.
\begin{cor}
There exist constants $0<c<C<\infty$ such that for each $a\in (1/2,1)$
the frame bounds $\tilde{A}(a)$,
$\tilde B(a)$ for the frame $\cG (g_1 , \Lambda(a) )$ satisfy
$$
c (1-a ^2)\le \tilde A(a ) \le C (1-a ^2)
$$
and
$$
c\le \tilde B(a) \le C.
$$
\end{cor}
The proof of Theorem~\ref{th:1} involves both time-frequency
methods and methods of complex analysis.
We use complex analysis in order to obtain the upper estimate for $A(a)$
and the Gabor analysis in order to obtain the rest of the statements in
Theorem 1.1 (though a pure complex-analytic proof is also available).
In particular we apply
Walnut's estimates for the norm of the frame
operator~\cite{walnut92}, and also precise decay estimates for the
dual window established in \cite{superframes}. The upper bound $A(a) \le C(1-a^2)$
will be established by the construction of a concrete example.
We produce a function $f_a$ (depending on
the lattice $\Lambda(a)$), such that
$$
\sum_{\lambda \in \Lambda(a) } |\langle f_a, \pi_\lambda \vf \rangle |^2
\le C(1-a^2) \|f_a\|_{\lr}^2 \, .
$$
By using the Bargmann transform, we translate our
problem into one of finding entire functions in
the Bargmann-Fock space whose restrictions to $\Lambda(a)$
are "small" with respect to
their Fock norms.
The paper is organized as follows. In the next section we discuss the estimates for $B(a)$.
Furthermore, we give the lower estimates for $A(a)$. Here we
mainly follow the arguments from \cite{superframes}.
In section 3 we recall the definition of the Fock space $\cF$ of entire functions, and discuss the
relations between the frame property of the system $\cG (\vf,\Lambda(a))$ and sampling in $\cF$. We also recall basic properties of the Weierstrass $\sigma$-function. In section 4 we use these facts and also special "atomization" techniques in order to construct the example which delivers the upper estimate in \eqref{eq:3}.
\textbf{Notation:} To avoid dealing with too many intermediate
constants, we use the standard notation $f \prec g$ to express
an inequality $f(x) \leq C g(x)$ for all $x$ with a constant $C$
independent of $x$ (and possibly other parameters). Likewise, $f
\asymp g$ means that there exist $A,B>0$ such that $A f(x) \leq g(x)
\leq Bf(x)$ for all $x$.
\section{Time-Frequency Methods To Estimate Frame Bounds}
The estimate \eqref{xlst} on the upper frame bound $B(a)$ can be obtained in various ways.
In particular, we can use Walnut's estimates, which give a sufficient condition for
the Gabor frame operator to be bounded~\cite{walnut92}. This result also follows from the Polya-Plancherel type
inequalities for functions in the Bargmann-Fock space, see below Section 3 for more details.
To obtain the lower estimates for $A(a)$ we need to show the invertibility of the Gabor frame operator
and to estimate the norm of the inverse operator. We will approach this problem by using information
about a suitable dual window $\gamma $ and then apply Walnut's estimates to the Gabor expansion~\eqref{eq:2}.
To state Walnut's result we need the following definitions.
Let $W$ be the the Wiener amalgam space of functions on the real line defined by the norm
$$
\|g\|_W = \sum _{k\in \bZ } \sup _{t\in [0,1]} |g(t+k)| \, .
$$
Given a function $g$ in $L^2(\mathbb R)$, consider the Gabor system $\cG
(g,\Lambda )$ and the corresponding synthesis operator $D_{g,\Lambda }$,
$$
D_{g,\Lambda } \mathbf{c} = \sum _{\lambda \in \Lambda } c_\lambda \pi_\lambda g,
$$
and the analysis operator $C_{g,\Lambda }$,
$$
(C_{g,\Lambda } f)(\lambda ) = \langle f, \pi_\lambda g\rangle,
\qquad \lambda \in \Lambda.
$$
If $D_{g,\Lambda }$ acts continuously from $\ell^2(\Lambda)$ to $L^2(\mathbb R)$, then
$C_{g,\Lambda }$ acts continuously from $L^2(\mathbb R)$ to $\ell^2(\Lambda)$,
and $C_{g,\Lambda } = D_{g,\Lambda }^*$.
The following lemma from \cite{walnut92} gives an estimate for
$\|D_{g,\Lambda}\|_{l^2\to L^2}$:
\begin{lem}\label{walnut}
If $g \in W$ and $\Lambda = a\bZ^2$, then $D_{g,\Lambda
}$ is bounded from $\ell ^2(\Lambda )$ to $L^2(\bR)$ and
$$
\|D_{g,\Lambda }\|_{L^2(\mathbb R)} \leq (1+a\inv ) \|g\|_W \, .
$$
\end{lem}
Since, obviously, in our situation $B(a)=\|C_{g,\Lambda}\|^2_{L^2\to l^2}=
\|D_{g,\Lambda}\|^2_{l^2\to L^2}$, we obtain
\begin{cor}
If $g\in W$ and $a>0$, then
\begin{equation}
\label{eq:c10}
B(a) \le (1+a\inv )^2 \|g\|_W^2 \, .
\end{equation}
\end{cor}
To treat Gabor frames with Gaussian window, we need to evaluate the amalgam space
norm of functions with Gaussian decay.
\begin{lem}\label{gaussdecay}
Assume that $\kappa>0$, $|\gamma (t)| \leq e^{-\pi \kappa t^2}$. Then
\begin{equation}
\label{eq:c11}
\|\gamma \|_W \le 2+\kappa ^{-1/2}.
\end{equation}
\end{lem}
\begin{proof}
For $n\ge 1$, $n\in \bZ$, we have
$$
\sup_{t\in [0,1]} |\gamma (n+t)| \le e^{-\pi \kappa n^2} \le \int_{n-1}^n e^{-\pi \kappa t^2}\, dt \, ,
$$
and likewise for $n<-1$, $n\in \bZ $, we have
$$
\sup _{t\in [0,1]} |\gamma (n+t)| \leq e^{-\pi \kappa (|n|-1)^2} \leq \int
_{|n|-2}^{|n|-1} e^{-\pi \kappa t^2}\, dt \, .
$$
Consequently,
$$
\|\gamma \|_W = \sum _{n\in \bZ } \sup _{t\in [0,1]} |\gamma(t+n)|
\leq 2+ \int _\bR e^{-\pi \kappa t^2}\, dt = 2+\kappa ^{-1/2} \, .
$$
\end{proof}
As a consequence we obtain an estimate on the upper frame bound of
Gaussian Gabor frames.
\begin{prop} \label{upper}
The upper frame bound $B(a)$ of $\cG (\vf , a\bZ^2 )$, $1/2<a<1$, satisfies the estimate
$$
1<B(a)< 100.
$$
\end{prop}
\begin{proof}
For the upper estimate we use \eqref{eq:c10} and \eqref{eq:c11} with $\kappa =1$.
To get the lower estimate we consider the sum \eqref{Eq:2.8} for $f= g = \vf $. Then
$$
\sum _{\lambda \in \Lambda (a)} |\langle \vf, \pi_\lambda \vf \rangle|^2 > \|\vf\|^2,
$$
which yields the desired estimate.
\end{proof}
The time-frequency methods also yield the lower estimate in \eqref{eq:3}. This estimate requires
the existence and some knowledge about a dual window. If $\cG (g, \Lambda )$ is a frame, then
by the frame theory there exists a dual window $\gamma \in L^2(\bR )$, such that every $f\in L^2(\bR )$
possesses a(n unconditionally convergent) series expansion (Gabor expansion) of the form
$$
f= \sum _{\lambda \in \Lambda } \langle f, \pi_\lambda g\rangle \pi_
\lambda \gamma = D_{\gamma , \Lambda } C_{g,\Lambda }f \, .
$$
For the square lattice $\Lambda(a)$,
Lemma~\ref{walnut} yields the following bound:
$$
\|f\|_{\lr}^2 \leq (a\inv +1)^2 \|\gamma \|_W^2 \, \sum
_{\lambda \in \Lambda } |\langle f, \pi_\lambda g\rangle |^2 \, .
$$
Consequently, the lower frame bound $A(a)$ can be estimated from below as
\begin{equation}
\label{eq:c17}
A(a) \ge \bigl((a\inv +1)^2 \|\gamma \|_W^2\bigl)\inv \, .
\end{equation}
\begin{prop}\label{lower}
For the square lattice $\Lambda(a)$, $1/2<a<1$, the lower
bound $A(a)$ of the Gaussian frame $\cG (\vf , \Lambda (a) )$ obeys
the estimate
$$
A(a) \succ 1-a^2.
$$
\end{prop}
\begin{proof}
In \cite{superframes} the authors consider
the Gaussian Gabor frame $\cG (\vf , \Lambda (a) )$. For this frame they
construct a dual window $\gamma$ such that
$$
|\gamma (t) | \leq C e^{-\pi \kappa t^2}
$$
with $\kappa \asymp 1-a^2$.
By Lemma~\ref{gaussdecay} we have
$$
\|\gamma \|_W \prec 2+\kappa ^{-1/2} \prec (1-a^2)^{-1/2},
$$
and the desired estimate follows now from \eqref{eq:c17}.
\end{proof}
\section{Complex Methods}
\subsection{Fock space\label{FM}}
We recall the definition and basic properties of the Fock space.
We refer the reader to \cite{Folland},
\cite{Charlybook} for detailed proofs and also for a discussion of numerous applications
of this space to signal analysis and quantum mechanics.
The {\em Fock space} $\cF$ is the Hilbert space of all entire functions such that
$$
\|F\|^2_\cF:=\int_ \bC |F(z)|^2 e^{-\pi |z|^2} dm_z <\infty,
$$
where $dm_z$ is Lebesgue measure on $\bC $.
The natural inner product in $\cF$ is denoted by $\lng \cdot,\cdot \rng_\cF$.
We will use the following well-known facts:
(a) The point evaluation is a bounded linear functional in $\cF$,
and the corresponding reproducing kernel is the function $w\mapsto
e^{\pi \bar{z} w }$, i.e.,
\beq
\label{reproducing}
F(z) = \lng F, e^{\pi \bar{z} \cdot } \rng_\cF \, , \qquad F\in \cF.
\eeq
(b) One defines the Bargmann transform of a function $f\in L^2(\bR)$ by
$$
f \mapsto \cB f(z)=F(z)= 2^{1/4}e^{-\pi z^2 /2}\int_\bR f(t)e^{-\pi
t^2}e^{2\pi t z} dt.
$$
The Bargmann transform is a unitary mapping from $L^2(\bR)$ onto $\cF$.
(c) In what follows we identify $\bC$ and $\bR^2$. In particular
for each $\zeta=\xi +i\eta \in \bC$ we write
$\pi_\zeta=\pi_{(\xi,\eta)}$.
Define the Fock space shift $\beta_\zeta: \cF \to \cF$
by
$$
\beta_\zeta F(z) = e^{i\pi \xi \eta} e^{-\pi |\zeta|^2/2}e^{\pi {\zeta} z} F(z-\bar{\zeta}).
$$
Then $\beta _\zeta $ is unitary on $\cF $, and the Bargmann transform intertwines
the Fock space shift and the time-frequency shift:
\beq
\label{intertwines}
\beta_\zeta \cB = \cB \pi_\zeta.
\eeq
(d)
\beq
\label{ed}
\cB \vf = 2^{-1/4},
\eeq
here as above $\vf$ is the Gaussian function.
(e) It follows from \eqref{intertwines} and \eqref{ed} that
$$
\cB{\pi_\zeta\vf} = 2^{-1/4} e^{i\pi \xi \eta} e^{-\pi |\zeta|^2/2} e^{\pi \zeta z}\, .
$$
Taking into account the reproducing property \eqref{reproducing},
we can rewrite the frame property \eqref{Eq:2.8} of $\cG (\vf , \Lambda)$ as the
sampling inequality
$$
A\|F\|^2_\cF \leq \frac 1 {2}\sum_{\lambda\in \Lambda} |F(\overline{\lambda})|^2 e^{-\pi |\lambda|^2}
\leq B\|F\|^2_\cF, \qquad F\in \cF.
$$
In the case of square lattice, $\Lambda$ is symmetric with respect to the real line, and we have
\beq
\label{sampling}
A\|F\|^2_\cF \leq \frac 1 {2}\sum_{\lambda\in \Lambda} |F(\lambda)|^2 e^{-\pi |\lambda|^2}
\leq B\|F\|^2_\cF, \qquad F\in \cF.
\eeq
(f)
Let $1/2< a < 2$, and let $w\in \bC$, $w\ne 0$. Consider the entire function
$\Phi_{a, w}(z)=e^{a\bar w z^2/w}$.
Then
$$
|\Phi_{a,w}(z)| \asymp e^{a|z|^2}, \qquad |z-w|< 1.
$$
This statement can be checked by
direct inspection.
\subsection{Reformulation of the main result}
The remaining part of Theorem \ref{th:1} can now be reformulated as follows.
\begin{th1}
\label{th:2}
Let $\Lambda(a)= a \bZ^2$, and let $A(a)$ be the best possible constant in the left hand side inequality of \eqref{sampling}.
Then for $1/2<a<1$ we have
$$
A(a ) \prec 1-a^2.
$$
\end{th1}
To prove this theorem we need to find a constant $K$ and functions $F=F_a \in \cF$
such that
\beq
\label{example}
K(1-a^2)\|F_a\|^2\ge \sum_{\lambda\in \Lambda(a)} |F_a(\lambda)|^2 e^{-\pi |\lambda|^2}.
\eeq
\subsection{Weierstrass $\sigma$-function.} The construction of the functions $F_a$ in the
next section is motivated by the properties of the classical Weierstrass $\sigma$-function.
Let us recall its definition and basic properties. We refer the reader to \cite{Ahiezer}
for a systematic study of this function and also to \cite{superframes}
for its applications in Gabor analysis.
Given a lattice $\Lambda \subset \bC$ we denote
$$
\sigma(\Lambda, z) = z \prod_{\lambda \in \Lambda \setminus \{0\}}
\Bigl(
1 - \frac z \lambda
\Bigr)
e^{\frac z \lambda + \frac 12 \left (\frac z \lambda \right )^2 }.
$$
This product converges uniformly on compact sets in $\bC$ to an entire function with
$\Lambda$ as the zero set. This is a function of order $2$; moreover there exists
$d_\Lambda \in \bC$ such that
$$
|\sigma(\Lambda, z)e^{d_\Lambda z^2}| \asymp e^{\frac \pi 2 s(\Lambda)^{-1}|z|^2}, \qquad
\dist(\Lambda, z)\ge \varepsilon>0.
$$
Here $s(\Lambda)$ is the area of the fundamental domain of $\Lambda$.
See \cite{Hayman} and also \cite{superframes}.
Once again, let $\Lambda(a)= a\bZ^2$. A direct inspection shows that $d_{\Lambda(a)}=0$, so that
\beq
\label{eq:s3}
|\sigma(\Lambda(a), z)| \asymp e^{\frac \pi 2 a^{-2}|z|^2},
\qquad \dist(\Lambda_a, z)\ge \varepsilon>0.
\eeq
This relation allows one to mimic the weight function $e^{\pi
|z|^2/2}$ in the definition of the Fock space by the absolute value
of an analytic function.
\section{Proof of \eqref{example}}
\subsection{Explicit construction}
If $a$ is in a compact subinterval of $(1/2,1)$, one can take $F_a=1$
and obtain \eqref{example} with some appropriate constant $K$.
Therefore, from now on, we assume that $a$ is sufficiently close to 1, say $0.999<a<1$.
Given such
$a$, we take
$R=R(a)$ such that
$$
2(1-a^2 ) < R^{-3/2} < 4( 1-a^2)
$$
and
$$
n_R:=\pi (1-R^{-3/2}) R^2 \in \mathbb N.
$$
We need some additional notation:
\begin{align*}
b^2&=1-R^{-3/2},
\\
\zeta_{m,n}&=b^{-1}(m+in),\notag\\
Q_{m,n}&=\{x+iy\in \bC: |x-b^{-1}m|< b^{-1}/2, |y-b^{-1}n|< b\inv /2\} , \notag\\
D_R&=\{ z \in \bC: |z| < R\} ,\notag\\
D'_R&= \cup \{ Q_{m,n}: |\zeta_{m,n}| < R- 3 \} , \ D''_R = D_R \setminus D'_R, \notag\\
\cN_R&=\{ (m,n)\in \bZ ^2: Q_{m,n} \subset D'_R\}, \notag\\
q_R&=\card \cN_R, \ p_R= n_R-q_R \, . \notag
\end{align*}
We have
\[
\{z: R-1< |z|< R \} \subset D''_R \subset \{z: R-4< |z|< R \}.
\]
Using the appropriate segments of radii of the disc $D_R$ we split $D''_R$ into the "sectors" $A_k$:
\[
D''_R=\bigcup_{k=1}^{p_R}A_k,
\]
such that
$$
\label{eq:20}
m< \diam A_k < M, \qquad \area A_k= b^{-2}
$$
for some $m,M$ independent of $a$.
Denote the center of mass of $A_k$ by
\beq
\label{eq:22}
\zeta_k= b^2\int_{A_k}\zeta dm_\zeta.
\eeq
We can find $c$ independent of $a$ such that
$$
\{w: |w-\zeta_k|< c\} \subset A_k, \qquad k=1,\ldots, p_R.
$$
We are going to verify the estimate \eqref{example} for the function
\beq
\label{eq:09}
F_a(z)= z \prod_{(m,n)\in \cN_R \setminus (0,0)} \left ( 1 - \frac z {\zeta_{m,n}} \right )
\prod_{k=1}^{p_R}\left (1 - \frac z{\zeta_k} \right ).
\eeq
The zero set of the function $F_a$ is
\begin{equation}
\label{eq:911}
\cZ_a=\{\zeta_{m,n}:|\zeta_{m,n}|<R-3\}\cup\{\zeta_k\}_{k=1}^{p_R}\, .
\end{equation}
By construction, the total number of zeros of $F_a$ is $n _R = \pi R^2b^2$.
In order to prove \eqref{example}, we need to estimate both $\|F_a\|_\cF^2$ and
$$
\|F_a\|^2_a:= \sum_{m,n\in \bZ} |F_a(a(m+in))|^2 e^{-\pi a^2(m^2+n^2)}.
$$
\subsection{Estimate of $\|F_a\|_\cF^2$. } To estimate the norm of $F_a$ in the Fock space, we compare
the logarithm of the modulus of the polynomial $F_a$ to a subharmonic function $u_R$ whose growth
is easy to control.
Consider the subharmonic function
$$
u_R(z)
= \int_{|\zeta|<R} \log \Bigl| 1- \frac z \zeta
\Bigr| dm_\zeta
= \left \{ \begin{array}{ll}
\frac \pi 2 |z|^2, & |z|<R, \\
\pi R^2\log |z| -\pi R^2\log R + \frac \pi 2 R^2, & |z|>R .
\end{array} \right .
$$
An easy estimate shows that
$$
u_R(z) < \frac \pi 2 |z|^2, \qquad |z| >R.
$$
We use the following approximation lemma.
\begin{lem}
\label{l:1nn}
For each $\varepsilon >0$ there exist constants $0<c(\varepsilon)<C(\varepsilon)< \infty$ such that \begin{equation}
\label{eq:09ab}
c(\varepsilon) |F_a(z)| < e^{ b^2 u_R(z)} < C(\varepsilon) |F_a(z)|, \qquad \dist(z,\cZ _a) > \varepsilon.
\end{equation}
and
\begin{equation}
\label{eq:09ac}
c(\varepsilon) |F_a(z)| < e^{ b^2 u_R(z)}, \qquad \dist(z,\cZ_a) \le \varepsilon.
\end{equation}
\end{lem}
\medskip
\noindent{\bf Remark.} Since the set $\cN$ is invariant with respect to rotation
by $\pi/2$ around the origin, we find that
$$
\sum _{(m,n)\in \cN_R \setminus (0,0)} \frac{1}{m+in} = \sum
_{(m,n)\in \cN_R \setminus (0,0)} \frac{1}{(m+in)^2} = 0 \, .
$$
So the first factor on the right-hand side of
\eqref{eq:09} in the definition of $F_a$ can be written as
\begin{multline}
V_R(z)=z \prod_{(m,n)\in \cN_R \setminus (0,0)} \Bigl( 1 - \frac z {\zeta_{m,n}} \Bigr) \\=
z\prod_{(m,n)\in \cN_R \setminus (0,0) } \Bigl ( 1 - \frac z {\zeta_{m,n}} \Bigr)
\exp \Bigl( {\frac z {\zeta_{m,n}}+ \frac 12 \Bigl(\frac z {\zeta_{m,n}}\Bigr)^2}
\Bigr).
\label{n68}
\end{multline}
Consequently, the function $V_R$ can be viewed as a truncated version of the Weierstrass
$\sigma$-function and estimates \eqref{eq:09ab} and \eqref{eq:09ac}
correspond to the
growth estimate \eqref{eq:s3} for the Weierstrass
$\sigma$-function.
We postpone the proof of this
technical lemma until subsection \ref{last}.
Assuming that Lemma \ref{l:1nn} is already proved, an estimate of $\|F_a\|_\cF^2$ is straightforward.
\begin{lem}
\label{l:02n}
$$
\|F_a\|^2_\cF \succ R^{3/2} \asymp (1-a^2)^{-1}.
$$
\end{lem}
\begin{proof}
Let $\Omega=\{z\in\mathbb C:|z|<R,\,\dist(z,\mathcal Z_a)>1/10\}.
$ Using Lemma~\ref{l:1nn}, we find that
$$
\|F_a\|^2_\cF \succ \int _\Omega
e^{2 b^{2}u_R(z)-\pi |z|^2} dm_z = I(a,R).
$$
We use that $1-b^2= R^{-3/2}$. Furthermore, for every $1<r<R$, the circle $z:|z|=r$ intersects with $\Omega$
on at most half of its length.
Therefore,
$$
I(a,R) \succ \int_1^{R}e^{-\pi R^{-3/2}t^2} t dt=
\frac{R^{3/2}}2\int_{R^{-3/2}}^{R^{1/2}}e^{-\pi u} du \asymp R^{3/2},
$$
and the statement of the Lemma now follows.
\end{proof}
\noindent{\bf Remark. } {\em A similar argument shows that }
\beq
\label{eq:24}
\int_{|z|>R-4}|F_a(z)|^2 e^{-\pi |z|^2} dm_z \to 0,
\eeq
{\em as $a\to1$ or equivalently, as $R\to \infty$.}
\subsection{Estimate of $\|F_a\|_a^2$. }
\begin{lem}
\label{l:03}
For $F_a$ as in \eqref{eq:09} we have
\begin{equation}
\label{eq:09c}
\|F_a \|^2_a= \sum_{m,n} |F_a(a(m+in))|^2 e^{-\pi a^2(m^2+n^2)}\asymp 1.
\end{equation}
\end{lem}
\begin{proof}
We have
$$
\|F_a \|^2_a=
\Bigl(
\sum_{(m,n)\in \cN_R} + \sum_{(m,n)\not\in \cN_R}
\Bigr) |F_a(a(m+in))|^2 e^{-\pi a^2(m^2+n^2)}
= \Sigma_1(a) + \Sigma_2(a).
$$
In order to estimate $ \Sigma_1(a)$, we observe that
$F_a(\zeta_{m,n})=0$, for $(m,n)\in \cN_R$ and
$$
|\zeta_{m,n}- a(m+in)| = (b^{-1}-a)|m+in| < 2R^{-3/2}|m+in|.
$$
Since $|m+in|< R$ for $(m,n)\in \cN_R$, and
$a$ is sufficiently close to $1$, we have
$$
|\zeta_{m,n}- a(m+in)| < \frac 15, \qquad (m,n)\in \cN_R.
$$
Denote $D_{m,n}=\{z\in \bC: |z -\zeta_{m,n}| < 1/4\}$. By
part (f) of subsection~\ref{FM} there exists a function
$\Phi_{m,n}(z)$ that is holomorphic on $D_{m,n}$ and satisfies
\beq
\label{eq:s4a}
|\Phi_{m,n}(z)| \asymp e^{b^2\frac \pi 2 |z|^2}, \qquad z\in D_{m,n}.
\eeq
Then for each $(m,n)\in \cN_R$ the function
$$
\Psi_{m,n}(z) = \frac {F_a(z)}{\Phi_{m,n}(z)}
$$
is holomorphic in $D_{m,n}$ and possesses the properties
$$
\Psi _{m,n}(\zeta_{m,n})= 0 \ \mbox{and} \ |\Psi _{m,n}(z)|\prec 1.
$$
By Cauchy's theorem, the functions $\Psi'_{m,n}$ are
uniformly bounded on $D^*_{m,n}=\{z\in \bC: |z -\zeta_{m,n}| < 1/5\}$, and hence
$$
|\Psi_{m,n}(a(m+in))| \prec |(a-b^{-1})(m+in)| \prec R^{-3/2}(m^2+n^2)^{1/2}, \quad (m,n)\in \cN_R.
$$
Returning to the function $F_a$ and using
\eqref{eq:s4a} once again, we obtain
$$
|F_a(a(m+in))|^2e^{-\pi a^2 (m^2+n^2)} \prec
R^{-3}(m^2+n^2) |\Phi_{m,n}(a(m+in))|^{2(1-b^{-2})}.
$$
The mean value inequality for $\bigl|\Phi_{m,n}(a(m+in))^{2(1-b^{-2})}\bigr|$ now yields
\begin{multline*}
|F_a(a(m+in))|^2e^{-\pi a^2 (m^2+n^2)} \prec
R^{-3}(m^2+n^2)
\int_{D_{m,n}} |\Phi_{m,n}(z)|^{2(1-b^{-2})} dm_z \\ \prec
R^{-3} \int_{D_{n,m}} |z^2| e^{\pi (b^2-1)|z|^2}dm_z.
\end{multline*}
Since all discs $D_{m,n}$ are disjoint we obtain
\beq
\label{eq:32}
\Sigma_1(a) \prec R^{-3}\int_{|z|<R}|z^2| e^{\pi (b^2-1)|z|^2}dm_z
\prec R^{-3}\int_0^R t^2 e^{-\pi R^{-3/2}t^2} tdt \prec 1.
\eeq
\medskip
Finally, for arbitrary $(m,n)$, the mean value theorem yields
\[
|F_a(a(m+in))|^2e^{-\pi a^2 (m^2+n^2)} \prec \\
\int_{D_{m,n}} |F_a(z)|^2 e^{-\pi |z|^2} dm_z,
\]
and, by \eqref{eq:24} we obtain
\beq
\label{eq:a14}
\Sigma_2(a) \prec \int_{|z|>R-4} |F_a(z)|^2 e^{-\pi |z|^2} dm_z \to 0, \
\mbox{as} \ a \nearrow 1.
\eeq
The estimates \eqref{eq:32} and \eqref{eq:a14} yield
\begin{equation}
\|F_a \|^2_a= \sum_{m,n} |F_a(a(m+in))|^2 e^{-\pi a^2(m^2+n^2)}\prec 1.
\label{e681}
\end{equation}
The opposite relation follows from Lemma~\ref{l:02n} and from the lower estimate on $A(a)$ established in
Proposition~\ref{lower}.
\end{proof}
\medskip
Relation \eqref{example} follows immediately from Lemma~\ref{l:02n} and the estimate \eqref{eq:09c} (or even \eqref{e681}).
\medskip
\subsection{Proof of the approximation lemma}
\label{last}
The proof of Lemma \ref{l:1nn} is based on atomization techniques, see e.g.
\cite{atomization}. First we rewrite \eqref{eq:09ab} and
\eqref{eq:09ac} in an additive form. We must prove that
\begin{equation}
\label{eq:09a}
\log |F_a(z)| = b^2 u_R(z) + O(1), \qquad \dist(z,\mathcal{Z}_a) > \varepsilon,
\end{equation}
and
$$
\log |F_a(z)| \leq b^2 u_R(z) +O(1), \qquad \dist(z,\cZ _a) \le \varepsilon,
$$
where $\cZ _a=\{\zeta_{m,n}: (m,n)\in \cN_R\}\cup
\{\zeta_k\}_{k=1}^{p_R}$ as in \eqref{eq:911},
and the quantities $O(1)$ in the right-hand sides of these relations are
bounded uniformly with respect to all $a\in (0.999,1)$ and depend only
on $\varepsilon$. It suffices to prove \eqref{eq:09a}, the second
relation will then follow by the maximum principle applied to $F_a\Phi_{a,w}^{-1}$ where
$\Phi_{a,w}$ is defined in part (f) of subsection~\ref{FM}.
Let $V_R$ be defined by \eqref{n68},
$$
v_R(z)= b^{2} \int_{D'_R} \log\left |1 - \frac z{\zeta} \right | dm_\zeta,
$$
and let
$$
w_R(z)= b^{2} \int_{D''_R} \log \left |1 - \frac z{\zeta} \right |dm_\zeta,
\qquad W_R(z)= \prod_1^{p_R}\left (1- \frac z {\zeta_k} \right ).
$$
We have
\begin{multline}
\label{eq:b06}
\log |F_a(z)| - b^2 u_R(z) = \\ \left (\log |V_R(z)| - v_R(z) \right ) +
\left (\log |W_R(z)| - w_R(z) \right ) = \mathfrak S_1(R,z)+ \mathfrak S_2(R,z),
\end{multline}
and we estimate separately each summand in the right-hand side of \eqref{eq:b06}.
Let
$ \dist(z,\cZ _a) > \varepsilon$. We have
\begin{multline*}
\mathfrak S_1(R,z) = \log |V_R(z)| - v_R(z)=
b^2\int_{Q_{0,0}} \left (
\log |z| -
\log\left |1 - \frac z \zeta\right |
\right ) dm_\zeta
\\
+ b^2\Bigl(
\sum_{\substack{
(m,n)\in \cN_R\setminus\{0,0\}, \\ \mbox{dist}(z,Q_{m,n})\leq 10
}} +
\sum_{\substack{(m,n)\in \cN_R\setminus\{0,0\}, \\ \mbox{dist}(z,Q_{m,n})> 10
} }
\Bigr)
\underbrace{
\int_{Q_{m,n}} \left (
\log \left |1-\frac z{\zeta_{m,n} } \right | -
\log\left |1 - \frac z \zeta\right |
\right ) dm_\zeta
}
_{j_{m,n}}.
\end{multline*}
It suffices to estimate just the second sum in the right-hand side because the first sum
contains only a finite number (at most $1000$, say) of uniformly bounded terms, and the first
integral is bounded uniformly in $z$.
Denote $L(\zeta)=\log(1 - z/\zeta)$. We then have
$$
j_{m,n}= \Re \Bigl [ \int_{Q_{m,n}}\bigl(
L(\zeta)-L(\zeta_{m,n})dm_\zeta
\bigr )\Bigr].
$$
We apply the second order Taylor expansion with the remainder term in the integral form:
\[
L(\zeta)-L(\zeta_{m,n}) = L'(\zeta_{m,n})(\zeta-\zeta_{m,n}) +
\frac 12 L''(\zeta_{m,n})(\zeta-\zeta_{m,n})^2 +
\frac 1 2 \int_{\zeta_{m,n}}^\zeta L'''(s) (\zeta-s)^2 ds.
\]
and use the fact that
\[
\int_{Q_{m,n}} (\zeta-\zeta_{m,n})dm_\zeta=
\int_{Q_{m,n}} (\zeta-\zeta_{m,n})^2dm_\zeta=0.
\]
Then
\begin{multline*}
|j_{m,n}|= \Bigl |
\int_{Q_{m,n}} \int_{\zeta_{m,n}}^\zeta
{(\zeta-s)^2} \left ( \frac 1 {(s-z)^3} - \frac 1{s^3} \right ) ds\, dm_\zeta
\Bigr|
\\ \prec \frac 1 {\mbox{dist}(z,Q_{m,n})^3}+ \frac 1 {\mbox{dist}(0,Q_{m,n})^3},
\end{multline*}
which implies that
$$
\mathfrak S_1(R,z) = O(1).
$$
\medskip
Finally,
\begin{multline*}
\mathfrak S_2(R,z)= | W_R(z) - w_R(z) |= \\ \Bigl | b^2\Bigl (
\sum_{ \mbox{dist}( z, A_k) \le M+10} +\sum_{\mbox{dist}( z, A_k) >M+10}
\Bigr)
\underbrace{
\int_{A_k} \left (\log \left |1 - \frac z{\zeta_k} \right |-
\log \left |1- \frac z{\zeta} \right | \right )dm_\zeta
}_{i_k}
\Bigr|,
\end{multline*}
with $M$ as in \eqref{eq:20}.
The first term in the right-hand side contains just a finite number of summands and
is always bounded.
In order to estimate each $i_k$ from the second term we use the Taylor formula (now of the
first order) with the same function $L(\zeta)=\log (1- z / \zeta)$.
The choice of $\zeta _k$ in \eqref{eq:22} implies that
$\int _{A_k} (\zeta -\zeta _k) dm_\zeta = 0$. Arguing as above,
we obtain
\[
|i_k| \prec \frac 1 {\dist (z, A_k)^2} + \frac 1 {\dist (0, A_k)^2},
\]
whence
$$
\mathfrak S_2(R,z) = O(1).
$$
This completes the proof of Lemma \ref{l:1nn}. $\Box$
| 48,038
|
Content
Setup ntopng on Centos 6
ntopng is a very powerful network traffic monitoring system. The interface has some awesome features like viewing of network traffic, including top hosts data, top flow talkers, application protocols in use, top flow senders data in live mode. Also using its web interface each and every node’s active flow can be viewed live.
1. Install EPEL/NTOP repo
Add EPEL repository by using wget command for download rpm file and then install it. If you have not installed wget then install it by using #yum install wget
Now download epel repository and install it using following commands
# cd ~ # wget # rpm -ivh epel-release-6-8.noarch.rpm
Once EPEL repository is installed, install NTOP repository.
# cd /etc/yum.repos.d/ # wget -O ntop.repo
2. Install Redis
Redis and Hiredis are the required packages for the installation.
# yum install redis hiredis
3. Install the Application
Install along with other packages.
# yum clean all # yum update # yum install pfring n2disk nprobe ntopng ntopng-data cento nbox
4. Enable Auto startup
# chkconfig redis on # chkconfig ntopng on
5. Configure Firewall
Configure firewall to allow traffic to port no 3000
# iptables -A INPUT -m state –state NEW -m tcp -p tcp –dport 3000 -j ACCEPT # iptables -A INPUT -m state –state NEW -m tcp -p tcp –dport 6379 -j ACCEPT # service iptables save # service iptables restart
6. Create configuration file
Now we will create a configuration files in /usr/local/etc/ntopng directory.
# cd /usr/local/etc # mkdir ntopng # cd ntopng # nano ntopng.start
Put these lines :
–local-network “172.31.0.0”
–interface 0
# nano ntopng.pid
Put this line :
-G=/var/run/ntopng.pid
7. Run the application
# service redis start # service ntopng start
Check the log file
8. Testing
Now you can test the application by typing . You will see the login page.
For the first time, you can use user ‘admin’ and password ‘admin’. You will be redirected to the dashboard.
Now click active flows
Click GEOMap
Click tree
Click autonomous system
7.Configuring flow collector to receive flow from another device such as Cisco Router.
Edit the config file as shown and add the following line at the end of the file and save the file [ You may choose other port number , in this tutorial we use port 5559 ]
nano /etc/ntopng/ntopng.conf -i=tcp://your-sender-ip-address:5559
Next we need to start the collector with the following command
nprobe –zmq “tcp://your-sender-ip-address:5559” =i none -n none –collector-port 2055
8.Cisco Router IP Flow Configuration Example
Global Configuration
config#ip flow-cache timeout active 1
config#ip flow-export source GigabitEthernet0/1
config#ip flow-export version 9
config#ip flow-export destination your-server-ip-address 2055
On the interface you want enable flow capturing so as to send it to the ntop server. [ This example illustrate using GigabitEthernet0/1]
config# interface GigabitEthernet0/1 config-if# ip flow ingress config-if# ip flow egress
Congratulation! Your server should be now receiving flow data from your wan device for traffics analysis.
| 242,362
|
\begin{document}
\title{Relative homological linking in critical point theory}
\author[A. Girouard]{Alexandre Girouard}
\address{D\'epartement de Math\'ematiques et
Statistique, Universit\'e de Montr\'eal, C. P. 6128,
Succ. Centre-ville, Montr\'eal, Canada H3C 3J7}
\subjclass[2000]{Primary 58E05}
\email{girouard@dms.umontreal.ca}
\begin{abstract}
A relative homological linking of pairs is proposed. It is shown to
imply homotopical linking, as well as earlier non-relative notion of
homological linkings. Using Morse theory we prove a simple
``homological linking principle'', thereby generalizing and
simplifying many well known results in critical point theory.
\end{abstract}
\maketitle
\section*{Introduction}
The use of linking methods in critical point theory is rather new. It
was implicitely present in the work of Ambrosetti and Rabinowitz
\cite{ar:1} in the early 70's as well as in the work of Benci and Rabinowitz \cite{benci:1}.
The first explicit definition was given by Ni in 1980 \cite{ni:1}.
\begin{definition}[Classical Homotopical Linking]
Let $A\subset B$ and $Q$ be subspaces of a topological space $X$
such that the pair $(B,A)$ is homeomorphic to $(D^n,S^{n-1})$. Then
\emph{$A$ homotopically links $Q$} if for each deformation
$\eta:[0,1]\times B\rightarrow X$ fixing $A$, $\eta(1,B)\cap Q \neq
\emptyset$.
\end{definition}
In the early 80's, homological linking was introduced in critical point theory
(see Fadell \cite{fadell:1}, Benci \cite{benci:2} and Chang \cite{chang:2} for instance).
\begin{definition}[Classical Homological Linking]
Let $A$ and $S$ be non-empty disjoint subspaces in a topological space $X$.
Then \emph{$A$ homologically links $S$} if the inclusion of $A$ in $X\setminus S$ induces a
non-trivial homomorphism in reduced homology.
\end{definition}
In her 1999's article \cite{frigon:1}, Frigon generalized homotopical linking to pairs of subspaces.
\begin{definition}[Relative Homotopical Linking]
Let $(B,A)$ and $(Q,P)$ be two pairs of subspaces in a topological space $X$
such that $B\cap P=\emptyset$ and $A\cap Q=\emptyset$. Then
\emph{$(B,A)$ homotopically links $(Q,P)$}
if for each deformation $\eta:[0,1]\times B\rightarrow
X$ fixing $A$ pointwise, $\eta(1,B)\cap Q=\emptyset\Rightarrow \exists t\in
]0,1], \eta(t,B)\cap P\neq \emptyset$.
\end{definition}
The classical definition corresponds to the case where
$(B,A)\cong (D^n,S^{n-1})$ and $P=\emptyset$.
The goal of this article is to propose a similar generalization for
homological linking.
In section 1.1 we explore the properties of this new homological
linking and in 1.2 we give some detailed examples.
In section 2 we interpret homotopical linking as an obstruction to
factoring certain homotopy through homotopically trivial pairs. It
becomes clear from this point of view that homological linking is
stronger than homotopical linking.
Our definition of homological linking fits very nicely with Morse theory. We exploit this
in section 3 to derive a new linking principle (see
\ref{homo_link_prin}) for detecting and locating critical
points. Despite its simplicity, the idea is quite fruitful. Close
analog to the Mountain Pass Theorem of Ambrosetti and Rabinowitz
\cite{ar:1} as well as to the Saddle Point Theorem of Rabinowitz
\cite{rabinowitz:1} are easy corollaries. In Proposition
\ref{FrigHomotop}, we also obtain a homological version of the
generalized saddle point theorem of Frigon \cite{frigon:1}. In section
4, some multiplicity results are studied.
Our approach has many advantages: each critical point is detected by a different linking,
stability type is directly available (i.e. critical groups are known) and last but not least,
the proofs are easy. However, it also has a disadvantage: working with Morse theory requires
more regularity than using a ``min-max'' method for example.
It might appear as if the content of this paper is extremely easy. We agree
with this point of view. In fact, it is rather surprising to see that so many of the
classical results of critical point theory are straightforward consequences of this new
definition of homological linking.
This paper is an extension of the author's master's thesis \cite{g:1}. He
would like to express his most sincere thanks to his advisor, Marlène
Frigon.
\tableofcontents
\section{Homological linking}
\subsection{Definition and properties}
The principal contribution of this article is the following definition.
\begin{definition}[Relative Homological Linking]
Let $(B,A)$ and $(Q,P)$ be pairs of subspaces in a topological
space $X$.
Then {\em $(B,A)$ homologically links $(Q,P)$ in $X$}
if $(B,A) \subset (X\setminus P, X\setminus Q)$
and if this inclusion induces
a non-trivial homomorphism in reduced homology.
Given integers $q, \beta \geq 0$, we say that
{\em $$(B,A)\ (q,\beta)\mbox{-links } (Q,P)\mbox{ in } X$$} if
the above inclusion induces a homomorphism of rank~$\beta$ on the
$q$-th reduced homology groups.
\end{definition}
\begin{remark}
For notational convenience, a topological pair $(B,\emptyset)$ will be
identified with the space $B$.
\end{remark}
\begin{remark}
The classical definition corresponds to the case
$A\ (q,\beta)\mbox{-links } (X,Q)$ and $\beta>0$.
\end{remark}
\begin{remark}\label{betti}
For any space $X$, $X\ (q,b_q(X))\mbox{-links } X$ in $X$, where $b_q(X)$ is the $q$-th reduced Betti number of $X$.
Thus our linking contains as much information as Betti numbers.
\end{remark}
The next proposition and it's corollary shows that in many situations,
it suffices to consider linking locally to deduce a global linking situation.
\begin{proposition}
Let $\mathcal{O}$ be an open subset of $X$. If\\ $A,B,P,Q\subset
\mathcal{O}$ with $Q$ closed, then
\begin{gather*}
(B,A)\ (q,\beta)\mbox{-links } (Q,P)\mbox{ in } X\\
\Leftrightarrow\\
(B,A)\ (q,\beta)\mbox{-links } (Q,P)\mbox{ in } \mathcal{O}.
\end{gather*}
\end{proposition}
\begin{proof}
Since $\mathcal{O}^c$ is closed and $X\setminus Q$ is open
in $X\setminus P$, the excision axiom applies to
$$\mathcal{O}^c \subset X\setminus Q \subset X\setminus P.$$
It follows that the the bottom line of the following commutative diagram is an isomorphism.
$$\xymatrix{
\tilde{H}_q(B,A)\ar[d]^i\ar[rd]^j & \\
\tilde{H}_q(\mathcal{O}\setminus P, \mathcal{O}\setminus Q)\ar[r]^{\cong} &
\tilde{H}_q(X\setminus P, X\setminus Q)
}$$
Hence, $\mbox{rank } j=\mbox{rank } i$.
\end{proof}
\begin{corollary}
Let $\mathcal{O}$ be the domain of a chart on a manifold $M$. If the
pair $(B,A)$ links the pair $(Q,P)$ in $\mathcal{O}$, with $Q$
closed, then $(B,A)$ also links the pair $(Q,P)$ in $M$.
\end{corollary}
The two following theorems show how some simple linking situations
lead to new linkings.
\begin{theorem}\label{thm_linking_1}
If $A\ (q,\beta)\mbox{-links } (X,Q)$ and $A\ (q, \delta)\mbox{-links } (X,X\setminus B)$ in $X$
for some $\delta<\beta$ then $(B,A)\ (q+1,\mu)\mbox{-links } Q$ in $X$ for some $\mu\geq\beta-\delta$.
\end{theorem}
\begin{proof}
It follows from the commutativity of
$$\xymatrix{
\tilde{H}_{q+1}(B,A) \ar[r]^{\Delta_1}\ar[d]^\alpha&\tilde{H}_q(A)\ar[r]^k\ar[d]^i&\tilde{H}_q(B)\ar[d]\\
\tilde{H}_{q+1}(X,X\setminus Q) \ar[r]^{\Delta_2}&\tilde{H}_q(X\setminus Q)\ar[r]&\tilde{H}_q(X)
}$$
that
\begin{align*}
\mu:=\mbox{rank }\alpha &\geq\mbox{rank } \Delta_2\circ \alpha =\mbox{rank }i\circ \Delta_1\\
&\geq \mbox{rank }\Delta_1 - \dim(\ker i)\\
&= \mbox{rang }\Delta_1 - (\dim \tilde{H}_q(A) - \mbox{rank }i)\\
&= \mbox{rank }i+\mbox{rank }\Delta_1 -\dim \tilde{H}_q(A)\\
&= \mbox{rank }i+\mbox{rank }\Delta_1 -(\mbox{rank }k + \dim(\ker k)).
\end{align*}
By exactness, $\mbox{rank }\Delta_1 = \dim(\ker k)$, thus
$$\mu \geq \mbox{rank }i - \mbox{rank }k = \beta - \delta.$$
\end{proof}
\begin{theorem}\label{thm_linking_2}
If $B\ (q,\beta)\mbox{-links } (X,P)$ and
$X\setminus Q\ (q,\delta)\mbox{-links } (X,P)$ for some
$\delta<\beta$, then $B\ (q,\mu)\mbox{-links }(Q,P)$ in $X$
for some $\mu\geq \beta-\delta$.
\end{theorem}
\begin{proof}
From the commutativity of
$$\xymatrix{
&\tilde{H}_q(B)\ar[r]^\cong\ar[d]^i&\tilde{H}_q(B,\emptyset)\ar[d]^\alpha\\
\tilde{H}_q(X\setminus Q)\ar[r]^k&\tilde{H}_q(X\setminus P)\ar[r]^j&\tilde{H}_q(X\setminus P,X\setminus Q)
}$$
it follows that
\begin{align*}
\mu=\mbox{rank }\alpha &= \mbox{rank }j\circ i\\
&\geq \mbox{rank }i -\dim(\ker j)\\
&=\mbox{rank }i - \mbox{rank }k\\
&= \beta - \delta.
\end{align*}
\end{proof}
\subsection{Examples of linking}
Our definition permits to obtain new situations of linking and to recover others already known.
In particular, in Propositions \ref{enlacement_1}, \ref{enlacement_2} and \ref{enlacement_3}
we present linking situations equivalent to those already studied by
Perera in \cite{perera:1} using a non relative definition of homological
linking.
Let $E$ be a Banach space. Given a direct sum decomposition
$E = E_1 \oplus E_2$, $B_i$ denotes the closed ball in $E_i$ and $S_i$
its relative boundary ($i=1,2$).
\begin{proposition}\label{enlacement_1}
Let $e\in E$, $\|e\|>1$.
Then $\{0, e\}\ (0,1)\mbox{-links }(E,S)$ in $E$.
\end{proposition}
\begin{proof}
The map $r:E\setminus S \rightarrow \{0,e\}$ defined
by
$$r(x)=\left\{
\begin{array}{ll}
0&\mbox{if } \|x\| < 1,\\
e&\mbox{if } \|x\| > 1.
\end{array}
\right.
$$
is a retraction. That is, the following diagram commutes
$$\xymatrix{
E\setminus S \ar[r]^r&\{0,e\}\\
\{0,e\}\ar[ru]_{\mbox{id}}\ar[u]&
}$$
It follows that the inclusion of $\{0,e\}$ in $E\setminus S$ is of
rank 1 in reduced homology.
\end{proof}
\begin{proposition}\label{enlacement_2}
Let $E=E_1\oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$.
Then $$S_1 \ (k-1,1)\mbox{-links } (E,E_2)$$ in $E$.
\end{proposition}
\begin{proof}
The long exact sequence induced by $S_1~\subset~E\setminus~E_2$ is
$$\cdots\rightarrow \tilde{H}_k(E\setminus E_2, S_1) \rightarrow
\tilde{H}_{k-1}(S_1) \stackrel{i}{\rightarrow}
\tilde{H}_{k-1}(E\setminus E_2)\rightarrow \cdots$$
Because $E\setminus E_2$ strongly retract on $S_1$,
$H_k(E\setminus E_2,S_1)=0$. It follows that
$\mbox{rank }i=\dim\tilde{H}_{k-1}(S_1)=1$.
\end{proof}
\begin{proposition}\label{enlacement_3}
Let $E=E_1\oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$
and let $e \in E_2$ be of unit length.
Let $A=\partial (B_1\oplus [0,2]e)$ in $E_1\oplus \mathbb{R}e$.
Then $A \ (k,1)\mbox{-links } (E,S_2)$ in $E$.
\end{proposition}
\begin{proof}
Let $P:E\rightarrow E_1$ be the projection on $E_1$ and
$r:E\setminus S_2 \rightarrow (E_1\oplus \mathbb{R}e) \setminus \{e\}$
be defined by $r(x)=P(x)+\|x-P(x)\|e$.
Let's make sure $\{e\}$ really is omitted by $r$.
Suppose $x\in E$ is such that $P(x)+\|x-P(x)\|e=e$. Then $P(x)=0$ and
$1=\|x-P(x)\|=\|x\|$. In other words, $x\in E_2$ and $\|x\|=1$ wich
is impossible for $x$ in the domain of $r$.
Let $i$ be the inclusion of $A$ in $E\setminus S_2$.
If $i_k:\tilde{H}_k(A)\rightarrow \tilde{H}_k(E\setminus S_2)$ is
null, then so is
$$r_k\circ i_k:\tilde{H}_k(A)\rightarrow \tilde{H}_k((E_1\oplus
\mathbb{R}e)\setminus \{e\}).$$
However, $r\circ i$ is the inclusion of $A$ in $(E_1\oplus
\mathbb{R}e)\setminus \{e\}$ and
$(E_1\oplus \mathbb{R}e)\setminus \{e\}$ strongly retract on
$A$. Thus $\tilde{H}_*((E_1\oplus \mathbb{R}e)\setminus
\{e\},A)\cong 0$. It then follows from the long exact sequence
induced by the inclusion $r\circ i$ of $A$ in $(E_1\oplus \mathbb{R}e)\setminus \{e\}$
$$0=\tilde{H}_{k+1}((E_1\oplus \mathbb{R}e)\setminus \{e\},A)\rightarrow
\tilde{H}_k(A)\overset{{r_k\circ i}_k}{\rightarrow}
\tilde{H}_k(E_1\oplus \mathbb{R}e\setminus \{e\})$$
that ${r_k\circ i}_k$ is not trivial because $\tilde{H}_k(A)\cong
\mathbb{K}$. Consequently $A\ (k,1)\mbox{-links } (E,S_2)$ in $E$, as
was to be proved.
\end{proof}
Theorem \ref{thm_linking_1} and the previous linking situations give
rise to other linkings which are in fact the classical situations
treated in the litterature. Observe that, in these classical
situations, the pair $(Q,P)$ is always of the form $(Q,\emptyset)$ and
the pair $(B,A)$ always has $A\neq\emptyset$.
\begin{corollary}\label{cor_enlacement_1}
Let $e\in E$ with $\|e\|>1$.
Then $([0,e],\{0, e\})\ (1,1)$-links $S$ in $E$.
\end{corollary}
\begin{corollary}\label{cor_enlacement_2}
Let $E= E_1 \oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$.
Then $$(B_1,S_1) \ (k,1)\mbox{-links } E_2$$ in $E$.
\end{corollary}
\begin{corollary}\label{cor_enlacement_3}
Let $E= E_1 \oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$
and let $e \in E_2$ be of unit length.
Let $B=B_1\oplus [0,2]e$ and $A=\partial B$ in
$E_1\oplus \mathbb{R}e$.
Then $(B,A)\ (k+1,1)\mbox{-links }S_2$ in $E$.
\end{corollary}
By combining the linking situations of proposition
\ref{enlacement_1}, \ref{enlacement_2} and \ref{enlacement_3} with theorem
\ref{thm_linking_2}, we get a new familly of linking situations.
These linking situation will be particularyly useful in applications to
critical point theory since they will allow us to relax the a priori estimates on $f$.
For these linking, the pair $(B,A)$ is always of the form $(B,\emptyset)$ and the pair $(Q,P)$
always has $P\neq\emptyset$.
\begin{corollary}\label{cor_enlacement_4}
Let $e\in E$, $\|e\|>1$.
Then $$\{0, e\}\ (0,1)\mbox{-links }(B,S)$$
in $E$.
\end{corollary}
\begin{corollary}\label{cor_enlacement_5}
Let $E=E_1\oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$ and let $e\in E_1$ be of unit
length.
Let $B=S_1, Q=E_2+[0,\infty[e$ and $P=E_2$.
Then $$B \ (k-1,1)\mbox{-links } (Q,P)$$ in $E$.
\end{corollary}
\begin{corollary}\label{cor_enlacement_6}
Let $E=E_1\oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$
and let $e \in E_2$ be of unit length.
Let $A=\partial (B_1\oplus [0,2]e)$ in $E_1\oplus \mathbb{R}e$.
Then $A \ (k,1)\mbox{-links } (B_2,S_2)$ in $E$.
\end{corollary}
The two following propositions exhibit new homological linking
situations. From a homotopical point of view, they where studied by
Frigon \cite{frigon:1}. These linking fully deserve to be called ``linking of pairs'' since for
both of them we have $A\neq\emptyset$ and $P\neq\emptyset$. A more geometrical argument is also
possible, but it is longuer.
\begin{proposition}\label{enlacement_4}
Let $E=E_1\oplus E_2\oplus \mathbb{R}e$ with
$e\in E$ of unit length and $k= \dim E_1 \in\ ]0,\infty[$. Let
$B=B_1+e$, $A = S_1+e$, $Q=E_2+[0,\infty[e$ et $P=E_2$
Then $(B,A) \ (k,1)\mbox{-links }(Q,P)$ in $E$.
\end{proposition}
\begin{proof}
Let $\epsilon \in ]0,1[$ and
\begin{gather*}
\hat{B}=B\cup (\epsilon B_1+]0,\infty[e+E_2),\\
\hat{A}=\hat{B}\setminus (]0,\infty[e+E_2).
\end{gather*}
Since $B$ (resp. $A$) is a strong deformation retract of $\hat{B}$
(resp. $\hat{A}$), the inclusion $(B,A)\rightarrow (\hat{B},\hat{A})$
induces an isomorphism $H_k(B,A)\cong H_k(\hat{B},\hat{A})$. Let
$$U=(E\setminus P)\setminus \hat{B} \subset E\setminus Q\subset
E\setminus P,$$
and observe that $\overline{U}\subset \mbox{int }(E\setminus Q)$ in
$E\setminus P$, $\hat{B}=(E\setminus P)\setminus U$ and
$\hat{A}=(E\setminus Q)\setminus U$. Hence, by excision, the inclusion
$(\hat{B},\hat{A})\rightarrow (E\setminus P,E\setminus Q)$ induces an
isomorphism $H_k(\hat{B},\hat{A})\cong H_k(E\setminus P,E\setminus
Q)$. The result follows from $H_k(B,A)\cong \mathbb{K}$.
\end{proof}
A similar argument leads to the following proposition.
\begin{proposition}\label{enlacement_5}
Let $E= E_1 \oplus E_2$ with $k=\dim E_1 \in\ ]0, \infty[$.\\Then
$(B_1, S_1) \ (k,1)\mbox{-links } (B_2,S_2)$ in $E$.
\end{proposition}
\section{Homotopical consequences of homological linking}
Let $(B,A)$ and $(Q,P)$ be pairs of subspaces in a topological
space $X$ such that $B\cap P=\emptyset$ and $A\cap Q=\emptyset$.
The following lemma shows that relative homotopical linking is an
obstruction to extension factoring through a homotopically trivial pair.
\begin{lemma}
The following statements are equivalent.
\begin{enumerate}
\item The pair $(B,A)$ homotopicaly links $(Q,P)$,
\item There exists no homotopy
$\eta:[0,1]\times (B,A)\rightarrow (X\setminus P, X\setminus Q)$
such that
$\eta=id$ on $\{0\}\times B\cup [0,1]\times A$ making the
following diagram commutative
$$\xymatrix{
(B,A)\ar[r]^{\eta_1}\ar[rd]_{\eta_1}& (X\setminus P, X\setminus Q)\\
&(X\setminus Q,X\setminus Q)\ar[u]
}$$
\end{enumerate}
\end{lemma}
\begin{corollary}
Homological linking implies homotopical linking.
\end{corollary}
\begin{remark}
To see that homotopical linking doesn't imply homological linking,
it is sufficient to consider $X=B=Q$ to be a singleton and
$A=P=\emptyset$.
\end{remark}
\section{Homological linking principle}
Let $H$ be a Hilbert space and let $f\in C^2(H,\mathbb{R})$.
The following notation is standard. Given $c\in \mathbb{R}$,
$f_c=\{p\in H \bigr| f(p)\leq c\}$ is a level set of $f$,
$K(f)=\{p\in H \bigr| f'(p)=0\}$ is the critical set of $f$,
$K_c(f)=K(f)\cap f^{-1}(c)$.
Throughout this section, the following hypothesis
are assumed,
\begin{itemize}
\item[(H1)] the Palais-Smale condition for $f$ holds. That is, each
sequence $(x_n)_{n\in\mathbb{N}}$ such that $(f(x_n))$ is bounded
and $f'(x_n)\rightarrow 0$ admits a convergent subsequence,
\item[(H2)] the set $K(f)$ of critical point of $f$ is discrete.
\end{itemize}
In particular, $f(K)$ is discrete and for each bounded interval $I$,
$K\cap I$ is compact.
Under these assumptions, there is a suitable Morse theory which is
well behaved (see \cite{CPA:1} for instance). We shall use the
following standard notation. Given $p\in K_c(f)$,
$$C_q(f,p):=H_q(f_c,f_c\setminus\{p\})$$
is the $q$-th critical group of $f$ at $p$. Let $a<b$ be two regular values
of $f$, $$\mu_q(f_b,f_a):=\underset{p\in K(f)\cap f^{-1}[a,b]}{\sum} \dim C_q(f,p)$$
is the Morse number of the pair $(f_b,f_a)$.
The function $f$ is said to be a Morse function if its critical
points are all non-degenerate.
\begin{remark}
Most of our results depend only on the Morse inequalities.
It is thus possible to use any other setting where they hold.
For example, in \cite{corvellec:1} a Morse theory for continuous functions
on metric spaces is presented. In applications to PDE, it may be necessary to use the
Finsler structure approach of Chang \cite{chang:2} to apply the
results in suitable Sobolev spaces.
\end{remark}
The following theorem is an easy exercise and was probably first
observed by Marston Morse himself.
\begin{theorem}[homological linking principle]\label{homo_link_prin}
Let $(B,A)$ and $(Q,P)$ be pairs of subspaces in $H$ and
let $a<b$ be regular values of $f$ such that
$(B,A) \subset (f_b,f_a) \subset (H\setminus P,H\setminus Q)$.
If $(B,A)\ (q,\beta)$-links $(Q,P)$ in $H$ for some $\beta\geq 1$ then
$f$ admits a critical point $p$ such that $a<f(p)<b$ and
$C_q(f,p)\neq 0$. Moreover, if $f$ is a Morse function then it
admits at least $\beta$ such points.
\end{theorem}
\begin{proof}
It follows from commutativity of
$$\xymatrix{
\tilde{H}_q(B,A) \ar[r]\ar[d]&\tilde{H}_q(H\setminus P,H\setminus Q)\\
\tilde{H}_q(f_b,f_a)\ar[ru]&
}$$
that $\dim \tilde{H}_q(f_b,f_a)\geq \beta$. Application of
the weak Morse inequalities leads to \break$\mu_q(f_b,f_a)~\geq~\beta$ and to
the first conclusion. The non-degeneracy condition leads to the
second one.
\end{proof}
\begin{remark}
From Remark \ref{betti} and our linking principle we recover the weak Morse inequalities. This shows that our homological linking
contains nearly as much information as classical Morse theory.
\end{remark}
\begin{lemma}
Let $(B,A)$ and $(Q,P)$ be pairs of subspaces in $H$ such that
\begin{gather*}
\sup f(B) < \inf f(P),\\
\sup f(A) \leq \inf f(Q).
\end{gather*}
If $(B,A)\ (q,\beta)$-links $(Q,P)$ in $H$ for some $\beta\geq 1$ then
$\inf f(Q) \leq \sup f(B)$.
\end{lemma}
\begin{proof}
Let the opposite be supposed: $\sup f(B)<\inf f(Q)$. For each
$n\in \mathbb{N}$, there exist regular values
$a_n < b_n$ in $]\sup f(B), \sup f(B)+1/n[$.
If $n$ is big enough, $\sup f(B)+1/n < \inf f(Q)\leq\inf f(P)$ so
that $$(B,A)\subset (f_{b_n},f_{a_n}) \subset (X\setminus P, X\setminus Q).$$
It follows from the homological linking principle that $f$ admits a
critical value $c_n\in\ ]a_n,b_n[$. The infinite sequence $(c_n)$
converges to $c=\sup f(B)$ which must therefore be critical because
the set of all critical values of $f$ is closed. This contradicts
the fact that critical values must be isolated.
\end{proof}
The next theorem will be usefull for applications. In the next section, it will be used to
prove some multiplicity results.
\begin{theorem}\label{app_linking_principle}
Let $(B,A)$ and $(Q,P)$ be pairs of subspaces in $H$ such that
\begin{gather*}
\sup f(B) < \inf f(P),\\
\sup f(A) < \inf f(Q).
\end{gather*}
If $(B,A)\ (q,\beta)$-links $(Q,P)$ in $H$ for some $\beta\geq 1$ then
$f$ admits a critical point $p$ such that
$$\inf f(Q)\leq f(p)\leq \sup f(B)$$ and $C_q(f,p)\neq 0$.
Moreover if $f$ is a Morse function then it admits at least $\beta$
such points.
\end{theorem}
\begin{proof}
By the preceding lemma,
$$\sup f(A) < \inf f(Q) \leq \sup f(B) < \inf f(P).$$
There exist regular values $a_n<b_n$ ($n\in \mathbb{N}$)
such that
$$\sup f(A) < a_n < \inf f(Q) \leq \sup f(B) < b_n < \inf f(P)$$ and
$a_n\rightarrow \inf f(Q)$, $b_n\rightarrow \sup f(B)$. By the
linking principle, there must exist a sequence $(p_n)$ of critical
points such that $C_q(f,p_n)\neq 0$ and such that the sequence
$(c_n)=(f(p_n))$ satisfies $a_n<c_n<b_n$. Because critical values
are isolated, $c_n\in [\inf f(Q), \sup f(B)]$ for $n$ big enough.
\end{proof}
The following result follows directly from Propositions~\ref{enlacement_5} and
Theorem~\ref{app_linking_principle}. As far as we know, this result is new.
\begin{theorem}\label{FrigHomotop}
Let $H=H_1\oplus H_2$ with $k=\dim H_1<\infty$.
If
\begin{gather*}
\sup f(S_1)<\inf f(B_2),\\
\sup f(B_1)< \inf f(S_2)
\end{gather*}
then $f$ admits a critical point $p$ such that
$$\inf f(S_2)\leq f(p) \leq \sup f(S_1)$$ and $C_k(f,p)\neq 0$.
\end{theorem}
\subsection{Multiplicity results}
By combining Corollaries \ref{cor_enlacement_3} and \ref{cor_enlacement_6} with Thorem
\ref{app_linking_principle}, we get a version of a well known multiplicity result
(see \cite{schechter:1} for instance). As before, we get extra information about the critical
groups.
\begin{proposition}
Let $H=H_1\oplus H_2$ with $k=\dim H_1 \in\ ]0, \infty[$
and $e \in H_2$ be of unit length.
Let $B=B_1\oplus [0,2]e$ and $A=\partial B$ in in $H_1\oplus \mathbb{R}e$.
If $f$ is bounded below on $B_2$ and if
$$\sup f(A)<\inf f(S_2)$$ then $f$ admits two critical points
$p_0\neq p_1$ such that
$$\inf(f(B_2)\leq f(p_0)\leq \sup f(A),$$
$$\inf f(S_2)\leq f(p_1)\leq \sup f(B)$$
and $C_k(f,p_0)\neq 0, C_{k+1}(f,p_1\neq 0)$.
\end{proposition}
\begin{proof}
Because
\begin{gather*}
\sup f(A) < \inf f(S_2)\\
\sup f(\emptyset)=-\infty < \inf f(B_2)
\end{gather*}
and $A\ (k,1)\mbox{-links } (B_2,S_2)$, it follows from
Theorem~\ref{app_linking_principle} that $f$ admits a critical point $p_0$
such that $\inf f(B_2)\leq f(p_0) \leq \sup f(A)$ and
$C_k(f,p_0)\neq 0$.
Also, Corrolary \ref{cor_enlacement_3} says
that $(B,A)\ (k+1,1)\mbox{-links } S_2$.
Since
\begin{gather*}
\sup f(B) < \infty=\inf f(\emptyset)\\
\sup f(A) < \inf f(S_2)
\end{gather*}
it follows from Theorem~\ref{app_linking_principle} that $f$ admits
a critical point $p_1$ such that
$\inf f(S_2)\leq f(p_1)\leq \sup f(B)$ and $C_{k+1}(f,p_1)\neq 0$.
The inequality $$f(p_0)\leq \sup f(A) < \inf f(S_2)\leq f(p_1)$$
insure that $p_0$ and $p_1$ are distinct.
\end{proof}
A similar argument using Corollaries \ref{cor_enlacement_2} and \ref{cor_enlacement_5}
leads to the next theorem. This result was already known to Perera \cite{perera:1}.
\begin{theorem}
Let $H=H_1\oplus H_2$ with $k=\dim H_1 \in\ ]0, \infty[$ and let $e\in H_1$ be of unit
length. If $f$ is bounded below on $H_1+[0,\infty[e$ and if
$$\sup f(S_1)<\inf f(H_2)$$ then $f$ admits two critical points $p_0\neq p_1$ such that
$$\inf(f(H_1+[0,\infty[e))\leq f(p_0)\leq \max f(S_1),$$
$$\inf f(H_2)\leq f(p_1)\leq \max f(B(0,1))$$
and $C_{k-1}(f,p_0)\neq 0, C_k(f,p_1\neq 0)$.
\end{theorem}
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\begin{document}
\title[]{Theta distinguished representations, inflation and the symmetric square $L$-function}
\author{Eyal Kaplan}
\email{kaplaney@gmail.com}
\maketitle
\begin{abstract}
Let $\Pi_0$ be a representation of a group $H$. We say that a representation $\tau$ is $(H,\Pi_0)$-distinguished, if it is
a quotient of $\Pi_0$. It is natural to ask whether this notion ``inflates" to larger groups, in the sense that a representation $\mathrm{I}(\tau)$
induced from $\tau$ and $H$ to a group $G$, is $(G,\Pi)$-distinguished. We study representations distinguished by theta representations:
$H=\GL_n$, $\Pi_0$ is a pair of the exceptional representations of Kazhdan and Patterson, $G=\GSpin_{2n+1}$ and $\Pi$ is a pair of
the small representations of Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation $\tau$ is distinguished
if and only if $\mathrm{I}(\tau)$ is distinguished, and the multiplicity in each model is the same.
If $\tau$ is supercuspidal and distinguished, we prove that the Langlands quotient of $\mathrm{I}(\tau)$ is distinguished. As a corollary, we
characterize supercuspidal distinguished representations, in terms of the pole of the local symmetric square $L$-function at $s=0$.
\end{abstract}
\section{Introduction}\label{section:introduction}
Let $\tau$ be an admissible representation of $\GL_n(F)$, where $F$ is a local non-Archimedean field.
Let $\theta_0$ and $\theta_0'$ be a pair of exceptional representations of a double cover $\widetilde{\GL}_n(F)$ of $GL_n(F)$, in the sense of Kazhdan and Patterson \cite{KP}.
We say that $\tau$ is distinguished if
\begin{align}\label{eq:homspace dist}
\Hom_{\GL_n(F)}(\theta_0\otimes\theta_0',\tau^{\vee})\ne0.
\end{align}
Here $\tau^{\vee}$ is the representation contragradient to $\tau$. Equivalently, the space
of $\GL_n(F)$-invariant trilinear forms on $\tau\times\theta_0\times\theta_0'$ is nonzero.
This space first appeared in a global context.
Let $\pi$ be a unitary cuspidal automorphic representation of $\GL_n(\Adele)$, where $\Adele$ is the
ad\`{e}les ring of a global field. Assume that $\pi$ has a trivial central character. Bump and Ginzburg \cite{BG} proved that if the partial symmetric square $L$-function
$L^S(s,\pi,\mathrm{Sym}^2)$ has a pole at $s=1$, the following period integral is nonvanishing
\begin{align}\label{int:BG period}
\int_{Z\GL_n(F)\backslash\GL_n(\Adele)}\varphi_{\pi}(g)\Theta(g)\Theta'(g)dg.
\end{align}
Here $Z$ is a subgroup of finite index in the center $C_{\GL_n}(\Adele)$ of $\GL_n(\Adele)$ ($Z=C_{\GL_n}(\Adele)$ when $n$ is odd);
$\varphi_{\pi}$ is a cusp form in the space of $\pi$; $\Theta$ and $\Theta'$ are automorphic forms in the space of a global exceptional representation of $\widetilde{\GL}_n(\Adele)$. For $n=2$, an earlier work
by Patterson and Piatetski-Shapiro \cite{PPS} showed that a similar integral characterizes the pole at $s=1$, for a global field of odd characteristic.
Periods of automorphic forms are often related to poles of $L$-functions and to questions of functoriality. Ginzburg, Jiang and Soudry \cite{GJS} described such relations in a general setup
and also considered several examples. Let $G_n=\GSpin_{2n+1}$ be the odd general spin group of rank $n+1$. Let $E(g;\rho,s)$ be the
Eisenstein series corresponding to an element $\rho$ in the space of the representation of
$G_n(\Adele)$ induced from $\tau|\det{}|^s\otimes1$ ($s\in\C$, $g\in G_n(\Adele)$). The residual representation
$E_{\pi}$ is the space spanned by the residues $E_{1/2}(\cdot;\rho)$ of $E(g;\rho,s)$ at $s=1/2$. The following result formulated in
\cite{GJS} (Theorem~3.3) was proved in a series of works (\cite{BG,BFG,GJS,me7}):
the following conditions are equivalent.
\begin{enumerate}
\item $L^S(s,\pi,\mathrm{Sym}^2)$ has a pole at $s=1$.
\item\label{item:period} The period integral \eqref{int:BG period} is nonzero.
\item\label{item:residual} The residual representation $E_{\pi}$ is nonzero.
\item $\pi$ is the Langlands functorial transfer of an irreducible generic cuspidal automorphic representation of the split $\mathrm{SO}_{n}(\Adele)$ (if $n$ is even) or $\Sp_{(n-1)/2}(\Adele)$ ($n$ is odd).
\end{enumerate}
Note that this was stated in \cite{GJS} with $G_n$ replaced by $\SO_{2n+1}$ (see below).
To prove that the nonvanishing of \eqref{int:BG period} implies the nontriviality of $E_{\pi}$, one can relate the period to the following co-period
integral
\begin{align*}
\mathcal{CP}(E_{1/2}(\cdot;\rho),\Theta,\Theta')=\int_{C_{G_{n}(\Adele)}G_{n}(F)\backslash G_{n}(\Adele)}E_{1/2}(g;\rho)\Theta(g)\Theta'(g)dg.
\end{align*}
Here $\Theta$ and $\Theta'$ belong to the exceptional, or small, representation of Bump, Friedberg and Ginzburg \cite{BFG},
whose analog for $G_n$ was described in \cite{me8}. This integral was studied in \cite{me7} for $\SO_{2n+1}$ (extended in \cite{me8} to $G_n$) and we proved
the implication $\eqref{item:period}\Rightarrow\eqref{item:residual}$. In the setting of $G_n$, one can also study the twisted symmetric square $L$-function. The Rankin-Selberg integral representation
for this function has recently been developed by Takeda \cite{Tk}.
The global unfolding of $\mathcal{CP}(E_{1/2}(\cdot;\rho),\Theta,\Theta')$ (in \cite{me7}) has a local counterpart. Assume that $\tau$ is a distinguished
representation. Let $\mathrm{I}(\tau)=\Ind_{Q_n}^{G_n}(\delta_{Q_n}^{1/2}\tau|\det{}|^{1/2}\otimes1)$, where $Q_n$ is the Siegel parabolic subgroup. Using an integral over $Q_n(F)\backslash G_n(F)$, we show (Proposition~\ref{proposition:upper heredity of dist 1})
that for a certain pair $\theta$ and $\theta'$ of exceptional representations of $\widetilde{G}_n(F)$ (\cite{BFG,me8}, see below),
\begin{align}\label{eq:homspace dist2}
\Hom_{G_n(F)}(\theta\otimes{\theta'},\mathrm{I}(\tau)^{\vee})\ne0.
\end{align}
In other words, $\mathrm{I}(\tau)$ is a distinguished representation of $G_n(F)$. In fact, depending on the central character of $\tau$, we may
need to replace $\tau|\det{}|^{1/2}\otimes1$ with $\tau|\det{}|^{1/2}\otimes\eta$ where $\eta$ is a character of $F^*$. (See the proposition for details.)
The local problem is to prove that the Langlands quotient
$\mathrm{LQ}(\mathrm{I}(\tau))$ of $\mathrm{I}(\tau)$ is also distinguished.
In the present study we consider an irreducible unitary supercuspidal $\tau$.
In this case $\mathrm{I}(\tau)$ is either irreducible and generic, or is of length two, has a unique irreducible
generic subrepresentation and $\mathrm{LQ}(\mathrm{I}(\tau))$ is non-generic.
Here is our main result, implying that if $\tau$ is distinguished, so is $\mathrm{LQ}(\mathrm{I}(\tau))$. For more details see Corollary~\ref{corollary:upper hered dist LQ}.
\begin{theorem}\label{theorem:tensor of small is non generic}
The space of $\theta\otimes\theta'$ as a representation of $G_n(F)$ does not afford a Whittaker functional.
\end{theorem}
A similar ``inflation" phenomena has already been observed by Ginzburg, Rallis and Soudry \cite{GRS5} (Theorem~2). Assume that $\tau$ is a supercuspidal and self-dual representation, such that the exterior square $L$-function $L(s,\tau,\wedge^2)$ has
a pole at $s=0$. This implies that $\tau$ has a Shalika model and then according to Jacquet and Rallis \cite{JR2}, $\tau$ admits a (nontrivial) $\GL_n\times\GL_n$ invariant functional.
In turn, the representation
parabolically induced from $\tau|\det|^{1/2}$ to $\Sp_{2n}$ has an $\Sp_n\times\Sp_n$ invariant functional.
Ginzburg, Rallis and Soudry \cite{GRS7} (Theorems~16-17) showed that an irreducible generic representation of $\Sp_{2n}$ does not admit such
a functional. It follows that the $\Sp_n\times\Sp_n$ functional factors through the Langlands quotient.
This inflation was one of the ingredients used
by Lapid and Mao for the proof of their conjecture on
Whittaker-Fourier coefficients, in the case of the metaplectic group (\cite{LM3,LM6,LM4,LM5}). Note that their conjecture actually applies
to any quasi-split group as well as the metaplectic group.
Our results here and their extension, in a forthcoming work, to an arbitrary irreducible generic distinguished representation,
are expected to be used in a proof of this conjecture for even orthogonal groups.
Here we have a similar relation between distinguished representations and the symmetric square $L$-function.
\begin{theorem}\label{theorem:supercuspidal dist GLn and pole}
Let $\tau$ be an irreducible unitary supercuspidal representation of $\GL_n$. Then $\tau$ is
distinguished if and only if $L(s,\tau,\mathrm{Sym}^2)$ has a pole at $s=0$.
\end{theorem}
Refer to Shahidi \cite{Sh5} (Theorem~6.2) for a description of the poles of $L(s,\tau,\mathrm{Sym}^2)$ in this setting.
The proof of Theorem~\ref{theorem:tensor of small is non generic} is essentially the local analog of the global computation of the co-period
in \cite{me7,me8}. Write $Q_n=M_n\ltimes U_n$ and let $C=C_{U_n}$ be the center of the unipotent radical $U_n$. The global unfolding argument involves a Fourier expansion
of $\Theta$ along $C$. Consider a non-generic character of $C$, this means that its stabilizer in $M_n$ contains a unipotent radical $V$ of a parabolic subgroup of $\GL_n$. The corresponding Fourier coefficient
is constant on $V$, then the cuspidality of $\pi$ is used to prove that the integral vanishes.
When $n$ is odd (and $n>1$), all characters of $C$ are non-generic. In the even case there is one generic orbit of characters. Its stabilizer
is ``almost" a Jacobi group, its reductive part is $Sp_{n/2}\times G_0$, where $Sp_{n/2}$ is a symplectic group in $n$ variables. One might attempt
to prove invariance of the product of Fourier coefficients under $Sp_{n/2}(\Adele)$, then use the fact that $\pi$ does not admit nontrivial
symplectic periods (\cite{JR}). Albeit the Fourier coefficients do not enjoy this invariance, a certain convolution against Weil theta
functions, introduced by Ideka \cite{Ik3}, can be used instead.
The local argument involves the computation of the twisted Jacquet modules $\theta_{C,\psi_k}$ of $\theta$ with respect to $C$ and
a representative $\psi_k$ of some orbit of characters. In contrast with the global setting, the generic character is the
crux of the proof. Roughly, this is because
$\theta_{C,\psi_k}$ is a tensor of an exceptional representation of $\widetilde{\GL}_k(F)$ and the Jacquet module of an exceptional representation of $\widetilde{G}_{2k}(F)$ along $C_{U_{2k}}$ and a generic character.
When the character is generic, the twisted Jacquet module is (by restriction) a representation of a Jacobi group.
The local theory of smooth representations of Jacobi groups \cite{Dijk,MVW,BSch} describes such a representation as a tensor $\kappa\otimes\omega_{\psi}$,
where $\omega_{\psi}$ is the Weil representation. The Heisenberg group acts trivially on the space of $\kappa$, while the action of the reductive part separates into
an action on the space of $\kappa$, and one on the space of $\omega_{\psi}$. We prove that $\kappa$ is a trivial representation of $Sp_{n/2}(F)$.
\begin{theorem}\label{theorem:small rep is weakly minimal}
Assume $n$ is even and let $\psi_{n/2}$ be a generic character of $C_{U_n}$. As a Jacobi representation $\theta_{C_{U_n},\psi_{n/2}}$ is the direct sum of (possibly infinitely many) copies of $\omega_{\psi}$.
\end{theorem}
Note that the action of the $G_0$ part of the stabilizer on $\theta_{C_{U_n},\psi_{n/2}}$ is given simply by the central character of $\theta$.
This result underlies Theorem~\ref{theorem:tensor of small is non generic}. Furthermore, it implies the following multiplicity property.
Assume that $\tau$ is irreducible and tempered. We prove that $\tau$ is distinguished if and only if $\mathrm{I}(\tau)$ is.
Moreover, the dimensions of \eqref{eq:homspace dist2} and \eqref{eq:homspace dist} are equal. See Proposition~\ref{proposition:one-dim}
for the precise statement. Note that Kable \cite{Kable} conjectured, and under a certain homogeneity assumption
proved (\cite{Kable} Corollary~6.1), that \eqref{eq:homspace dist} enjoys multiplicity one. These results motivate the introduction of ``exceptional models" over $\GL_n$ and $G_n$.
Theorem~\ref{theorem:small rep is weakly minimal} may have additional applications. Explicit descriptions of Jacquet modules of exceptional representations have had numerous applications (see below).
Note that when $n=2$, $Q_n$ is the Heisenberg parabolic subgroup in the notation of Gan and Savin \cite{GanSavin}. In their terminology, Theorem~\ref{theorem:small rep is weakly minimal}
shows that $\theta$ is ``weakly minimal". In this case it is the minimal representation and $\theta_{C_{U_n},\psi_{n/2}}\isomorphic\omega_{\psi}$ (\cite{GanSavin} Section~3).
Bump, Friedberg and Ginzburg \cite{BFG} constructed the small representation $\theta^{\SO_{2n+1}}$ for the special odd orthogonal group. This is a representation of
a ``double cover" $\widetilde{SO}_{2n+1}(F)$, obtained by restricting the $4$-fold
cover of $\SL_{2\glnidx+1}(F)$ of Matsumoto \cite{Mats}. For the low rank cases $n=2,3$, it is the minimal representation. In fact for $n=3$,
this representation was already developed by Roskies \cite{Roskies}, Sabourin \cite{Sabourin} and Torasso \cite{Torasso}.
For $n>3$, there is no minimal representation for a group of type $B_n$ (\cite{V}).
Bump, Friedberg and Ginzburg \cite{BFG2} showed that when $n>3$, $\theta^{\SO_{2n+1}}$ is attached to one of the
possible coadjoint orbits, which is smallest next to the minimal one. This translates into the vanishing of a large class
of Fourier coefficients, called generic in \cite{BFG} (see also \cite{CM,Cr,G2}). Locally, this means that a large class
of twisted Jacquet modules vanish. The representation $\theta^{\SO_{2n+1}}$ was used by these authors to construct a lift with certain
functorial properties, between covers of orthogonal groups \cite{BFG2}. The representation $\theta^{SO_{7}}$ was used to construct an integral representation
(\cite{BFG3}).
There is a technical issue when working with $\widetilde{SO}_{2n+1}(F)$: the underlying field must contain all $4$ $4$-th roots of unity.
This can be remedied using $G_n$. Indeed, one obtains a nontrivial double cover of $G_n(F)$ by restricting the $2$-fold cover of $Spin_{2n+3}(F)$
of Matsumoto \cite{Mats}. The theory of Bump, Friedberg and Ginzburg \cite{BFG} can be extended to $\widetilde{G}_n(F)$, mainly because
both groups are of type $B_n$ and in particular, the unipotent subgroups are isomorphic. The details were carried out in \cite{me8}.
Our results here are stated for $G_n$, but apply similarly to $\SO_{2n+1}$ and $\theta^{\SO_{2n+1}}$ (see Section~\ref{section:relevance to SO(2n+1)}).
Minimal representations have been studied and used extensively, by numerous authors.
Knowledge of Fourier coefficients, or Jacquet modules, has proved very useful for applications
\cite{GRS6,G2,GJS2}. They have played a fundamental role in the theta correspondence, the descent method and Rankin-Selberg integrals.
See, for example \cite{V,KZ,KS,Pd,Savin2,BK,Savin,GRS,BFG3,GRS3,KP3,BFG,JSd1,GanSavin,Soudry4,LokeSavin,LokeSavin2,RGS}.
We mention that Bump and Ginzburg \cite{BG} also developed a local theory, where they considered a similar space of equivariant trilinear forms, except that
$\theta_0'$ was replaced by a certain induced representation.
For $n=3$, Savin \cite{Savin3} determined the dimension
of \eqref{eq:homspace dist} for an arbitrary irreducible quotient of a principal series representation. He also conjectured (and proved for $n=3$),
that the class of distinguished spherical representations of $\GL_n(F)$ is precisely those representations, which are lifts from
a certain prescribed classical group. Kable \cite{Kable2} proved that these lifts are distinguished, the other
direction was proved in \cite{me9}.
The group $G_n$ has been the focus of study of a few recent works, among which are the works of
Asgari \cite{Asg2,Asg} on local $L$-functions, Asgari and Shahidi \cite{AsgSha,AsgSha2} on functoriality
and Hundley and Sayag \cite{HS} on the descent construction.
The rest of this work is organized as follows. In Section~\ref{section:preliminaries} we provide notation and definitions. In particular we
describe the construction of exceptional representations. Section~\ref{section:Heisenberg jacquet modules} contains the proof of
Theorem~\ref{theorem:small rep is weakly minimal}. Theorems~\ref{theorem:tensor of small is non generic}
and \ref{theorem:supercuspidal dist GLn and pole} are proved in Section~\ref{section:Distinguished representations}.
Section~\ref{section:relevance to SO(2n+1)} provides the formulation of our results for $\SO_{2n+1}$.
\subsection*{Acknowledgments}
I wish to express my gratitude to Erez Lapid for suggesting this project to me. I would like to thank Eitan Sayag, for explaining
to me how to use his results with Offen (\cite{OS}, Proposition~1).
Lastly, I thank Jim Cogdell for his kind encouragement and useful remarks.
\section{Preliminaries}\label{section:preliminaries}
\subsection{General notation}\label{subsection:general notation}
Let $F$ be a local \nonarchimedean\ field of characteristic different from $2$.
For an integer $r\geq1$, let $\mu_r$ be the subgroup of $r$-th roots of unity in $F$. Set $F^{*r}=(F^*)^r$.
We usually fix a nontrivial additive character $\psi$ of $F$. Then the normalized Weil factor (\cite{We} Section~14)
is denoted by $\gamma_{\psi}$ ($\gamma_{\psi}(a)$ is
$\gamma_F(a,\psi)$ of \cite{Rao}, $\gamma_{\psi}(\cdot)^4=1$). The Hilbert symbol of order $r$ is $(,)_r$.
If $G$ is a group, $C_G$ denotes its center. For $x,y\in G$ and $Y<G$, $\rconj{x}y=xyx^{-1}$ and
$\rconj{x}Y=\setof{\rconj{x}y}{y\in Y}$. Hereby we omit references to the field, e.g., $\GL_n=\GL_n(F)$.
\subsection{The group $\GL_{n}$ and its cover}\label{subsection:GL_n and its cover}
Fix the Borel subgroup $B_{\GL_n}=T_{\GL_n}\ltimes N_{\GL_n}$ of upper triangular matrices, where $T_{\GL_n}$ is the diagonal torus.
For any $k_1,k_2\geq0$ such that $k_1+k_2=n$, denote by $P_{k_1,k_2}$
the maximal parabolic subgroup whose Levi part is isomorphic to $\GL_{k_1}\times\GL_{k_2}$.
Its unipotent radical is $Z_{k_1,k_2}=\{\left(\begin{smallmatrix}I_{k_1}&z\\&I_{k_2}\end{smallmatrix}\right)\}$. The ``mirabolic" subgroup
$P_{n-1,1}^{\circ}$ is the subgroup of $\GL_n$ of matrices with the last row $(0,\ldots,0,1)$.
Let $I_n$ be the identity matrix of $\GL_n$ and $J_{n}$ be the matrix with $1$ on the anti-diagonal and $0$ elsewhere. For $g\in\GL_{n}$, $\transpose{g}$ is the transpose of $g$.
We will use the metaplectic double cover $\widetilde{\GL}_n$ of $\GL_n$, as constructed by
Kazhdan and Patterson \cite{KP}.
Let $\widetilde{SL}_{n+1}$ be the double cover of $\mathrm{SL}_{n+1}$ of Matsumoto \cite{Mats} and let
$\sigma_{\mathrm{SL}_{n+1}}$ be the corresponding cocycle of Banks, Levi and Sepanski \cite{BLS} (Section~3).
We define a $2$-cocycle $\sigma_{\GL_n}$ of $\GL_{n}$ via
\begin{align*}
\sigma_{\GL_n}(a,a')=\sigma_{\mathrm{SL}_{n+1}}(\mathrm{diag}(\det{a}^{-1},a),\mathrm{diag}(\det{a'}^{-1},a')).
\end{align*}
This cocycle is related to the cocycle $\sigma_{n}$ of $\GL_n$ defined in \cite{BLS} by
\begin{align*}
\sigma_{\GL_n}(a,a')=c(\det{a},\det{a'})\sigma_{n}(a',a).
\end{align*}
In particular $\sigma_{\GL_1}(a,a')=(a,a')_2$.
\subsection{The group $\GSpin_{2n+1}$}\label{subsection:GSpin}
We start by defining the special odd orthogonal group
\begin{align*}
\SO_{2n+1}=\setof{g\in \SL_{2n+1}}{\transpose{g}J_{2n+1}g=J_{2n+1}}.
\end{align*}
Select its Borel subgroup $B_{\SO_{2n+1}}=B_{\GL_{2n+1}}\cap \SO_{2n+1}$.
Let $\Spin_{2n+1}$ be the simple split simply-connected algebraic group of type $B_{n}$. It is the algebraic double cover of $\SO_{2n+1}$. We will take the Borel subgroup, which is the preimage of $B_{\SO_{2n+1}}$. The set of simple roots of $\Spin_{2n+1}$ is
$\Delta_{\Spin_{2n+1}}=\setof{\alpha_i}{1\leq i\leq n}$, where $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for $1\leq i\leq n-1$ and $\alpha_{n}=\epsilon_{n}$.
The group $G_n=\GSpin_{2n+1}$ is an $F$-split connected reductive algebraic group, which can be defined using a based root
datum as in \cite{Asg,AsgSha,HS}. It is also embedded in $G'_{n+1}=\Spin_{2n+3}$ as the Levi part of the parabolic subgroup
corresponding to $\Delta_{G'_{n+1}}\setdifference\{{\alpha_1}\}$ (see \cite{Mt}). We adapt this identification, which
is more natural for the purpose of cover groups, because we will obtain a cover of $G_n$ by restricting
the cover of $G'_{n+1}$.
Let $\delta_{Q}$ be the modulus character of a parabolic subgroup $Q<G_n$.
In the degenerate case $G_0=\GL_{1}$.
The Borel subgroup of $G_n$ is denoted $B_n=T_{n+1}\ltimes N_{n}$, where
$N_{n}$ is the unipotent radical (the rank of the torus is $n+1$). For $0\leq k\leq n$, denote by $Q_k=M_k\ltimes U_k$ the standard maximal parabolic subgroup of $G_{n}$ with $M_k\cong\GL_{k}\times G_{n-k}$. This isomorphism is not canonical. We describe the choice used in \cite{me8}, which is convenient for certain computations (see below).
The derived group $\SL_k$ of $\GL_{k}$ is generated by the root subgroups of $\setof{\alpha_i}{2\leq i\leq k}$. Let $\eta_1^{\vee},\ldots,\eta_{k}^{\vee}$ be the standard cocharacters of
$T_{\GL_k}$ and map $\eta_i^{\vee}\mapsto \epsilon_{i+1}^{\vee}-\epsilon_{1}^{\vee}$ for $1\leq i\leq k$.
Regarding $G_{n-k}$, the set $\setof{\alpha_i}{k+2\leq i\leq n+1}$ identifies $G'_{n-k}$ and if $\theta_1,\ldots,\theta_{n-k+1}$ are the characters of $T_{n-k+1}$, define $\theta_1^{\vee}\mapsto\epsilon_1^{\vee}$ and for $2\leq i\leq n-k+1$, $\theta_i^{\vee}\mapsto \epsilon_{k+i}^{\vee}$.
The projection $G'_{n}\rightarrow \SO_{2n+1}$ is an isomorphism between unipotent subgroups, hence
we can identify unipotent subgroups of $G_{n}$ with those of $\SO_{2n+1}$.
We use the character $\epsilon_1$ to define a ``canonical" character $\Upsilon$ of $G_{n}$. Namely $\Upsilon$ is the extension of $-\epsilon_1$ to $G_{n}$ (the only other choice would be to use $\epsilon_1$).
The aforementioned embedding of $\GL_k\times G_{n-k}$ in $M_k$ has a few properties, suitable for computations. To compute the conjugation of $b\in\GL_k$ on $U_k$, we can simply look at this action in $\SO_{2n+1}$, where $b$ takes the
form $\mathrm{diag}(b,I_{2(n-k)+1},J_k\transpose{b}^{-1}J_k)$. The image of $G_0$ is $C_{G_n}$.
The restriction of $\Upsilon$ to $\GL_k$ is $\det$.
\subsection{The double cover of $\GSpin_{2n+1}$}\label{subsection:metaplectic GSpin}
Let $\cover{G}'_{n+1}$ be the double cover of $G'_{n+1}$, constructed by Matsumoto \cite{Mats} using $(,)_{2}$ as the Steinberg symbol. Restricting the cover to $G_{n}$, we obtain the exact sequence
\begin{align*}
1\rightarrow{\mu_2}
\rightarrow\cover{G}_{n}\xrightarrow{p}G_{n}\rightarrow 1.
\end{align*}
Then $\cover{G}_{n}$ is a double cover of $G_{n}$. For a subset $X\subset G_n$, $\widetilde{X}=p^{-1}(X)$.
Let $\mathfrak{s}:G'_{n+1}\rightarrow\cover{G}'_{n+1}$ be the block-compatible section constructed by Banks, Levi and Sepanski \cite{BLS}
and $\sigma_{G'_{n+1}}$ be the corresponding cocycle. Denote the restriction of $\sigma_{G'_{n+1}}$ to $G_n\times G_n$ by $\sigma_{G_n}$. In \cite{me8} we proved that $\sigma_{G_n}$ satisfies the following block compatibility property: if $(a,g),(a',g')\in\GL_k\times G_{n-k}\isomorphic M_k$,
\begin{align}\label{eq:block-compatibility}
\sigma_{G_n}((a,g),(a',g'))=\sigma_{\GL_{k}}(a,a')\sigma_{G_{n-k}}(g,g')(\Upsilon(g),\det{a'})_2.
\end{align}
We also mention that $C_{\widetilde{G_n}}=\widetilde{C_{G_n}}$.
\subsection{Representations}\label{subsection:representations}
Let $G$ be an $l$-group (\cite{BZ1} 1.1). Representations of $G$ will be complex and smooth.
For a representation $\pi$ of $G$, $\pi^{\vee}$ is the representation contragradient to $\pi$.
We say that $\pi$ is glued from representations $\pi_1,\ldots,\pi_k$, if $\pi$ has a filtration, whose quotients (which may be isomorphic or zero) are, after a permutation, $\pi_1,\ldots,\pi_k$.
Regular induction is denoted $\Ind$ and $\ind$ is the compact induction.
Induction is not normalized.
Let $\pi$ be as above and let $U<G$ be a unipotent subgroup, exhausted by its compact subgroups (always the case here). Let $\psi$ be a character of $U$. The Jacquet module of $\pi$ with respect to $U$ and $\psi$ is denoted $\pi_{U,\psi}$. It is a representation of the stabilizer of $\psi$ (and normalizer of $U$). The action is not normalized. We have an exact sequence
\begin{align*}
0\rightarrow \pi(U,\psi)\rightarrow \pi\rightarrow \pi_{U,\psi}\rightarrow0.
\end{align*}
The kernel $\pi(U,\psi)$ can be characterized by the Jacquet-Langlands lemma:
\begin{lemma}\label{lemma:Jacquet kernel as integral}(see e.g. \cite{BZ1} 2.33)
a vector $v$ in the space of $\pi$ belongs to $\pi(U,\psi)$ if and only if
\begin{align*}
\int_O\pi(u)v\ \psi^{-1}(u)\ du=0,
\end{align*}
for some compact subgroup $O<U$.
\end{lemma}
When $\psi=1$, we simply write $\pi(U)$ and $\pi_U$.
Assume that $\widetilde{G}$ is a given $r$-th cover of $G$. Let $\varepsilon:\mu_r\rightarrow\C^*$ be a faithful character. A representation $\pi$ of $\widetilde{G}$ is called $\varepsilon$-genuine if it restricts to $\varepsilon$ on $\mu_r$. When $r=2$, such a representation is simply called genuine.
Let $\varphi:G\rightarrow\widetilde{G}$ be a section and assume $\pi$ and $\pi'$ are representations of $\widetilde{G}$, such that
$\pi$ is $\varepsilon$-genuine and $\pi'$ is $\varepsilon^{-1}$-genuine. Then $\pi\otimes \pi'$ (outer tensor product) is a representation of $G$ via $g\mapsto\pi(\varphi(g))\otimes\pi'(\varphi(g))$. The actual choice of $\varphi$ does not matter, whence
we omit it.
\subsection{Representations of Levi subgroups}\label{subsection:representations of Levi subgroups}
Levi subgroups of classical groups are direct products. The tensor of representations of the direct factors is usually
used, to describe their representations. In passing to a cover group, these factors do not necessarily commute and then the tensor construction fails.
Except for the case of $k=n$, the preimages of $\GL_k$ and $G_{n-k}$ in $\widetilde{M}_k$ do not commute. The same phenomena occurs in $\GL_n$.
The following discussion describes a replacement for the usual tensor product. For more details see \cite{me8}.
The metaplectic tensor product in the context of $\GL_{n}$ has been studied by various authors \cite{FK,Su2,Kable,Mezo,Tk2}.
Our definitions were motivated by the construction of Kable \cite{Kable} (see Remark~\ref{remark:diff Kable tensor} below).
For any $H<G_n$, put $H^{\square}=\setof{h\in H}{\Upsilon(h)\in {F^{*2}}}$. The subgroup $H^{\square}$ is normal in $H$ and the quotient is a finite
abelian group. If $\xi$ is a representation of $\cover{H}$, let $\xi^{\square}=\xi|_{\cover{H}^{\square}}$. Assume $0<k<n$. According to \eqref{eq:block-compatibility}, the preimages of $\GL_k^{\square}$ and $G_{n-k}^{\square}$ are commuting in the cover. Then if $\rho$ and $\pi$ are genuine representations of $\widetilde{H}_1<\widetilde{\GL}_k$ and $\widetilde{H}_2<\widetilde{G}_{n-k}$, the representation $\rho^{\square}\otimes\pi^{\square}$ is a genuine representation of
\begin{align*}
p^{-1}(H_1^{\square}\times H_2^{\square})\isomorphic \lmodulo{\setof{(\epsilon,\epsilon)}{\epsilon\in\mu_2}}{(\widetilde{H}_1^{\square}\times\widetilde{H}_1^{\square})}.
\end{align*}
Put $H=H_1H_2$ and define
\begin{align*}
\mathcal{I}^{\square}(\rho,\pi)=\ind_{p^{-1}(H_1^{\square}\times H_2^{\square})}^{\cover{H}}{(\rho^{\square}\otimes\pi^{\square})}.
\end{align*}
When $k=n$, $\widetilde{\GL}_n$ and $\widetilde{G}_0$ are commuting, then the metaplectic tensor is defined as usual.
\begin{remark}\label{remark:diff Kable tensor}
The arguments in \cite{Kable} do not readily extend to $G_n$, mainly because $C_{G_n}<G_n^{\square}$ for all $n$, and then $C_{G_n}$ does not play a role similar to that of $C_{\GL_n}$ in the cover.
\end{remark}
We will use Mackey Theory to relate this induced representation to $\rho$ and $\pi$. We reproduce the following result from \cite{me8}, which mimics \cite{Kable} (Theorem~3.1) in our context.
\begin{lemma}\label{lemma:induced representation composition factors}
The representation $\mathcal{I}^{\square}(\rho,\pi)$ is a direct sum of $[F^*:{F^{*2}}]$ copies of
\begin{align*}
\ind_{p^{-1}(H_1^{\square}\times H_2)}^{\widetilde{H}}(\rho^{\square}\otimes\pi).
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:induced representation composition factors}]
Since $p^{-1}(H_1^{\square}\times H_2^{\square})$ is normal of finite index in $\widetilde{H}$ and $p^{-1}(H_1^{\square}\times H_2)$ modulo $p^{-1}(H_1^{\square}\times H_2^{\square})$ is abelian,
\begin{align}\label{eq:claim tensor following Kable 1}
\mathcal{I}^{\square}(\rho,\pi)=\bigoplus_{a\in\lmodulo{H_1^{\square}}{H_1}}\ind_{p^{-1}(H_1^{\square}\times H_2)}^{\widetilde{H}}(\rho^{\square}\otimes\omega_a\pi).
\end{align}
Here $\omega_a(h)=(\Upsilon(h),\det{a})_2$ ranges over the finite set of characters of $H_2$, which are trivial
on $H_2^{\square}$. By \eqref{eq:block-compatibility} if $a_0\in\widetilde{\GL}_k$ and $h_0\in\widetilde{G}_{n-k}$, $\rconj{a_0^{-1}}h_0=(\Upsilon(h_0),\det{a_0})_2h_0$. Hence $\rho^{\square}\otimes\omega_a\pi=\rconj{a}(\rconj{a^{-1}}(\rho^{\square})\otimes\pi)$ and the result follows.
\end{proof}
\subsection{The Weil representation}\label{subsection:the Weil representation}
We introduce the Weil representation, which plays an important role in this work. Let $n=2k$ and $\lambda$ be the symplectic bilinear form on $F^{n}$ defined by
$\lambda(u,v)=u\left(\begin{smallmatrix}&J_{k}\\-J_{k}\end{smallmatrix}\right)\transpose{v}$, where $u$ and $v$ are regarded as rows.
Let $H_{n}$ be the $(\glnidx+1)$-dimensional Heisenberg group, with the group operation given by
\begin{align*}
(u_1,u_2;z_1)\cdot (v_1,v_2;z_2)=(u_1+v_1,u_2+v_2,z_1+z_2+\lambda((u_1,u_2),(v_1,v_2))),
\end{align*}
where $u_i,v_i\in F^k$ and $z_i\in F$.
Let $\Sp_k$ be the symplectic group defined with respect to $\lambda$, i.e., the group of
$g\in\GL_n$ such that $\lambda(ug,vg)=\lambda(u,v)$ for all $u,v\in F^n$. Let $\widetilde{\Sp}_k$ be the metaplectic double cover of $\Sp_k$, realized using the normalized Rao cocycle \cite{Rao}. The group $\Sp_k$ acts on $H_n$ via $g^{-1}(u_1,u_2;z)g=((u_1,u_2)g;z)$.
Fix a nontrivial additive character $\psi$ of $F$. Let $\omega_{\psi}$ be the Weil representation of $H_{n}\rtimes\widetilde{\Sp}_k$, realized on the space $\mathcal{S}(F^{k})$ of Schwartz-Bruhat functions on $F^{k}$. Recall the following formulas for $\omega_{\psi}$ (see \cite{P}): for $\varphi\in\mathcal{S}(F^{k})$,
\begin{align}
&\omega_{\psi}((u_1,0;z))\varphi(x)=\psi(z)\varphi(x+u_1),\label{eq:Weil X action}\\
&\omega_{\psi}((0,u_2;0))\varphi(x)=\psi(x J_{k}\transpose{u_2})\varphi(x),\label{eq:Weil R action}\\
&\omega_{\psi}((\left(\begin{smallmatrix}I_{k}&u\\&I_{k}\end{smallmatrix}\right),\epsilon))\varphi(x)=\epsilon\psi(\half x J_{k}\transpose{u}\transpose{x})\varphi(x)\qquad (\epsilon\in\mu_2).\label{eq:Weil Y action}
\end{align}
Let $R=\{(0,u_2;0)\}<H_n$ and $U=\{\left(\begin{smallmatrix}I_{k}&u\\&I_{k}\end{smallmatrix}\right)\}<\Sp_k$. Since $U$ normalizes (in fact, commutes with) $R$, $(\omega_{\psi})_{R}$ is a $U$-module.
We will use the following simple observation.
\begin{claim}\label{claim:Jacquet of Weil 1 dim}
The vector spaces $(\omega_{\psi})_{R}$ and $(\omega_{\psi})_{RU}$ are one dimensional.
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:Jacquet of Weil 1 dim}]
According to Lemma~\ref{lemma:Jacquet kernel as integral} and \eqref{eq:Weil R action}, the space of $\omega_{\psi}(R)$ is $\mathcal{S}(F^k\setdiff 0)$.
Hence $(\omega_{\psi})_{R}$ is one dimensional. Then $(\omega_{\psi})_{RU}=((\omega_{\psi})_{R})_U$ is nonzero, because by Lemma~\ref{lemma:Jacquet kernel as integral} and \eqref{eq:Weil Y action}, a function $\varphi\in \mathcal{S}(F^k)$ such that
$\varphi(0)\ne0$ does not belong to $\omega_{\psi}(RU)$.
\end{proof}
We will also encounter the tensor $\omega_{\psi}\otimes\omega_{\psi^{-1}}$ of two Weil representations. This is a representation of $H_n$, trivial
on $C_{H_n}$, and a representation of $\Sp_k$. Regarding it as a representation of $\lmodulo{C_{H_n}}{H_n}$, we can compute its twisted Jacquet modules.
The group $\Sp_k$ acts transitively on the nontrivial characters of $\lmodulo{C_{H_n}}{H_n}$, hence we can consider only one nontrivial character.
\begin{claim}\label{claim:Jacquet modules along Hn of double Weil reps}
Let $\mu$ be a character of $H_n$, which is trivial on $C_{H_n}$.
\begin{enumerate}
\item If $\mu=1$, $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n,\mu}$ is the trivial one-dimensional representation of $\Sp_k$.
\item If $\mu(u_1,u_2;z)=\psi((u_1)_1)$,
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n,\mu}$ is the trivial one-dimensional representation of
$P_{n-1,1}^{\circ}\cap \Sp_k$.
\end{enumerate}
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:Jacquet modules along Hn of double Weil reps}]
First assume $\mu=1$.
Equality~\eqref{eq:Weil R action} implies that elements $\varphi\otimes\varphi'$ in the space of $\omega_{\psi}\otimes\omega_{\psi^{-1}}$, such that
the supports of $\varphi$ and $\varphi'$ (as functions in $\mathcal{S}(F^k)$) are different, vanish under the Jacquet module along $R$ (use Lemma~\ref{lemma:Jacquet kernel as integral}).
Hence the space of $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_R$ is isomorphic to $\mathcal{S}(F^k)$. Since the action of $C_{H_n}$ is trivial
on $\omega_{\psi}\otimes\omega_{\psi^{-1}}$, $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_R=(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}$.
Now $RC_{H_n}$ is a normal subgroup of $H_n$, whence $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$ is a quotient of $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}$.
It then follows
from \eqref{eq:Weil Y action} that the action of $U$ is trivial on
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}$ and in particular, on
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$. The latter is a representation
of $\Sp_k$, and because $\Sp_k$ is generated (as an abstract group) by the subgroups $\rconj{w}U$, $w\in\Sp_k$ (it is enough
to take Weyl elements $w$), $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$ must be a trivial representation of $\Sp_k$.
Moreover
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$ is one-dimensional. This can be seen as follows. Replace
a function $f\in \mathcal{S}(F^k)=(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}$ with its Fourier transform
$\widehat{f}(x)=\int_{F^k}f(y)\psi(x(\transpose{y}))dy$. This changes the module structure, but the action of $\Sp_k$ remains trivial. The action of
$(u_1,0;0)\in H_n$ is now given by
\begin{align*}
(u_1,0;0)\cdot\widehat{f}(x)=\psi^{-1}(x\transpose{u_1})\widehat{f}(x)
\end{align*}
(instead of \eqref{eq:Weil X action}). Next apply Lemma~\ref{lemma:Jacquet kernel as integral}, the space of
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}(\lmodulo{(RC_{H_n})}{H_n})$ is $\mathcal{S}(F^k-0)$.
We conclude that $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$ is the trivial one-dimensional representation of $\Sp_k$.
Now consider the case of the nontrivial $\mu$, given in the statement of the claim. Since $\mu|_{R}=1$,
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n,\mu}$ is a quotient of
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}$ and in particular, a trivial representation of $U$. In coordinates,
\begin{align*}
P_{n-1,1}^{\circ}\cap \Sp_k=\left\{\left(\begin{array}{ccc}1&u&v\\&g&*\\&&1\end{array}\right):g\in\Sp_{k-1}\right\}.
\end{align*}
(In particular it stabilizes $\mu$.) We see that $U<P_{n-1,1}^{\circ}\cap \Sp_k$ and using conjugations of $U$ by
elements from $\Sp_{k-1}$, it follows that $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n,\mu}$ is the trivial representation of
$P_{n-1,1}^{\circ}\cap \Sp_k$.
To show this is a one-dimensional representation, argue as above using the Fourier
transform, the space of
$(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{RC_{H_n}}(\lmodulo{(RC_{H_n})}{H_n,\mu})$ is $\mathcal{S}(F^k-(-1,0,\ldots,0))$.
\end{proof}
\subsection{Exceptional or small representations}\label{subsection:The exceptional representations}
We describe the exceptional representations that appear in this work. Kazhdan and Patterson \cite{KP} introduced and studied these
representations for $\GL_n$. For $G_n$ these are essentially the small representations of Bump, Friedberg and Ginzburg \cite{BFG},
who developed them using a cover of $\SO_{2n+1}$ (see Section~\ref{section:relevance to SO(2n+1)}). Their construction was extended to $G_n$ in \cite{me8}.
Let $G$ be either $\GL_n$ or $G_n$. Let $B$ be the Borel subgroup of $G$, $T$ be the maximal torus and $\Delta$ be the subset of simple roots. For $\alpha\in\Delta$,
if $\alpha$ is a long root and $n>1$, put $\mathfrak{l}(\alpha)=2$, otherwise $\mathfrak{l}(\alpha)=1$. Denote by $\alpha^{\vee}$ the coroot of $\alpha$.
Let $\xi$ be a genuine character of $C_{\cover{T}}$. We say that $\xi$ is exceptional if
\begin{align*}
\xi(\mathfrak{s}(\alpha^{\vee}(x^{\mathfrak{l}(\alpha)})))=|x|,\qquad\forall \alpha\in\Delta,x\in F^*.
\end{align*}
The character $\xi$ corresponds to a genuine irreducible representation $\rho(\xi)$ of $\widetilde{T}$ (when $n>1$, $\widetilde{T}$ is a $2$-step nilpotent subgroup). Since $\xi$ is exceptional,
the representation $\Ind_{\widetilde{B}}^{\widetilde{G}}(\delta_{B}^{1/2}\rho(\xi))$ has a unique irreducible quotient $\theta$, called an
exceptional representation. Note that $\theta$ is admissible. Occasionally, we use the notation
$\theta^{G}$ to record the group.
We appeal to the following explicit description of $C_{\widetilde{T}}$:
$C_{\widetilde{T}_{\GL_n}}=\widetilde{T}_{\GL_n}^2C_{\widetilde{\GL}_n}$ with
$T_{\GL_n}^2=\setof{t^2}{t\in T_{\GL_n}}$,
$C_{\widetilde{\GL}_n}=\setof{z\cdot I_n}{z\in F^{*\mathe}}$, where
$\mathe$ is $1$ if $n$ is odd, otherwise $\mathe=2$; $C_{\widetilde{T}_{n+1}}=C_{\widetilde{T}_{\GL_n}}\widetilde{G}_0$, and if
$T_{n+1}^2=T_{\GL_n}^2G_0$, $C_{\widetilde{T}_{n+1}}=\widetilde{T}_{n+1}^2C_{\widetilde{\GL}_n}$ (see \cite{KP} p.~57 and \cite{me8} Section~2.1.6).
Note that $G_0^{\square}=G_0$. Furthermore, the cocycle $\sigma_G$ satisfies $\sigma_G(z\cdot I_n,z'\cdot I_n)=(z,z')^{\lceil n/2\rceil}$.
The exceptional characters can be parameterized in the following way. Start by fixing a character $\xi_0$ of
$p(C_{\widetilde{T}})$, trivial on $p(C_{\widetilde{G}})$. This does not determine $\xi_0$ uniquely.
For $\GL_n$, take $\xi_0=\delta_{B_{\GL_n}}^{1/4}$. For $G_n$ take $\xi_0$ whose restriction to
$p(C_{\widetilde{T}_{\GL_n}})$ is $\delta_{B_{\GL_n}}^{1/4}\cdot|\det{}|^{(n+1)/4}$.
Given that $\xi_0$ is trivial on $p(C_{\widetilde{G}_n})$, this determines $\xi_0$ uniquely for $G_n$ (see \cite{me8} Section~2.3.3 for
an explicit formula).
Now for any given $\xi$, there is a character $\chi$ of $F^*$ (called ``determinantal character" in \cite{BG}) and a nontrivial additive
character $\psi$ of $F$ such that
\begin{align}\label{eq:explicit construction exceptional characters}
\xi(\epsilon\mathfrak{s}(td))=\epsilon\xi_0(td)\chi(\Upsilon(td))\gamma_{\psi}^{\lceil n/2\rceil}(z),\qquad
\forall t\in T^2,d=z^{\mathe}\cdot I_n\in T,\epsilon\in\mu_2.
\end{align}
The corresponding exceptional
representation will be denoted $\theta_{\chi,\psi}$.
\begin{claim}\label{claim:taking characters out of exceptional}
We have $\theta_{\chi,\psi}=\chi\theta_{1,\psi}$, where on the right-hand side we
pull back $\chi$ to a non-genuine character of $\widetilde{G}$ via $g\mapsto\chi(\Upsilon(p(g)))$.
Additionally, if $\psi_0$ is another additive character of $F$,
$\theta_{\chi,\psi}=\eta\theta_{\chi,\psi_0}$ for some square trivial character $\eta$ of $F^*$.
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:taking characters out of exceptional}]
The first assertion is clear. Write $\psi(x)=\psi_0(\alpha x)$ for some
$\alpha\in F^*$. Then $\gamma_{\psi}=\eta_0\gamma_{\psi_0}$, where $\eta_0(z)=(\alpha,z)_2$. Since $\eta_0$ is trivial
on $F^{*2}$ and
\begin{align*}
\eta_0(\Upsilon(d))=\eta_0(\det{d})=\eta_0(z^n)=\eta_0(z)
\end{align*}
(the last equality is trivial if $n$ is even, because then $z\in F^{*2}$), we obtain
\begin{align*}
\chi(\Upsilon(td))\gamma_{\psi}^{\lceil n/2\rceil}(z)=
(\eta_0^{\lceil n/2\rceil}\chi)(\Upsilon(td))\gamma_{\psi_0}^{\lceil n/2\rceil}(z).
\end{align*}
Hence $\theta_{\chi,\psi}=\theta_{\eta_0^{\lceil n/2\rceil}\chi,\psi_0}=\eta_0^{\lceil n/2\rceil}\theta_{\chi,\psi_0}$.
\end{proof}
Exceptional representations have a useful inductive property.
Let $\theta$ be an exceptional representation of $\widetilde{G}_{n}$. Following the arguments of Bump, Friedberg and Ginzburg \cite{BFG} (Theorem~2.3),
we computed $\theta_{U_k}$ (\cite{me8}). For $0<k<n$,
\begin{align}\label{eq:containment of Jacquet module along maximal unipotent}
(\theta_{\chi,\psi})_{U_k}\subset\mathcal{I}^{\square}(\theta^{\GL_k}_{|\cdot|^{(2n-k-1)/4}\chi,\psi},\theta^{G_{n-k}}_{\chi,\psi}).
\end{align}
If $k$ (resp. $n-k$) is odd, the exceptional representation of $\widetilde{\GL}_k$ (resp. $\widetilde{G}_{n-k}$) is unique only up
to varying the character $\psi$, or multiplying $\chi$ by a square trivial character of $F^*$. In any case, the space
on the \rhs\ of \eqref{eq:containment of Jacquet module along maximal unipotent} is unique, because by Claim~\ref{claim:taking characters out of exceptional},
the exceptional representation obtained by such a change to $\psi$ or $\chi$, has the same restriction to $\widetilde{\GL}_k^{\square}$ (resp. $\widetilde{G}_{n-k}^{\square}$)
as the original representation.
If $k=n$,
\begin{align}\label{eq:containment of Jacquet module along maximal unipotent 2}
(\theta_{\chi,\psi})_{U_n}=\theta^{\GL_n}_{|\cdot|^{(n-1)/4}\chi,\psi}\otimes\theta^{G_0}_{\chi,1}.
\end{align}
In \cite{me8} we did not find the precise exceptional representations appearing on the \rhs\ of \eqref{eq:containment of Jacquet module along maximal unipotent 2}, but this is
simple to obtain:
since $C_{\widetilde{T}_{n+1}}=C_{\widetilde{T}_{\GL_n}}\widetilde{G}_0$, there is an exceptional character $\xi_1$ of $C_{\widetilde{T}_{\GL_n}}$
such that $\rconj{w_0}\xi=\rconj{w_0'}\xi_1\otimes\xi|_{\widetilde{G}_0}$,
where $w_0$ and $w_0'$ are the longest Weyl elements in the Weyl groups of $G_n$ and $\GL_n$ (see \cite{me8} Claim~2.18 for details). It remains to write $\xi_1$
using \eqref{eq:explicit construction exceptional characters}.
\begin{remark}
The reason for the imprecise result when $k<n$ is the
lack of a definition for a tensor product. These results are sufficient for our applications.
\end{remark}
For $\GL_n$, Kable proved a result similar to \eqref{eq:containment of Jacquet module along maximal unipotent 2} for all standard unipotent radicals,
with the tensor replaced by his metaplectic tensor (\cite{Kable} Theorem~5.1 (4)).
Exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules.
The following result is the extension of Theorem~2.6 of \cite{BFG} and Proposition~3 of \cite{BFG2} to $G_n$ (this extension
appeared in \cite{me8}). It was used in \cite{BFG,BFG2} (for $\SO_{2n+1}$) to deduce all vanishing properties.
The unipotent radical $U_1$ is abelian. A character $\psi^{(1)}$ of $U_1$ takes the form
\begin{align*}
\psi^{(1)}(\left(\begin{array}{ccc}1&u&*\\&I_{2n-1}&*\\&&1\end{array}\right))=\psi(ua),
\end{align*}
where $a\in F^{2n-1}$ is a column. The length of $a$ is defined to be $\transpose{a} J_{2n-1}a$.
When $\psi$ is fixed and clear from the context, we also refer to $\transpose{a}J_{2n-1}a$ as the length of $\psi^{(1)}$.
\begin{theorem}\label{theorem:main vanishing Jacquet of small rep}
If the length of $\psi^{(1)}$ is nonzero, $\theta_{U_1,\psi^{(1)}}=0$.
\end{theorem}
See \cite{me8} (Lemma~2.25) for the details.
\section{Twisted Jacquet modules of $\theta^{G_n}$}\label{section:Heisenberg jacquet modules}
In this section we describe the twisted Jacquet modules of $\theta=\theta^{G_n}$ with respect to the center of $U_n$. These modules appear in a filtration of $\theta$ as a $\widetilde{Q}_n$-module and will be used in Section~\ref{section:Distinguished representations}.
The group $\GL_n$, embedded in $M_n$, acts on the set of characters of $C=C_{U_n}$ with finitely many orbits. Let $\psi$ be a nontrivial
additive character of $F$. For any $0\leq k\leq \lfloor n/2 \rfloor$, define a character of $C$ by
\begin{align*}
\psi_k(c)=\psi(\sum_{i=1}^kc_{n-2i+1,n+2i})
\end{align*}
($c$ is regarded as a $(2n+1)\times(2n+1)$ unipotent matrix in $\SO_{2n+1}$).
The stabilizer of $\psi_k$ in $\widetilde{M}_n$ is $\widetilde{\mathrm{St}}_{\psi_k}$, with
\begin{align*}
\mathrm{St}_{\psi_k}=\left\{\left(\begin{array}{cc}a&z\\0&b\end{array}\right):a\in\GL_{n-2k},b\in \Sp_{k}\right\}\times G_0.
\end{align*}
Here $\Sp_{k}$ is the symplectic group in $2k$ variables, corresponding to a symplectic bilinear form defined according to $\psi_{k}$.
Then $\theta_{C,\psi_k}$ is a representation of $\widetilde{\mathrm{St}}_{\psi_k}\ltimes U_n$.
Regarding
$\GL_{n-2k}$ and $\Sp_k$ as subgroups of $\mathrm{St}_{\psi_k}$,
\begin{align*}
\mathrm{St}_{\psi_k}=((\GL_{n-2k}\times \Sp_k)\ltimes Z_{n-2k,2k}) \times G_0.
\end{align*}
($Z_{n-2k,2k}$ was given in Section~\ref{subsection:GL_n and its cover}.)
We turn to the proof of Theorem~\ref{theorem:small rep is weakly minimal}.
Namely, for $n=2k$, $\theta_{C,\psi_{k}}$ is the direct sum of copies of the Weil representation $\omega_{\psi}$.
Here is an outline of the proof.
The theory of smooth representations of Jacobi groups (\cite{Dijk,MVW,BSch}) implies that any such representation, with a central character $\psi$, takes the form
$\kappa\otimes\omega_{\psi}$, where the Heisenberg group acts trivially on the space of $\kappa$, and the action of the
symplectic group separates into an action on the space of $\kappa$, and one on the space of $\omega_{\psi}$. The vanishing properties of $\theta$ - Theorem~\ref{theorem:main vanishing Jacquet of small rep}, will show that
$\kappa$ is trivial.
\begin{proof}[Proof of Theorem~\ref{theorem:small rep is weakly minimal}]
Put $k=n/2$. The image of $G_0$ in $G_n$ is $C_{G_n}$. Hence $\mathrm{St}_{\psi_{k}}$ is the direct product of a Jacobi group and
$G_0$. Moreover, $\widetilde{G}_0=C_{\widetilde{G_n}}$ (see Section~\ref{subsection:metaplectic GSpin}) and because
$\theta$ is an irreducible representation, $\widetilde{G}_0$ acts by the central character of $\theta$. Therefore in the proof we ignore the $G_0$ part of
$\mathrm{St}_{\psi_k}$.
We may replace $\psi_k$ by any character of $C$ in the same $\GL_n$-orbit, since the Jacquet module will be isomorphic. For convenience,
we redefine $\psi_{k}(c)=\psi(\sum_{i=1}^kc_{i,{n}+1+i})$. We use the notation of Section~\ref{subsection:the Weil representation}. The stabilizer $\mathrm{St}_{\psi_k}$ is now the symplectic group defined with respect to the form $\lambda$. The cover $\widetilde{\mathrm{St}}_{\psi_k}$ is a nontrivial double cover.
We have an epimorphism $\ell:\widetilde{\mathrm{St}}_{\psi_k}\ltimes U_n\rightarrow \widetilde{\Sp}_k\ltimes H_n$:
\begin{align*}
&\ell(\left(\begin{array}{ccc}I_{{n}}&u&z\\&1&-\transpose{u}J_{{n}}\\&&I_{{n}}\end{array}\right))=(\transpose{u}J_{{n}};\half(\sum_{i=1}^kz_{i,i}-
\sum_{i=k+1}^{{n}}z_{i,i}))\in H_n,\\
&\ell((\left(\begin{array}{ccc}g&&\\&1&\\&&J_{{n}}\transpose{g}^{-1}J_{{n}}\end{array}\right),\epsilon))=(J_{{n}}\transpose{g}^{-1}J_{{n}},\epsilon)\in\widetilde{\Sp}_k \qquad(\text{$g$ preserves $\lambda$},\quad\epsilon\in\mu_2).
\end{align*}
The kernel of $\ell$ is contained in the kernel of $\psi_k$ \footnote{In \cite{me7} p.~25 it was incorrectly stated they are equal.}. Therefore we can regard $\theta_{C,\psi_k}$ as a genuine representation
of $\widetilde{\Sp}_k\ltimes H_n$. As such, it is isomorphic to $\kappa\otimes\omega_{\psi}$ (see e.g. \cite{BSch} p.~28), where
$\kappa$ is a non-genuine representation on a space $\mathbb{V}$, and the action is
given by
\begin{align*}
(g,\epsilon)h\cdot(\mathfrak{f}\otimes\varphi)=\kappa(g)\mathfrak{f}\otimes \omega_{\psi}((g,\epsilon)h)\varphi,\qquad g\in \Sp_k,\quad h\in H_n,\quad \mathfrak{f}\in\mathbb{V},\quad\varphi\in\mathcal{S}(F^k).
\end{align*}
We must show that $\kappa$ is trivial. Consider the subgroup
\begin{align*}
Y=\left\{\left(\begin{array}{ccc}1&0&y\\&I_{2(n-1)}&0\\&&1\end{array}\right)\right\}<\Sp_k.
\end{align*}
Since $\Sp_k$ is generated
by the conjugates $\rconj{x}Y$ where $x\in\Sp_k$, it is enough to prove
invariance under $Y$. That is, we show
\begin{align}\label{eq:invariancy under Y to prove}
\kappa_{Y}=\kappa.
\end{align}
A character of $Y$ takes the form $y\mapsto\psi(\alpha y)$ for some $\alpha\in F$ (on the \lhs\ $y$ is regarded as a matrix, on the \rhs\ as an element of $F$). The action of the torus of $\Sp_k$ on
the nontrivial characters of $Y$ has finitely many orbits, namely the different square classes in $F^*$. Each of these orbits is open. Therefore the kernel of the Jacquet functor $\kappa_{Y}$ is filtered by representations induced from $\kappa_{Y,\psi(\alpha\cdot)}$, where $\alpha$
ranges over the square classes (\cite{BZ1} 5.9-5.12). Hence \eqref{eq:invariancy under Y to prove} follows if we prove $\kappa_{Y,\psi(\alpha\cdot)}=0$ for any $\alpha\ne0$.
Consider the subgroup $X=Y\cdot\setof{(0,u_2;0)\in H_n}{u_2=(0,\ldots,0,r)}$ (a direct product). The
epimorphism $\ell$ splits over $X$, hence
there is a subgroup $U<\widetilde{\mathrm{St}}_{\psi_k}U_n$ isomorphic to $X$. In fact,
\begin{align*}
U=\left\{\left(\begin{array}{ccccccc}1&0&y&r&0&0&-r^2/2\\&I_{n-2}&&&&&0\\&&1&&&&0\\&&&1&&&-r\\&&&&1&&-y\\&&&&&I_{n-2}&0\\&&&&&&1\end{array}\right)\right\}<U_1.
\end{align*}
The pullback of $\psi(\alpha\cdot)$ to $U$ is $\psi^{\star}(u)=\psi(-\alpha u_{1,n})$. Observe that $(\theta_{C,\psi_k})_{U,\psi^{\star}}=0$. Indeed, $(\theta_{C,\psi_k})_{U,\psi^{\star}}=\theta_{CU,\psi_k\psi^{\star}}$, which is a quotient of $\theta_{U(C\cap U_1),\psi_k\psi^{\star}}$. Since for $u\in U(C\cap U_1)$, $\psi_k\psi^{\star}(u)=\psi(-\alpha u_{1,n}+u_{1,n+2})$, any extension of $\psi_k\psi^{\star}$ to a character of $U_1$ is a character of nonzero length. Thus Theorem~\ref{theorem:main vanishing Jacquet of small rep} yields $\theta_{U(C\cap U_1),\psi_k\psi^{\star}}=0$ whence $(\theta_{C,\psi_k})_{U,\psi^{\star}}=0$. Therefore by Lemma~\ref{lemma:Jacquet kernel as integral}, for any $\mathfrak{f}\otimes\varphi$ there is a compact $\mathcal{O}<Y\cdot R$ ($R=\{(0,u_2;0)\}$, see Section~\ref{subsection:the Weil representation}) such that
\begin{align}\label{eq:Y acts trivially relation to twisted Jacquet vanishing of theta}
\int_{\mathcal{O}}yr\cdot(\mathfrak{f}\otimes \varphi)\ \psi^{-1}(\alpha y)\ dr\ dy=0.
\end{align}
According to Claim~\ref{claim:Jacquet of Weil 1 dim} and Lemma~\ref{lemma:Jacquet kernel as integral} there is
$\varphi\in\mathcal{S}(F^k)$ such that for all compact subgroups $\mathcal{Y}<Y$ and $\mathcal{R}<R$,
\begin{align}\label{eq:def of varphi to take for tensor vanishing}
\varphi^{\mathcal{Y},\mathcal{R}}=\int_{\mathcal{Y}}\int_{\mathcal{R}}\omega_{\psi}(yr)\varphi\ dy\ dr\ne0.
\end{align}
Take $\mathfrak{f}\in\mathbb{V}$. We will show that for large enough $\mathcal{Y}$ and $\mathcal{R}$,
\begin{align}\label{eq:tensor vanishes to show}
\int_{\mathcal{Y}}\kappa(y_1)\mathfrak{f}\ \psi^{-1}(\alpha y_1)\ dy_1\ \otimes\ \varphi^{\mathcal{Y},\mathcal{R}}=0.
\end{align}
This along with \eqref{eq:def of varphi to take for tensor vanishing} imply that $\mathfrak{f}$ belongs to the space of $\kappa(Y,\psi(\alpha\cdot))$.
Plugging \eqref{eq:def of varphi to take for tensor vanishing} into \eqref{eq:tensor vanishes to show} and changing variables leads to
\begin{align*}
\int_{\mathcal{Y}}\left(\int_{\mathcal{Y}}\int_{\mathcal{R}} \kappa(y)\mathfrak{f}\ \otimes\ \omega_{\psi}(yry_1^{-1})\varphi\ \psi^{-1}(\alpha y)\ dr\ dy\right)\ dy_1.
\end{align*}
We will show that the inner $drdy$-integration vanishes for all $y_1\in\mathcal{Y}$. Fix $y_1$. Since $\kappa|_{H_n}$ is trivial,
this inner integration equals
\begin{align}\label{eq:part 00}
\int_{\mathcal{Y}}\int_{\mathcal{R}}yr\cdot(\mathfrak{f}\otimes\omega_{\psi}(y_1^{-1})\varphi)\ \psi^{-1}(\alpha y)\ dr\ dy.
\end{align}
Again resorting to Claim~\ref{claim:Jacquet of Weil 1 dim},
\begin{align}\label{eq:part 1}
\omega_{\psi}(y_1^{-1})\varphi=c_{y_1}\varphi+\varphi_{y_1}^{\circ},
\end{align}
where $c_{y_1}\in\C$ and $\varphi_{y_1}^{\circ}$ belongs to the space of $\omega_{\psi}(R)$. Since $y_1$ varies in a compact subgroup,
there is a large enough $\mathcal{R}$ such that
\begin{align}\label{eq:part 2}
\int_{\mathcal{R}}\omega_{\psi}(r)\varphi_{y_1}^{\circ}\ dr=0,\qquad\forall y_1\in\mathcal{Y}.
\end{align}
Furthermore \eqref{eq:Y acts trivially relation to twisted Jacquet vanishing of theta} implies that for large
$\mathcal{Y}$ and $\mathcal{R}$,
\begin{align*}
\int_{\mathcal{Y}}\int_{\mathcal{R}} yr\cdot(\mathfrak{f}\otimes \varphi)\ \psi^{-1}(\alpha y)\ dr\ dy=0
\end{align*}
and then for any $c\in\C$,
\begin{align}\label{eq:part 3}
\int_{\mathcal{Y}}\int_{\mathcal{R}} yr\cdot(\mathfrak{f}\otimes c\varphi)\ \psi^{-1}(\alpha y)\ dr\ dy=0.
\end{align}
Combining \eqref{eq:part 1}-\eqref{eq:part 3} we conclude that for sufficiently large $\mathcal{R}$ and $\mathcal{Y}$,
the inner $drdy$-integration \eqref{eq:part 00} vanishes. Note the order of selecting the compact subgroups: first, choose $\mathcal{R}$ and $\mathcal{Y}$
which ensure \eqref{eq:part 3}, they depend only on $\mathfrak{f}$ and $\varphi$. Then, increase $\mathcal{R}$ to have \eqref{eq:part 2}, it will depend on $\varphi$ and $\mathcal{Y}$. This completes the proof of \eqref{eq:tensor vanishes to show} and thereby \eqref{eq:invariancy under Y to prove}. We conclude that $\kappa$ is trivial.
\end{proof}
In the more general case, for arbitrary $k$, we are less precise.
\begin{proposition}\label{proposition:Jacquet module C and k}
There are exceptional representations $\theta^{\GL_{n-2k}}$ and $\theta^{G_{2k}}$ such that
$\theta_{C,\psi_k}$ is embedded in a finite direct sum of copies of the representation
\begin{align*}
\vartheta\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k},\qquad
\vartheta=\begin{dcases}\theta^{\GL_{n}}&k=0,\\
\ind_{\widetilde{\GL}_{n-2k}^{\square}}^{\widetilde{\GL}_{n-2k}}((\theta^{\GL_{n-2k}})^{\square})&k>0.
\end{dcases}
\end{align*}
Here $\vartheta\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$ is
regarded as a representation of $\widetilde{\mathrm{St}}_{\psi_k}\ltimes U_n$ by extending it trivially on $U_{n-2k}$.
The representations $\theta^{\GL_{n-2k}}$ and $\theta^{G_{2k}}$ are related to
$\theta$ via \eqref{eq:containment of Jacquet module along maximal unipotent} and \eqref{eq:containment of Jacquet module along maximal unipotent 2}.
If $k=0$, the
embedding is in fact an isomorphism and there is only one summand, otherwise there are $[F^*:F^{*2}]$ summands.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{proposition:Jacquet module C and k}]
For $n=1$, we have $C=U_1$ and $k=0$, whence $\theta_{C,\psi_k}=\theta_{U_1}$ and the result follows immediately
from \eqref{eq:containment of Jacquet module along maximal unipotent 2}.
Assume $n>1$. Further assume $k<n/2$, otherwise there is nothing to prove.
The main part of the proof is to show that $U_{n-2k}$ acts trivially on $\theta_{C,\psi_k}$. Of course, this holds for
$U_{n-2k}\cap C$. Let $V_k=U_{n-2k}\cap U_n$ and note that $Z_{n-2k,2k}=U_{n-2k}\cap M_n$. Clearly $U_{n-2k}=V_k\rtimes Z_{n-2k,2k}$.
The following claims imply that the action of $U_{n-2k}$ is trivial.
\begin{claim}\label{claim:V_k acts trivially}$\theta_{C,\psi_k}=\theta_{V_kC,\psi_{k}}$.
\end{claim}
\begin{claim}\label{claim:Z_n-2k,k acts trivially}$\theta_{V_kC,\psi_{k}}=\theta_{U_{n-2k}C,\psi_{k}}$.
\end{claim}
Before proving the claims, let us deduce the proposition. Clearly
$\theta_{U_{n-2k}C,\psi_{k}}=(\theta_{U_{n-2k}})_{C_{U_{2k}},\psi_{k}}$. Assume $k>0$. Then
by \eqref{eq:containment of Jacquet module along maximal unipotent},
\begin{align}\label{eq:equality to correct if k = 0}
\theta_{U_{n-2k}C,\psi_{k}}\subset\mathcal{I}^{\square}(\theta^{\GL_{n-2k}},\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}.
\end{align}
According to Lemma~\ref{lemma:induced representation composition factors}, $\mathcal{I}^{\square}(\theta^{\GL_{n-2k}},\theta^{G_{2k}})$ is the finite direct sum of $[F^*:F^{*2}]$ copies of
\begin{align*}
\ind_{p^{-1}(\GL_{n-2k}^{\square}\times G_{2k})}^{\widetilde{M}_{n-2k}}((\theta^{\GL_{n-2k}})^{\square}\otimes\theta^{G_{2k}}).
\end{align*}
Let $\mathrm{St}'_{\psi_k}$ be the stabilizer of $\psi_k$ in $M_{2k}$, when $\psi_k$ is regarded as a character of $C_{U_{2k}}$ and $M_{2k}<G_{2k}<G_n$.
The double coset space
\begin{align*}
\rmodulo{\lmodulo{\GL_{n-2k}^{\square}\times G_{2k}}{M_{n-2k}}}{\GL_{n-2k}\times\mathrm{St}'_{\psi_k}}
\end{align*}
is trivial. Then by virtue of the Geometric Lemma of Bernstein and Zelevinsky \cite{BZ2} (Theorem~5.2),
\begin{align*}
\ind_{p^{-1}(\GL_{n-2k}^{\square}\times G_{2k})}^{\widetilde{M}_{n-2k}}((\theta^{\GL_{n-2k}})^{\square}\otimes\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}
=\ind_{p^{-1}(\GL_{n-2k}^{\square}\times \mathrm{St}'_{\psi_k})}^{p^{-1}(\GL_{n-2k}\times \mathrm{St}'_{\psi_k})}((\theta^{\GL_{n-2k}})^{\square}\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}).
\end{align*}
Equality~\eqref{eq:block-compatibility} implies the subgroups $\widetilde{\GL}_{n-2k}$ and $\widetilde{\Sp}_k$ commute. Therefore
\begin{align*}
\ind_{p^{-1}(\GL_{n-2k}^{\square}\times \mathrm{St}'_{\psi_k})}^{p^{-1}(\GL_{n-2k}\times \mathrm{St}'_{\psi_k})}((\theta^{\GL_{n-2k}})^{\square}\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k})
=(\ind_{\widetilde{\GL}_{n-2k}^{\square}}^{\widetilde{\GL}_{n-2k}}(\theta^{\GL_{n-2k}})^{\square})\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}.
\end{align*}
Thus $\mathcal{I}^{\square}(\theta^{\GL_{n-2k}},\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$ is the direct sum of representations $\vartheta\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$.
The proposition follows from this. Note that for $k=0$,
$\theta_{U_{n-2k}C,\psi_{k}}=\theta_{U_{n}}$ and we can apply \eqref{eq:containment of Jacquet module along maximal unipotent 2} instead of
\eqref{eq:containment of Jacquet module along maximal unipotent}, then
\eqref{eq:equality to correct if k = 0} becomes $\theta_{U_{n}C}=\theta^{\GL_{n}}\otimes\theta^{G_{0}}$.
\begin{proof}[Proof of Claim~\ref{claim:V_k acts trivially}]
A character $\psi_k^{\star}$ of $V_kC$ extending $\psi_k$ is defined by its restriction to the nontrivial coordinates on the $(n+1)$-th column of $v\in V_k$. We call $\psi_k^{\star}$ nontrivial if this restriction is nontrivial.
We prove that the Jacquet module of $\theta_{C,\psi_k}$ with respect to $V_k$ and $\psi^{\star}$ vanishes for any nontrivial $\psi^{\star}$. The group $\GL_{n-2k}<\mathrm{St}_{\psi_k}$ acts transitively on these characters. Therefore, it is enough to show
\begin{align*}
(\theta_{C,\psi_k})_{V_k,\psi_{k}^{\star}}=\theta_{V_kC,\psi_{k}^{\star}}=0,\qquad \psi_k^{\star}(v)=\psi(v_{1,n+1}), \quad v\in V_k.
\end{align*}
This follows immediately from Theorem~\ref{theorem:main vanishing Jacquet of small rep}, because any character of $U_1$ extending $\psi_{k}^{\star}|_{U_1}$ has a nonzero length. Thus $\theta_{C,\psi_k}=\theta_{V_kC,\psi_{k}}$.
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim:Z_n-2k,k acts trivially}]
For $k=0$ there is nothing to prove ($V_0=U_n)$. Assume $k>0$.
The claim follows once we show that for any nontrivial character $\mu$ of $Z_{n-2k,2k}$,
\begin{align}\label{eq:lemma first vanishing k < n/2 action of Z}
(\theta_{V_kC,\psi_k})_{Z_{n-2k,2k},\mu}=0.
\end{align}
The group
$Z_{n-2k,2k}$ is abelian and $\GL_{n-2k}\times \Sp_k$ acts on the characters of $Z_{n-2k,2k}$. Write an element $z\in Z_{n-2k,2k}$ in the form
\begin{align*}
z=z(z_1,z_2,z_3,z_4)=\left(\begin{array}{cccc}I_{n-2k-1}&&z_1&z_2\\&1&z_3&z_4\\&&1\\&&&I_{2k-1}\end{array}\right).
\end{align*}
We may assume that $\mu$ does not depend on the coordinates of $z_1$ and $z_4$, and depends on $z_3$. For simplicity, also assume
$\mu(z(0,0,z_3,0))=\psi(z_3)$.
We use the local analog of ``exchanging roots", proved by Ginzburg, Rallis and Soudry \cite{GRS5} (Lemma~2.2). (For the global setting see \cite{G,GRS3,Soudry5,RGS}.) Let $Z_{1}<Z_{n-2k,2k}$ be the subgroup of elements $z(z_1,0,0,0)$ and $Z_{2,3,4}<Z_{n-2k,2k}$ be the subgroup
consisting of elements $z(0,z_2,z_3,z_4)$. Clearly $Z_{n-2k,2k}=Z_{1}\cdot Z_{2,3,4}$ (a direct product). Also consider the subgroup
\begin{align*}
E=\left\{\left(\begin{array}{ccc}I_{n-2k-1}\\e&1\\&&I_{2k}\end{array}\right)\right\}.
\end{align*}
In general, if $\pi$ is a smooth representation of $Z_{n-2k,2k}\rtimes E$, then by \cite{GRS5} (Lemma~2.2), as $Z_{2,3,4}$-modules
\begin{align*}
\pi_{Z_{n-2k,2k},\mu}=\pi_{Z_{2,3,4}\rtimes E,\mu}.
\end{align*}
Indeed, it is simple to check that the list of properties stated in the lemma are satisfied in this setting (in the notation
of \cite{GRS5}, $C=Z_{2,3,4}$, $X=Z_1$ and $Y=E$).
\begin{comment}
Indeed, it is simple to check that the list of properties stated in the lemma are satisfied in this setting. Namely,
\begin{enumerate}
\item $Z_{1},Z_{2,3,4}$ and $E$ subgroups of a unipotent subgroup $A$ of $\GL_n$.
\item $Z_{1}$ and $E$ are abelian, normalize $Z_{2,3,4}$ and fix $\mu$.
\item $z_1^{-1}e^{-1}z_1e\in Z_{2,3,4}$ for all $z\in Z_{1}$ and $e\in E$.
\item $A=Z_{n-2k,2k}\rtimes E=(Z_{2,3,4}E)\rtimes Z_1$.
\item The set $z_1\mapsto\mu(z_1^{-1}e^{-1}z_1e)$ as $e$ varies in $E$ is the group of characters of $Z_1$.
\item If $z_1=\mathrm{exp}(L)$ and $e=\mathrm{exp}(M)$, where $L$ and $M$ belong to the Lie algebras $\mathcal{Z}_1$ and $\mathcal{E}$ of $Z_1$ and $E$, then
$\mu(z_1^{-1}e^{-1}z_1e)=\psi((L,M))$. Here $(,)$ is a non degenerate, bilinear pairing between $\mathcal{Z}_1$ and $\mathcal{E}$.
\end{enumerate}
\begin{remark}
Lemma~2.2 of \cite{GRS5} was stated for unipotent subgroups of symplectic groups, but the arguments are general and hold in our setting.
\end{remark}
\end{comment}
\begin{remark}
Lemma~2.2 of \cite{GRS5} was stated for unipotent subgroups of symplectic groups, but the arguments are general and hold in our setting.
See also Section~2.3 of \cite{GRS5}.
\end{remark}
It follows that as $Z_{2,3,4}$-modules
\begin{align*}
(\theta_{V_kC,\psi_k})_{Z_{n-2k,2k},\mu}=\theta_{(V_kC)\rtimes(Z_{2,3,4}E),\psi_k\mu}.
\end{align*}
Conjugating the \rhs\ by a Weyl element of $G_n$, whose
action on $N_n$ is given by the action of
\begin{align*}
\mathrm{diag}(\left(\begin{array}{cc}&I_{2k+1}\\I_{n-2k-1}\end{array}\right),1,
\left(\begin{array}{cc}&I_{n-2k-1}\\I_{2k+1}\end{array}\right)),
\end{align*}
we obtain that $\theta_{(V_kC)\rtimes(Z_{2,3,4}E),\psi_k\mu}$ is a quotient of
\begin{align*}
(\theta_{U_1,\psi_1})_{U_2',\psi_2}.
\end{align*}
Here $\psi_1(u)=\psi(u_{1,2})$, $U_2'$ is a certain subgroup of $U_2$ (obtained from the conjugation of $C$) and $\psi_2(u)=\psi(u_{2,2n-1})$. Note that $\psi_1$ corresponds
to $\mu$ and the coordinate $z_3$ while $\psi_2$ corresponds to the
character $\psi_k$ of $C$ and the $(n-2k+1,n+2k)$-th coordinate of $c\in C$. The character $\psi_2$ is nontrivial on $U_2'$. Finally, by Proposition~4 of Bump, Friedberg and Ginzburg \cite{BFG2} (which is easily extended to $G_n$, given the analog of Theorem~\ref{theorem:main vanishing Jacquet of small rep} in \cite{me8}),
$\theta_{U_1,\psi_1}$ is a quotient of $\theta_{U_2}$ (this is valid for $n\geq 3$, here $0<k<n/2$ whence $n\geq3$). Hence, $U_2$ acts trivially on
$\theta_{U_1,\psi_1}$ and therefore $(\theta_{U_1,\psi_1})_{U_2',\psi_2}=0$ and \eqref{eq:lemma first vanishing k < n/2 action of Z} follows.
\end{proof}
\end{proof}
\section{Distinguished representations}\label{section:Distinguished representations}
Let $G$ be either $\GL_n$ or $G_n$. Let $\tau$ be an admissible representation of $G$ with a central character $\omega_{\tau}$.
Assume that $\theta$ and $\theta'$ are a pair of exceptional representations of $\widetilde{G}$.
We say that $\tau$ is $(\theta,\theta')$-distinguished if
\begin{align*}
\Hom_{G}(\theta\otimes\theta',\tau^{\vee})\ne0.
\end{align*}
The following result describes the upper hereditary property of a distinguished representation of $\GL_n$, when induced to a representation of
$G_n$. Following the notation of Section~\ref{subsection:The exceptional representations},
we denote the exceptional representation of $\widetilde{G}$ corresponding to $\chi$ and $\psi$ by
$\theta^G_{\chi,\psi}$.
For any representation $\sigma$ of $\GL_n$, $s\in\C$ and a character $\mu$ of $F^*$, one forms a representation $\sigma|\det{}|^s\otimes\mu$ of $M_n$. Put
\begin{align*}
\mathrm{I}(\sigma,s,\mu)=\Ind_{Q_n}^{G_n}(\delta_{Q_n}^{1/2}\sigma|\det{}|^s\otimes\mu).
\end{align*}
\begin{proposition}\label{proposition:upper heredity of dist 1}
Let $\tau$ be a $(\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})$-distinguished representation of $\GL_n$ and set
$\eta=(\chi\chi')^{-1}$. Then $\mathrm{I}(\tau,1/2,\eta)$ is a
$(\theta^{G_n}_{\chi,\psi},\theta^{G_n}_{\chi',\psi'})$-distinguished representation of $G_n$.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{proposition:upper heredity of dist 1}]
By definition the space
\begin{align*}
\Tri_{\GL_n}(\tau,\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})
\end{align*}
of $\GL_n$-equivariant trilinear forms on $\tau\times\theta^{\GL_n}_{\chi,\psi}\times\theta^{\GL_n}_{\chi',\psi'}$ is nonzero. Therefore
\begin{align}\label{space:heredity proof lower space}
\Tri_{M_n}(\tau\otimes\eta,\theta^{\GL_n}_{\chi,\psi}\otimes\theta^{G_0}_{\chi,1},\theta^{\GL_n}_{\chi',\psi'}\otimes\theta^{G_0}_{\chi',1})\ne0.
\end{align}
According to \eqref{eq:containment of Jacquet module along maximal unipotent 2},
\begin{align*}
(\theta_{\chi,\psi}^{G_n})_{U_n}=\theta^{\GL_n}_{|\cdot|^{(n-1)/4}\chi,\psi}\otimes\theta^{G_0}_{\chi,1}.
\end{align*}
Applying Frobenius reciprocity we see that $\theta_{\chi,\psi}^{G_n}$ is a subrepresentation of
\begin{align}\label{eq:tensor in heredity proof 0}
\Ind_{\widetilde{Q}_{n}}^{\widetilde{G}_n}(\delta_{Q_{n}}^{\frac{n-1}{4n}}\theta^{\GL_n}_{\chi,\psi}\otimes\theta^{G_0}_{\chi,1}).
\end{align}
A similar result holds for $\theta^{G_n}_{\chi',\psi'}$.
We define
\begin{align*}
T\in\Tri_{G_n}(\mathrm{I}(\tau,1/2,\eta),\theta^{G_n}_{\chi,\psi},\theta^{G_n}_{\chi',\psi'})
\end{align*}
and prove it is nonzero.
Let $\varphi$ belong to the space $\theta^{G_n}_{\chi,\psi}$, regarded as an element of \eqref{eq:tensor in heredity proof 0}, and similarly let $\varphi'$ belong to the space of $\theta^{G_n}_{\chi',\psi'}$. Also take $f$ in the space of $\mathrm{I}(\tau,1/2,\eta)$.
Now if $L\ne0$ belongs to \eqref{space:heredity proof lower space},
\begin{align*}
L(f(q),\varphi(q),\varphi'(q))=\delta_{Q_n}(q)L(f(1),\varphi(1),\varphi'(1)),\qquad q\in Q_n.
\end{align*}
Thus the following integral is (formally) well defined (see e.g. \cite{BZ1} 1.21),
\begin{align*}
T(f,\varphi,\varphi')=\int_{\lmodulo{Q_n}{G_n}}L(f(g),\varphi(g),\varphi'(g))\ dg.
\end{align*}
It is absolutely convergent according to the Iwasawa decomposition.
Since $T$ satisfies the necessary equivariance properties, it remains to show $T\ne0$.
Assume $L(x,y,y')\ne0$ for suitable data. Take $f$ supported on $Q_{n}\mathcal{N}$,
for a small compact open neighborhood $\mathcal{N}$ of the identity in $G_n$, and such that
\begin{align}
f((a,b)uv)=\delta_{Q_{n}}^{1/2}(a)|\det{a}|^{1/2}\eta(b)\tau(a)x,\qquad\forall (a,b)\in \GL_n\times G_0, u\in U_{n}, v\in\mathcal{N}.
\end{align}
We may assume $\varphi(1)=y$ (because $\theta^{\GL_n}_{\chi,\psi}\otimes\theta^{G_0}_{\chi,1}$ is irreducible) and $\varphi'(1)=y'$.
Using the Iwasawa decomposition and $Q_{n}\mathcal{N}\cap K=(Q_{n}\cap K)\mathcal{N}$ then yields
\begin{align*}
T(f,\varphi,\varphi')=
\int_{(Q_{n}\cap K)\mathcal{N}}L(f(k),\varphi(k),\varphi'(k))\ dk.
\end{align*}
Since $L$ is invariant with respect to $Q_{n}\cap K$, taking a sufficiently small $\mathcal{N}$
(with respect to $\varphi$ and $\varphi'$), the $dk$-integration reduces to a nonzero constant multiple of
$L(f(1),\varphi(1),\varphi'(1))$, which is nonzero.
We conclude that $\mathrm{I}(\tau,1/2,\eta)$ is $(\theta^{G_n}_{\chi,\psi},\theta^{G_n}_{\chi',\psi'})$-distinguished.
\end{proof}
Let $\tau$ be a representation of $G$ as above.
Write $\theta=\theta_{\chi,\psi}$ and $\theta'=\theta_{\chi',\psi'}$. Since $\theta_{\chi,\psi}=\chi\theta_{1,\psi}$,
we may assume $\chi=\chi'=1$, perhaps twisting $\tau$ by a character. For simplicity, we then say that $\tau$ is $(\psi,\psi')$-distinguished.
If $n$ is even, the characters $\psi$ and $\psi'$ can be ignored, because
$\theta_{1,\psi}$ does not depend on $\psi$. If $n$ is odd and $\tau$ is $(\psi_0,\psi_0')$-distinguished,
then for any $\psi$ there is $\psi'$ such that $\tau$ is $(\psi,\psi')$-distinguished. Indeed,
write $\psi(x)=\psi_0(\alpha x)$ for some $\alpha\in F^*$ and put $\psi'(x)=\psi_0'(\alpha x)$, then by
Claim~\ref{claim:taking characters out of exceptional} and its proof,
$\theta_{1,\psi_0}\otimes\theta_{1,\psi_0'}=\theta_{1,\psi}\otimes\theta_{1,\psi'}$.
In light of these observations, we say that
$\tau$ is distinguished if for any $\psi$ there is $\psi'$ such that $\tau$ is $(\psi,\psi')$-distinguished. Proposition~\ref{proposition:upper heredity of dist 1} implies,
\begin{corollary}\label{corollary:upper hered dist}
Let $\tau$ be a distinguished representation of $\GL_n$. Then $\mathrm{I}(\tau,1/2,1)$ is distinguished.
\end{corollary}
Now we prove Theorem~\ref{theorem:tensor of small is non generic}. Namely, for any pair $\theta$ and $\theta'$ of exceptional representations of $\widetilde{G}_n$,
and a generic character $\psi$ of $N_n$,
\begin{align*}
(\theta\otimes\theta')_{N_n,\psi}=0.
\end{align*}
We consider the filtrations of $\theta$ and $\theta'$ corresponding to the Jacquet functor along $C=C_{U_n}$.
The kernel of this functor is glued from representations induced from the Jacquet modules described in Section~\ref{section:Heisenberg jacquet modules}.
Taking the twisted Jacquet functor along $N_n$ truncates some of these quotients and, essentially, reduces the problem to a representation induced from
$(\theta^{G_{2k}})_{C_{U_{2k}},\psi_{k}}\otimes({\theta'}^{G_{2k}})_{C_{U_{2k}},\psi_{k}^{-1}}$.
Theorem~\ref{theorem:small rep is weakly minimal} then enables us to further reduce the problem, to the vanishing of
$\ind_{\Sp_k U_n}^{\GL_{2k}}(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{N_n,\psi}$, which essentially
follows from the results of Offen and Sayag on Klyachko models (\cite{OS}, see also \cite{Klyachko}).
\begin{proof}[Proof of Theorem~\ref{theorem:tensor of small is non generic}]
By an analog of the Geometric Lemma of Bernstein and Zelevinsky (\cite{BZ2} Theorem~5.2 and \cite{BZ1} 5.9-5.12), as a $\widetilde{Q}_n$-module, $\theta$ is glued from
\begin{align*}
\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k}), \qquad 0\leq k\leq \lfloor n/2 \rfloor.
\end{align*}
(See Section~\ref{section:Heisenberg jacquet modules} for the notation.)
Then
as a $Q_n$-module $\theta\otimes\theta'$ is glued from
\begin{align}\label{eq:filtration quotients}
\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k})\otimes
\ind_{\widetilde{\mathrm{St}}_{\psi_{k'}}U_n}^{\widetilde{Q}_n}(\theta'_{C,\psi_{k'}^{-1}}),\qquad 0\leq
k,k'\leq\lfloor n/2\rfloor.
\end{align}
We prove
that the Jacquet functor with respect to $N_n$ and $\psi$ vanishes on each of these representations.
Since functions in $\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k})$ are compactly supported modulo $\widetilde{\mathrm{St}}_{\psi_k}U_n$, and
$C$ is normal in $Q_n$, by Lemma~\ref{lemma:Jacquet kernel as integral},
\begin{align}\label{eq:123 k and k'}
(\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k})\otimes
\ind_{\widetilde{\mathrm{St}}_{\psi_{k'}}U_n}^{\widetilde{Q}_n}(\theta'_{C,\psi_{k'}^{-1}}))_{N_n,\psi}=\begin{dcases}
(\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k}\otimes\theta'_{C,\psi_{k}^{-1}}))_{N_n,\psi}&k=k',\\0&k\ne k'.
\end{dcases}
\end{align}
To see this consider $f$ in the space of $\ind_{\widetilde{\mathrm{St}}_{\psi_k}U_n}^{\widetilde{Q}_n}(\theta_{C,\psi_k})$ and $f'$ in the space of
$\ind_{\widetilde{\mathrm{St}}_{\psi_{k'}}U_n}^{\widetilde{Q}_n}(\theta'_{C,\psi_{k'}^{-1}})$, and look at the Jacquet-Langlands integral
\begin{align*}
&\int_{\mathcal{C}}c\cdot(f\otimes f')(g,g')\ dc=\int_{\mathcal{C}}f(gc)f'(g'c)\ dc=f(g)f'(g')\int_{\mathcal{C}}\psi_{k}(\rconj{g}c)\psi_{k'}^{-1}(\rconj{g'}c)\ dc,
\end{align*}
where $\mathcal{C}<C$ is a compact subgroup.
Since $\theta_{C,\psi_k}\otimes\theta'_{C,\psi_{k}^{-1}}$ is a non-genuine representation of $\widetilde{\mathrm{St}}_{\psi_k}$, we can replace the representation on the
\rhs\ of \eqref{eq:123 k and k'} with
\begin{align*}
(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}(\theta_{C,\psi_k}\otimes\theta'_{C,\psi_{k}^{-1}}))_{N_n,\psi}.
\end{align*}
Define $\vartheta$ with respect to $\theta^{\GL_{n-2k}}$ as in Proposition~\ref{proposition:Jacquet module C and k} and similarly, define $\vartheta'$ with respect to ${\theta'}^{\GL_{n-2k}}$.
According to the proposition
, $\theta_{C,\psi_k}$ is embedded in a finite direct sum of representations
$\vartheta\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$, which are trivial on $U_{n-2k}$. Put
\begin{align*}
\Pi_k=\vartheta\otimes\vartheta'\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}\otimes({\theta'}^{G_{2k}})_{C_{U_{2k}},\psi_k^{-1}}.
\end{align*}
It is enough to prove that for all $0\leq k\leq \lfloor n/2\rfloor$,
\begin{align}\label{eq:vanishing result to prove}
(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}\Pi_k)_{N_n,\psi}=0.
\end{align}
This holds for $k=0$, simply because $U_n$ is normal in $Q_n$, $\Pi_0$ is trivial on $U_n$ while $\psi$ is not.
The case of $k=n/2$ is handled by the following claim, whose proof is deferred to below.
\begin{claim}\label{claim:induced from double Weil claim 0}
Equality~\eqref{eq:vanishing result to prove} holds for $k=n/2$.
\end{claim}
Lastly, assume $0<k<n/2$. Set $Q=Q_{n-2k}\cap Q_n$. By transitivity of induction
\begin{align*}
(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}\Pi_k)_{N_n,\psi}=(\ind_{Q}^{Q_n}(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q}\Pi_k))_{N_n,\psi}.
\end{align*}
The representation $\ind_{\mathrm{St}_{\psi_k}U_n}^{Q}\Pi_k$ is trivial on $U_{n-2k}$.
By virtue of the Geometric Lemma of Bernstein and Zelevinsky (\cite{BZ2} Theorem~5.2), the representation on the \rhs\ is glued from
Jacquet modules of $\ind_{\mathrm{St}_{\psi_k}U_n}^{Q}\Pi_k$. Note that in general, the quotients are representations induced from Jacquet modules, here the induction is trivial because the stabilizer of the character $\psi$ of $N_n$ is $N_n\times G_0$.
Let $\mathcal{W}$ be a set of representatives to the double cosets $\rmodulo{\lmodulo{Q}{Q_n}}{(N_nG_0)}$. That is, $Q_n=\coprod_{w\in\mathcal{W}}Qw^{-1}N_nG_0$. We can take the elements $w$ to be Weyl elements of $\GL_n$. When
$\psi|_{\rconj{w}U_{n-2k}\cap N_n}\ne1$, the quotient corresponding to $w$ vanishes. This implies there is only one quotient, corresponding to $w_0=\left(\begin{smallmatrix}&I_{2k}\\I_{n-2k}\end{smallmatrix}\right)$, namely
\begin{align*}
\delta\cdot(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q}\Pi_k)_{N_{\GL_{n-2k}}\times N_{2k},\psi}.
\end{align*}
Here $\delta$ is some modulus character, hereby ignored, and $N_{\GL_{n-2k}}\times N_{2k}<M_{n-2k}$. As a $G_0$-module, this representation is isomorphic to
\begin{align*}
(\vartheta\otimes\vartheta')_{N_{\GL_{n-2k}},\psi}\otimes\ind_{\mathrm{St}_{\psi_k}U_{2k}}^{Q_{2k}}((\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}\otimes({\theta'}^{G_{2k}})_{C_{U_{2k}},\psi_k^{-1}})_{N_{2k},\psi}.
\end{align*}
Here $\psi$ is regarded as a generic character of $N_{\GL_{n-2k}}$ and $N_{2k}$. Since the case $k=n/2$ implies
\begin{align*}
\ind_{\mathrm{St}_{\psi_k}U_{2k}}^{Q_{2k}}((\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}\otimes({\theta'}^{G_{2k}})_{C_{U_{2k}},\psi_k^{-1}})_{N_{2k},\psi}=0,
\end{align*}
Equality~\eqref{eq:vanishing result to prove} follows.
\begin{proof}[Proof of Claim~\ref{claim:induced from double Weil claim 0}]
For $k=n/2$, $\Pi_k=\theta_{C,\psi_k}\otimes\theta'_{C,\psi_k^{-1}}$.
Now we apply Theorem~\ref{theorem:small rep is weakly minimal}. For simplicity of computations, we can replace
$\psi_k$ with the character defined in the proof of the theorem, then use the epimorphism
$\ell:\widetilde{\mathrm{St}}_{\psi_k}\ltimes U_n\rightarrow \widetilde{\Sp}_k\ltimes H_n$ given there. By
Theorem~\ref{theorem:small rep is weakly minimal}, $\theta_{C,\psi_k}$ is isomorphic to the direct sum of copies
of $\omega_{\psi}$. Pull $\omega_{\psi}$ back to a representation of $\widetilde{\mathrm{St}}_{\psi_k}\ltimes U_n$. Note that
the $G_0$ part of $\mathrm{St}_{\psi_k}$ was ignored in the proof of Theorem~\ref{theorem:small rep is weakly minimal}, since it acts
by a character, so we can ignore this here as well. Equality~\eqref{eq:vanishing result to prove} will follow from
\begin{align}\label{claim:induced from double Weil claim 0 main equality}
(\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\omega_{\psi}\otimes\omega_{\psi^{-1}}))_{N_{n},\psi}=0.
\end{align}
We need some notation. Put
\begin{align*}
\pi_0=\omega_{\psi}\otimes\omega_{\psi^{-1}},\qquad G=P_{n,1}^{\circ},\qquad V=Z_{n,1},\qquad P=\Sp_k\ltimes V,\qquad X=\lmodulo{P}{G}.
\end{align*}
Note that $\lmodulo{C}{Q_n}\isomorphic G=\GL_n\ltimes V$ (in fact, $\lmodulo{C}{Q_n}\isomorphic G\times G_0$ but $G_0$ was ignored), this isomorphism restricts to an isomorphism $\lmodulo{C}{U_n}\isomorphic V$. In this manner $\psi$ is also a character of $V$,
$\psi(v)=\psi(v_{n,n+1})$.
Since $\pi_0$ is trivial on $C$, we can regard it as a representation of $P$ and if
$\pi=\ind_{P}^{G}(\pi_0)$, $\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\pi_0)\isomorphic\pi$
as $G$-modules.
We apply the theory of $l$-sheafs of Bernstein and Zelevinsky (\cite{BZ1} 1.13 and Section~6). In the following, we freely use their
terminology and definitions. Let $(X,\mathcal{F})$ be the $l$-sheaf
corresponding to $\pi$ (\cite{BZ1} 2.23). The group $N_n$ acts on $X$ by $u\cdot x=xu^{-1}$ and on $\mathcal{F}$ by
$u\cdot\varphi(x)=\psi^{-1}(u)\varphi(u^{-1}\cdot x)$. An $N_n$-invariant $\mathcal{F}$-distribution on $X$ is an element of
$\Hom_{N_n}(\pi,\psi)$. Since $\Hom_{N_n}(\pi,\psi)$ is the algebraic dual of $\pi_{N,\psi}$, to prove
\eqref{claim:induced from double Weil claim 0 main equality} we will show that there are no
nonzero $N_n$-invariant $\mathcal{F}$-distributions on $X$.
The action of $N_n$ on $X$ is constructive (\cite{BZ1} Theorem~A). If $x\in X$, let $P^x=\rconj{x^{-1}}P\cap N_n$ be the stabilizer of $x$ in $N_n$. The orbit
of $x$ is $N_n\cdot x$.
The mapping $u\cdot x\mapsto (P^x)u^{-1}$ induces a homeomorphism $N_n\cdot x\isomorphic\lmodulo{P^x}{N_n}$ (\cite{BZ1} 1.6).
The restriction of $\mathcal{F}$ to the orbit of $x$ (this restriction is an $l$-sheaf, because the action is constructive) is isomorphic to
$\ind_{P^x}^{N_n}(\rconj{x^{-1}}\pi_0)$, where $\rconj{x^{-1}}\pi_0$ is the representation
of $P^x$ acting in the space of $\pi_0$ by $\rconj{x^{-1}}\pi_0(z)=\pi_0(\rconj{x}z)$.
By virtue of Theorem~6.9 of \cite{BZ1}, to show there are no nonzero $N_n$-invariant $\mathcal{F}$-distributions on $X$, it is enough to
prove that for each representative $x$,
\begin{align*}
\Hom_{N_n}(\ind_{P^x}^{N_n}(\rconj{x^{-1}}\pi_0),\psi)=\Hom_{P^x}(\rconj{x^{-1}}\pi_0,\psi)=0.
\end{align*}
We can take representatives $x\in P_{n-1,1}^{\circ}<\GL_n$. Then $P^x=\Sp_k^x\ltimes V$, where $\Sp_k^x=\rconj{x^{-1}}\Sp_k\cap N_{\GL_n}$.
In addition, because $P_{n-1,1}^{\circ}$ fixes $\psi|_{V}$, $(\rconj{x^{-1}}\pi_0)_{V,\psi}=\rconj{x^{-1}}({(\pi_0)}_{V,\psi})$. Hence
\begin{align*}
\Hom_{P^x}(\rconj{x^{-1}}\pi_0,\psi)=\Hom_{\Sp_k^x}(\rconj{x^{-1}}({(\pi_0)}_{V,\psi}),\psi).
\end{align*}
Note that $(\pi_0)_{V,\psi}=(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n,\mu}$ for the nontrivial $\mu$ given in
Claim~\ref{claim:Jacquet modules along Hn of double Weil reps}. According to that claim $(\pi_0)_{V,\psi}$ is the trivial one-dimensional
representation of $\ell^{-1}(\Sp_k\cap P_{n-1,1}^{\circ})$. Since
$\rconj{x}N_{\GL_n}<P_{n-1,1}^{\circ}$ for any $x\in P_{n-1,1}^{\circ}$,
$\rconj{x^{-1}}({(\pi_0)}_{V,\psi})$ is trivial on $\Sp_k^x$ (the epimorphism $\ell$ is easily seen to be harmless here).
But Offen and Sayag (\cite{OS} Proposition~2,
we use $\mathcal{H}^{r,r'}$ with $r=0$ and $r'=n$, in their notation) proved that
$\psi|_{\Sp_k^x}\ne1$ for any $x\in \GL_n$. This implies $\Hom_{\Sp_k^x}(\rconj{x^{-1}}({(\pi_0)}_{V,\psi}),\psi)=0$, as required.
\end{proof}
\end{proof}
\begin{corollary}\label{corollary:upper hered dist LQ}
Let $\tau$ be an irreducible unitary supercuspidal $(\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})$-distinguished representation of $\GL_n$.
Then the Langlands quotient of $\mathrm{I}(\tau,1/2,(\chi\chi')^{-1})$ is
$(\theta^{G_n}_{\chi,\psi},\theta^{G_n}_{\chi',\psi'})$-distinguished. In particular, if $\tau$ is
distinguished, so is the Langlands quotient of $\mathrm{I}(\tau,1/2,1)$.
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{corollary:upper hered dist LQ}]
According to Proposition~\ref{proposition:upper heredity of dist 1},
\begin{align*}
\Hom_{G_n}(\theta^{G_n}_{\chi,\psi}\otimes\theta^{G_n}_{\chi',\psi'},\mathrm{I}(\tau,1/2,(\chi\chi')^{-1})^{\vee})\ne0.
\end{align*}
Theorem~\ref{theorem:tensor of small is non generic} implies $\mathrm{I}(\tau,1/2,(\chi\chi')^{-1})$ is reducible (because $\tau$ is generic),
then by Casselman and Shahidi \cite{CSh} (Theorem~1), the Langlands quotient $\mathrm{LQ}(\mathrm{I}(\tau,1/2,(\chi\chi')^{-1}))$ is non-generic, and
the unique irreducible subspace of $\mathrm{I}(\tau,1/2,(\chi\chi')^{-1})$ is generic. Also note that the length of $\mathrm{I}(\tau,1/2,(\chi\chi')^{-1})$
is $2$ (because it is reducible and $\tau$ is supercuspidal, see \cite{BZ2}~2.8). Now the result follows from
Theorem~\ref{theorem:tensor of small is non generic} and the left exactness of the $\Hom$ functor.
\end{proof}
As a corollary, we can now prove Theorem~\ref{theorem:supercuspidal dist GLn and pole}. Namely,
for an irreducible unitary supercuspidal $\tau$, being distinguished is equivalent to the occurrence of a pole at $s=0$ of
$L(s,\tau,\mathrm{Sym}^2)$.
\begin{proof}[Proof of Theorem~\ref{theorem:supercuspidal dist GLn and pole}]
If $\tau$ is distinguished, as in the proof of Corollary~\ref{corollary:upper hered dist LQ} we see that $\mathrm{I}(\tau,1/2,1)$ is reducible,
then according to Casselman and Shahidi \cite{CSh} (Proposition~5.3, their Conjecture 1.1 was proved for $G_n$ in \cite{Asg}) $L(s,\tau,\mathrm{Sym}^2)$ has a pole at $s=0$.
In the other direction, assume $L(s,\tau,\mathrm{Sym}^2)$ has a pole at $s=0$. As an application of the descent method of Ginzburg, Rallis and Soudry
(see e.g., \cite{GRS2,GRS5,GRS7,GRS3,GRS8,JSd1,JSd2,Soudry5,Soudry4,RGS}), one can globalize $\tau$ to a cuspidal automorphic representation
$\pi$ of $\GL_n(\Adele)$, such that $L^S(s,\pi,\mathrm{Sym}^2)$ has a pole at $s=1$ (see the appendix of \cite{PR}).
Therefore the nonvanishing of \eqref{int:BG period} implies $\tau$ is distinguished (\cite{BG} Theorem~7.6).
\end{proof}
\begin{corollary}
If $\tau$ is an irreducible unitary supercuspidal distinguished representation, it must be self-dual.
\end{corollary}
\begin{remark}\label{remark:applicability to twisted}
The analogous result for an irreducible supercuspidal $(\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})$-distinguished $\tau$ should
also hold. One needs to verify the applicability of the globalization argument (see \cite{HS}).
\end{remark}
Given exceptional representations $\theta$ and $\theta'$ of $\widetilde{G}$ (as in the beginning of this section), we can consider the space of $\theta\otimes\theta'$ as a model for representations of $G$.
We refer to the dimension of $\Hom_G(\theta\otimes\theta',\tau^{\vee})$ as the multiplicity of $\tau$. The next proposition relates the multiplicities
of $\tau$ and $\mathrm{I}(\tau,1/2,\eta)$.
In the case of $\GL_n$, Kable \cite{Kable} conjectured that the multiplicity of an irreducible representation is at most one. He proved this for $n\leq 3$, and for arbitrary $n$
under a certain homogeneity condition (\cite{Kable} Corollary~6.1).
It is reasonable to believe multiplicity one also holds in the context of $G_n$ (see
\cite{LM6} Remark~4.2).
\begin{proposition}\label{proposition:one-dim}
Let $\tau$ be an irreducible tempered representation of $\GL_n$, put $\eta=(\chi\chi')^{-1}$ and assume $|\eta|=1$. Then
\begin{align}\label{eq:proposition one-dim statement}
\Dim\ \Hom_{G_n}(\theta^{G_n}_{\chi,\psi}\otimes\theta^{G_n}_{\chi',\psi'},\mathrm{I}(\tau,1/2,\eta)^{\vee})=\Dim\
\Hom_{\GL_n}(\theta^{\GL_n}_{\chi,\psi}\otimes\theta^{\GL_n}_{\chi',\psi'},\tau^{\vee}).
\end{align}
In particular, the representation $\tau$ is $(\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})$-distinguished if and only if
$\mathrm{I}(\tau,1/2,\eta)$ is $(\theta^{G_n}_{\chi,\psi},\theta^{G_n}_{\chi',\psi'})$-distinguished.
\end{proposition}
\begin{remark}\label{remark:uniqueness current state}
If $\tau$ is $(\theta^{\GL_n}_{\chi,\psi},\theta^{\GL_n}_{\chi',\psi'})$-distinguished,
in particular $\omega_{\tau}(z^2\cdot I_n)=\eta(z^{2n})$ for all $z\in F^*$ ($\omega_{\tau}$ - the central character of $\tau$). It follows that $|\eta|=1$, because a tempered representation is
unitary. In this case by
\eqref{eq:proposition one-dim statement}
$\tau$ enjoys multiplicity one if and only if $\mathrm{I}(\tau,1/2,\eta)$ does.
\end{remark}
\begin{proof}[Proof of Proposition~\ref{proposition:one-dim}]
Set $\theta_0=\theta^{\GL_n}_{\chi,\psi}$, $\theta=\theta^{G_n}_{\chi,\psi}$ and similarly $\theta_0'$ and $\theta'$ (with $\chi'$ and $\psi'$).
For brevity, put $\mathrm{d}=|\det{}|$. Applying Frobenius reciprocity,
\begin{align*}
\Hom_{G_n}(\theta\otimes\theta',\mathrm{I}(\tau,1/2,\eta)^{\vee})=
\Hom_{\GL_n}((\theta\otimes\theta')_{U_n}|_{\GL_n},\mathrm{d}^{(n-1)/2}\tau^{\vee}).
\end{align*}
Using the notation of Section~\ref{section:Heisenberg jacquet modules}, the representation $(\theta\otimes\theta')_{U_n}$ is filtered by representations
\begin{align*}
\mathbb{W}_k=(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}(\theta_{C,\psi_k}\otimes\theta'_{C,\psi_{k}^{-1}}))_{U_n},\qquad 0\leq k\leq \lfloor n/2 \rfloor.
\end{align*}
The representation
$\mathbb{W}_0$ is a quotient of $(\theta\otimes\theta')_{U_n}$ and the kernel of the mapping
$(\theta\otimes\theta')_{U_n}\rightarrow\mathbb{W}_0$ is filtered by $\mathbb{W}_k$ with $1\leq k\leq \lfloor n/2 \rfloor$. Regard $\mathbb{W}_k$ as a representation of $\GL_n$ (by restriction from $M_n$).
According to
Proposition~\ref{proposition:Jacquet module C and k},
$\mathbb{W}_0=\mathrm{d}^{(n-1)/2}\theta_0\otimes\theta_0'$. Then
\begin{align*}
\Hom_{\GL_n}(\mathbb{W}_0
,\mathrm{d}^{(n-1)/2}\tau^{\vee})=
\Hom_{\GL_n}(\theta_0\otimes\theta_0',\tau^{\vee}).
\end{align*}
Thus \eqref{eq:proposition one-dim statement}
will follow, once we prove that for $k>0$,
\begin{align}\label{rep:homspace quotient in filtration one-dim}
\Hom_{\GL_n}(\mathbb{W}_k,
\mathrm{d}^{(n-1)/2}\tau^{\vee})=0.
\end{align}
If $n$ is even, we claim the following.
\begin{claim}\label{claim:induced from double Weil claim 1}
For any irreducible generic representation $\sigma$ of $\GL_n$, $\Hom_{\GL_n}(\mathbb{W}_{n/2},\sigma)=0$. In particular
Equality~\eqref{rep:homspace quotient in filtration one-dim} holds for $k=n/2$.
\end{claim}
The proof will be given below.
Assume $0<k<n/2$. The representation $\theta_{C,\psi_k}$ is embedded in a finite direct sum of representations
$\vartheta\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$ (Proposition~\ref{proposition:Jacquet module C and k}). The representation $\vartheta$ is semisimple, and
by Theorem~\ref{theorem:small rep is weakly minimal} the representation $(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}$ is also semisimple.
Hence $\theta_{C,\psi_k}\otimes\theta'_{C,\psi_{k}^{-1}}$ is embedded in a semisimple representation, which is the finite direct sum of
$\Pi_k$ ($\Pi_k=\vartheta\otimes\vartheta'\otimes(\theta^{G_{2k}})_{C_{U_{2k}},\psi_k}\otimes({\theta'}^{G_{2k}})_{C_{U_{2k}},\psi_k^{-1}}$).
Therefore, using the exactness of induction and Jacquet functors, $\mathbb{W}_k$ is a quotient of
$(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}(\bigoplus\Pi_k))_{U_n}$. We conclude that we may replace $\mathbb{W}_k$ in \eqref{rep:homspace quotient in filtration one-dim} with $(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}\Pi_k)_{U_n}$.
An application of \cite{BZ2} (Theorem~5.2) yields
\begin{align*}
(\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}\Pi_k)_{U_n}=
\ind_{P_{n-2k,2k}}^{\GL_n}(\vartheta\otimes\vartheta'\otimes\mathbb{W}'),
\end{align*}
where
\begin{align*}
\mathbb{W}'=
(\ind_{\mathrm{St}_{\psi_k}U_{2k}}^{Q_{2k}}(\theta^{G_{2k}}_{C_{U_{2k}},\psi_k}\otimes{\theta'}^{G_{2k}}_{C_{U_{2k}},\psi_{k}^{-1}}))_{U_{2k}}|_{\GL_{2k}}.
\end{align*}
Note that the relevant double coset space (for \cite{BZ2} Theorem~5.2) is $\rmodulo{\lmodulo{(Q_{n-2k}\cap Q_n)}{Q_n}}{Q_n}$, there is only one representative to consider.
Hence to deduce \eqref{rep:homspace quotient in filtration one-dim}, it is enough to prove
\begin{align*}
\Hom_{\GL_n}(\ind_{P_{n-2k,2k}}^{\GL_n}(\vartheta\otimes\vartheta'\otimes
\mathbb{W}'),\mathrm{d}^{(n-1)/2}\tau^{\vee})=0.
\end{align*}
The \lhs\ equals
\begin{align}\nonumber
&\Hom_{\GL_n}(\mathrm{d}^{(1-n)/2}\tau,\ind_{P_{n-2k,2k}}^{\GL_n}(\delta_{P_{n-2k,2k}}(\vartheta\otimes\vartheta'\otimes\mathbb{W}')^{\vee}))\nonumber
\\&=\Hom_{P_{n-2k,2k}}(\mathrm{d}^{(1-n)/2}\tau_{Z_{n-2k,2k}},\delta_{P_{n-2k,2k}}(\vartheta\otimes\vartheta'\otimes\mathbb{W}')^{\vee})\nonumber
\\\label{eq:last space}&=\Hom_{P_{n-2k,2k}}(\mathrm{d}^{(1-n)/2}\delta_{P_{n-2k,2k}}^{-1/2}\vartheta\otimes\vartheta'\otimes\mathbb{W}',j(\tau)^{\vee}).
\end{align}
Here $j(\tau)=\delta_{P_{n-2k,2k}}^{-1/2}\tau_{Z_{n-2k,2k}}$ (the normalized Jacquet functor). It suffices to show that
the last space \eqref{eq:last space} vanishes, when $j(\tau)$ is replaced by any of its irreducible subquotients.
Let $\tau_1\otimes\tau_2$ be one such subquotient. Then \eqref{eq:last space} vanishes if either
\begin{align}
\label{eq:last tensor 1}&\Hom_{\GL_{2k}}(\mathbb{W}',\mathrm{d}^{(2k-1)/2}\tau_2^{\vee})\text{ or}\\
\label{eq:last tensor 2}&\Hom_{\GL_{n-2k}}(\mathrm{d}^{(1-n-2k)/2}\vartheta\otimes\vartheta',\tau_1^{\vee})
\end{align}
are zero. As explained in the proof of Proposition~4.1 of Lapid and Mao \cite{LM6}, since $\tau$ is tempered, either $\tau_2$ is generic,
or the central character $\omega_{\tau_1}$ of $\tau_1$ satisfies $|\omega_{\tau_1}|=|\det{}|^{\alpha}$ for some $\alpha>0$. In the former case
\eqref{eq:last tensor 1} vanishes by Claim~\ref{claim:induced from double Weil claim 1}. In the latter case \eqref{eq:last tensor 2} vanishes. Indeed,
$z^2\cdot I_{n-2k}$ acts on $\vartheta\otimes\vartheta'$ by $\mathrm{d}^{(-1+n+2k)/2}\eta^{-1}(z^{2(n-2k)})$ and since
$|\eta|=1$, this action is unitary on
$\mathrm{d}^{(1-n-2k)/2}\vartheta\otimes\vartheta'$, but $\omega_{\tau_1}$ is not unitary.
\begin{proof}[Proof of Claim~\ref{claim:induced from double Weil claim 1}]
Put $k=n/2$ and $\pi=\ind_{\mathrm{St}_{\psi_k}U_n}^{Q_n}(\theta_{C,\psi_k}\otimes\theta'_{C,\psi_k^{-1}})$. We need to prove
\begin{align*}
\Hom_{\GL_n}(\pi_{U_n},
\sigma)=0.
\end{align*}
We will show that $\pi_{U_n}$ has a filtration,
whose quotients are all isomorphic to $\ind_{\Sp_{n/2}}^{\GL_n}1$.
Then since $\sigma$ is irreducible and generic, $\Hom_{\GL_n}(\ind_{\Sp_{n/2}}^{\GL_n}1,\sigma)=0$
(\cite{OS} Proposition~1, take $\mathcal{H}^{0,n}$) and the result follows.
We turn to prove the filtration of $\pi_{U_n}$. As in the proof of Claim~\ref{claim:induced from double Weil claim 0},
Theorem~\ref{theorem:small rep is weakly minimal} implies that $\pi_{U_n}$ is filtered by copies of the representation
$(\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\omega_{\psi}\otimes\omega_{\psi^{-1}}))_{U_n}$. We prove
\begin{align}\label{eq:claim desp 1}
(\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\omega_{\psi}\otimes\omega_{\psi^{-1}}))_{U_n}=\ind_{\Sp_{n/2}}^{\GL_n}1.
\end{align}
(Recall that the \lhs\ is regarded as a representation of $\GL_n$.)
Since $U_n$ is normal in $Q_n$, Lemma~\ref{lemma:Jacquet kernel as integral} implies
\begin{align*}
(\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\omega_{\psi}\otimes\omega_{\psi^{-1}}))_{U_n}=
\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}((\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}).
\end{align*}
In more detail, if $f$ belongs to the space of $\ind_{\mathrm{St}_{\psi_k}U_{n}}^{Q_{n}}(\omega_{\psi}\otimes\omega_{\psi^{-1}})$,
the Jacquet-Langlands integral takes the form
\begin{align*}
\int_{\mathcal{U}}f(gu)\ du=\int_{\mathcal{U}}
(\omega_{\psi}\otimes\omega_{\psi^{-1}})(\rconj{g}u)f(g)\ du,
\end{align*}
for a compact subgroup $\mathcal{U}<U_n$. Because $f$ is compactly supported modulo $\mathrm{St}_{\psi_k}U_{n}$, this integral vanishes for all $g\in Q_n$ if and only if
$f(g)$ belongs to the space of $(\omega_{\psi}\otimes\omega_{\psi^{-1}})({H_n})$ for all $g$. It remains to use the exactness of induction.
According to Claim~\ref{claim:Jacquet modules along Hn of double Weil reps}, $(\omega_{\psi}\otimes\omega_{\psi^{-1}})_{H_n}$ is the trivial one-dimensional
representation of $\Sp_k$ and
\eqref{eq:claim desp 1} holds.
\end{proof}
\end{proof}
We can now improve Corollary~\ref{corollary:upper hered dist} for tempered representations:
\begin{corollary}\label{corollary:upper hered dist for tempered}
Let $\tau$ be an irreducible tempered representation of $\GL_n$. Then
$\tau$ is distinguished if and only if $\mathrm{I}(\tau,1/2,1)$ is distinguished.
\end{corollary}
\section{The small representation of $\SO_{2n+1}$}\label{section:relevance to SO(2n+1)}
Bump, Friedberg and Ginzburg \cite{BFG} constructed and studied the
small representations for $\SO_{2n+1}$. In this section we briefly recall their results and formulate our results for $\SO_{2n+1}$.
The cover $\widetilde{SO}_{2n+1}$ was obtained by restricting the $4$-fold cover of
$\SL_{2n+1}$ of Matsumoto \cite{Mats}. It is a ``double cover"
in the sense that the square of the cocycle is trivial on the kernel of the spinor norm.
We use the same notation of $G_n$, e.g., $B_n=B_{\SO_{2n+1}}$ (see Section~\ref{subsection:GSpin}), $T_n$ is the diagonal torus and $Q_k$ is a standard maximal parabolic subgroup.
If $(a,g)\in\GL_k\times SO_{2(n-k)+1}$, $(a,g)$ is embedded in $M_k$ as $\mathrm{diag}(a,g,J_k\transpose{a}^{-1}J_k)$.
The block compatibility formula now reads (see \cite{BFG} (2.20))
\begin{align*}
&\sigma_{\SL_{2n+1}}((a,g),(a',g'))\\&=\sigma_{\SL_{k+1}}(\mathrm{diag}(a,\det{a}^{-1}),\mathrm{diag}(a',\det{a'}^{-1}))^2(\det{a},\det{a'})_4\sigma_{\SL_{2(n-k)+1}}(g,g').
\end{align*}
The benefit of this cover is that the preimages
of $\GL_k$ and $SO_{2(n-k)+1}$ commute, thus
the tensor can be used to describe representations of Levi subgroups. Restriction of the cover to $\GL_k$ is a double cover.
The small representation $\theta=\theta^{\SO_{2n+1}}$ is unique; it is the representation of $\widetilde{SO}_{2n+1}$ corresponding to the
exceptional character
\begin{align*}
\xi(\mathfrak{s}(\mathrm{diag}(t_1^2,\ldots,t_n^2,1,t_n^{-2},\ldots,t_1^{-2})))=\prod_{i=1}^n|t_i|^{n-i+1}.
\end{align*}
According to \cite{BFG} (Theorem~2.3), $\theta_{U_k}=\theta^{\GL_k}\otimes\theta^{SO_{2(n-k)+1}}$, where $\theta^{\GL_k}$ was explicitly
given, and by \cite{BFG} (Theorem~2.6) and \cite{BFG2} (Proposition~3), $\theta_{U_1,\psi^{(1)}}=0$ if the length of $\psi^{(1)}$ is nonzero
(using the notation of Section~\ref{subsection:The exceptional representations}).
Theorems~\ref{theorem:tensor of small is non generic} and \ref{theorem:small rep is weakly minimal} remain valid as stated. Proposition~\ref{proposition:Jacquet module C and k} now takes the form
\begin{align*}
\theta_{C,\psi_k}=\theta^{\GL_{n-2k}}\otimes(\theta^{SO_{4k+1}})_{C_{U_{2k}},\psi_k}.
\end{align*}
Here $\theta^{\GL_{n-2k}}$ is uniquely determined.
Indeed, this equality replaces \eqref{eq:equality to correct if k = 0} because $\theta_{U_{n-2k}}$ is a tensor of representations.
Regarding Proposition~\ref{proposition:upper heredity of dist 1}, assume $\tau$ is $(\psi,\psi')$-distinguished. Twisting $\tau$ by some square trivial character,
we obtain a $(\psi,\psi)$-distinguished representation. Then, perhaps after using another twist of $\tau$, it becomes $(\theta^{\GL_n},\theta^{\GL_n})$-distinguished
for the exceptional representation $\theta^{\GL_n}$ such that $\theta_{U_n}=\theta^{\GL_n}$. Now the proof of the proposition proceeds as in the case
of $G_n$. So, in this case one must start with a $(\theta^{\GL_n},\theta^{\GL_n})$-distinguished representation of $\GL_n$, in order to obtain
a distinguished representation $\mathrm{I}(\tau,1/2,1)$ of $\SO_{2n+1}$. Corollary~\ref{corollary:upper hered dist LQ} is applicable
for $\tau$ such that Proposition~\ref{proposition:upper heredity of dist 1} is valid. Proposition~\ref{proposition:one-dim}
remains valid.
| 49,188
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\begin{document}
\maketitle
\begin{abstract} We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.\end{abstract}
\section{Introduction}
Considering the importance of heat kernels in Riemannian geometry, it appears very natural to investigate the links between geometry and asymptotics of Brownian paths in a Lorentzian setting.
Extending Dudley's seminal work \cite{dudley1}, J. Franchi and Y. Le Jan constructed in \cite{flj}, on the unitary tangent bundle $T^1 \mathcal M$ of an arbitrary Lorentz manifold $\mathcal M$, a diffusion process which is Lorentz-covariant. This process, that we will simply call \textit{relativistic diffusion} in the sequel, is the Lorentzian
analogue of the classical Brownian motion on a Riemannian manifold. It can be seen either as a random perturbation of the timelike geodesic flow on the unitary tangent bundle, or as a stochastic development of Dudley's diffusion in a fixed tangent space.
\par
\medskip
In Minkowski space-time, and more generally in Lorentz manifolds of constant curvature, the long-time asymptotics of the relativistic diffusion is well understood, see \cite{dudley1,dudley3, flj,ismael,camille}. But, as in the Riemannian case, there is no hope to fully determinate the asymptotic behavior of the relativistic diffusion on an arbitrary Lorentzian manifold: it could depend heavily on the base space, see e.g. \cite{marcanton} and its references in the ``simple'' case of Cartan--Hadamard manifolds.
Recently in \cite{angstannIHP}, we studied in details the long-time asymptotic behavior of the relativistic diffusion in the case when the underlying space-time belong to a large class of curved Lorentz manifolds: Robertson-Walker space-times, or RW space-times for short, whose definition is recalled in Sect. \ref{sec.RW} below. We show in particular that the relativistic diffusion's paths converge almost surely to random points of a natural geometric compactification of the base manifold $\mathcal M$, namely its causal boundary $\partial \mathcal M_c^+$ introduced in the reference \cite{geroch}.
\newpage
\begin{theo}[Theorem 3.1 in \cite{angstannIHP}]
Let $\mathcal M$ be a RW space-time, $(\xi_0, \dot{\xi}_0) \in T^1 \mathcal M$ and let $(\xi_s, \dot{\xi}_s)_{0 \leq s \leq \tau}$ be the relativistic diffusion starting from $(\xi_0, \dot{\xi_0})$. Then, almost surely as $s$ goes to the explosion time $\tau$ of the diffusion, the first projection $\xi_s$ converges to a random point $\xi_{\infty}$ of the causal boundary $\partial \mathcal M_c^+$ of $\mathcal M$.
\end{theo}
The purpose of this paper is to push the analysis further by showing that, in the case of a RW space-time with exponential growth, the Poisson boundary of the diffusion is precisely generated by the single random variable $\xi_{\infty}$ of the causal boundary $\partial \mathcal M_c^+$, which in that case can be identified with a spacelike copy of the Euclidean space $\mathbb R^3$ (see \cite{angstannIHP} and Theorem 4.3 of \cite{flores}). Namely, we have:
\begin{theo}[Theorems \ref{the.asymp} and \ref{the.poisson} below]
Let $\mathcal M:=(0, +\infty) \times_{\alpha} \mathbb R^3$ be a RW space-time where $\alpha$ has exponential growth. Let $(\xi_s, \dot{\xi}_s)_{s \geq 0}=(t_s,x_s, \dot{t}_s, \dot{x}_s)_{s \geq 0}$ be the relativistic diffusion starting from $(\xi_0, \dot{\xi}_0) \in T^1 \mathcal M$. Then, almost surely as $s$ goes to infinity, the process $x_s$ converges to a random point $x_{\infty}$ in $\mathbb R^3$, and the invariant sigma field of the whole diffusion $(\xi_s, \dot{\xi}_s)_{s \geq 0}$ coincides almost surely with $\sigma(x_{\infty})$.
\end{theo}
The above result is the first computation of the Poisson boundary of the relativistic diffusion in the case of a Lorentz manifold with non-constant curvature. It can be seen as a complementary result of those of \cite{ismael, camille} in the flat cases. The difficulty here lies in the following facts: classical coupling techniques are hardly implemented in hypoelliptic settings, classical Lie group methods or explicit conditionning (Doob transform) do not apply in the presence of curvature. Our approach is purely probabilistic, it is first based on the existence of natural subdiffusions due to symmetries of the base manifold, namely the processes
$(t_s,\dot{t}_s)_{s \geq 0}$ and $(t_s, \dot{t}_s, \dot{x}_s/|\dot{x}_s|)_{s \geq 0}$ are subdiffusions of the whole process. Using successive shift-couplings, we then show that the Poisson boundaries of these two subdiffusions are trivial. Finally, we conclude via an abstract conditionning argument, allowing us to show that
the invariant sigma field of the whole diffusion is indeed generated by the invariant sigma field of the subdiffusion $(t_s, \dot{t}_s, \dot{x}_s/|\dot{x}_s|)_{s \geq 0}$
and the single extra information $\sigma(x_{\infty})$.
\par
\medskip
This last argument is new and non-trivial, it takes into account the hypoellipticity of the infinitesimal generator of the relativistic diffusion and it's equivariance with respect to Euclidean spatial translations of the base space. It is actually the starting point of the very recent work \cite{AT1} in collaboration with C. Tardif, where our main motivation is to exhibit a general setting and some natural conditions that allow to compute the Poisson boundary of a diffusion starting from the Poisson boundary of a subdiffusion. It is interesting to note that the resulting devissage method can be used to recover Bailleul's result \cite{ismael} in a more direct way, see Section 4.2 of \cite{AT1}.
\par
\medskip
The article is organized as follows. In the first section, we briefly recall the geometrical background on RW space-times and the definition of the relativistic diffusion in this setting. In Section \ref{sec.results}, we then state our results concerning the asymptotic behavior of the relativistic diffusion and its Poisson boundary. The last section is dedicated to the proofs of these results. For the sake of self-containedness and in order to provide an easily readable article, some results of \cite{angstannIHP} and their proofs are recalled here.
\newpage
\section{Geometric and probabilistic backgrounds}
\label{sec.RW}\label{sec.background}
The Lorentz manifolds we consider here are RW space-times. They are the natural geometric framework to formulate the theory of Big-Bang in General Relativity theory.
The constraint that a space-time satisfies both Einstein's equations and the cosmological principle implies it has a warped product structure, see e.g. \cite{Weinberg} p. 395--404. A RW space-time, classically denoted by $\mathcal M:=I \times_{\alpha} M$, is thus defined as a Cartesian product of a open real interval $(I,-dt^2)$ (the base) and a 3-dimensional Riemannian manifold $(M, h)$ of constant curvature (the fiber), endowed with a Lorentz metric of the following form $g := -dt^2 + \alpha^2(t)h,$
where $\alpha$ is a positive function on $I$, called the \textit{expansion function}. Classical examples of RW space-times are the (half)$-$Minkowski space-time, Einstein static universe, de Sitter and anti-de Sitter space-times etc.
\par
\medskip
A detailed study of the relativistic diffusion in a general RW space-time has been led in \cite{angstannIHP} where we characterized the almost-sure long-time behavior of the diffusion. We focus here on the case when the real interval $I$ is unbounded and the Riemannian fiber is Euclidean, see Remark \ref{rem.courb} below.
Namely, we consider RW space-times $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$, where the expansion function $\alpha$ satisfies the following
hypotheses:
\begin{hyp}\noindent
\begin{enumerate}
\item The function $\alpha$ is of class $C^2$ on $(0, +\infty)$ and it is increasing and $\mathrm{log}-$concave, \emph{i.e.} the Hubble function $H:=\alpha'/\alpha$ is non-negative and non-increasing.
\item The function $\alpha$ has exponential growth, \emph{i.e.} the limit $H_{\infty}:=\displaystyle{\lim_{t \to + \infty} H(t)}$ is positive.
\end{enumerate}
\end{hyp}
\begin{rmk}\label{rem.courb}
The hypothesis of log-concavity of the expansion function is classical, it appears natural from both physical \cite{hawell} and mathematical points of view \cite{alias}. Note also that we are working in dimension $3+1$ because physically relevent space-times have dimension 4, but our results apply verbatim in dimension $d+1$ if $d \geq 3$. Finally, as noticed in Remark 3.5 of \cite{angstannIHP}, in a RW space-time $\mathcal M:=I \times_{\alpha} M$, if the expansion is exponential (the inverse of $\alpha$ is thus integrable at infinity), whatever the curvature of the Riemannian manifold $M$, the process $\dot{x}_s/|\dot{x}_s|$ asymptotically describes a recurrent time-changed spherical Brownian motion in the limit unitary tangent space, i.e. it does not ``see" the curvature $M$, that is why we concentrate here on the case $M=\mathbb R^3$.
\end{rmk}
\begin{rmk}
A RW space-time $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$ is naturally endowed with a global chart $\xi=(t,x)$ where $x=(x^1, x^2,x^3)$ are the canonical coordinates in $\mathbb R^3$. At a point $(t,x)$, the scalar curvature is given by $R= -6 ( \alpha''(t)/\alpha(t)+ {\alpha'}^2(t)/\alpha^2(t))$. In particular, although spatially flat, such a space-time is not globally flat in general. In the case of a ``true'' exponential expansion, i.e. when $\alpha(t)=\exp(H \times t)$ for a positive constant $H$, the space-time $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$ is an Einstein manifold: its Ricci tensor is proportional to its metric.
\end{rmk}
On a general Lorentzian manifold $\mathcal M$, the sample paths $(\xi_s, \dot{\xi}_s)$ of the relativistic diffusion introduced in \cite{flj} are time-like curves that are future-directed and parametrized by the arc length $s$ so that the diffusion actually lives on the positive part of the unitary tangent bundle of the manifold, that we denote by $T^1_+ \mathcal M$.
The infinitesimal generator of the diffusion is the following hypoelliptic operator
\[
\mathcal L:= \mathcal L_0 + \frac{\sigma^2}{2 } \Delta_{\mathcal V},
\]
where $\mathcal L_0$ generates the geodesic flow on $T^1 \mathcal M$, $\Delta_{\mathcal V}$ is the vertical Laplacian, and $\sigma$ is a real parameter. Equivalently, if $\xi^{\mu}$ is a local chart on $\mathcal M$ and if $\Gamma_{\nu \rho}^{\mu}$ are the usual Christoffel symbols, the relativistic diffusion is the solution of the following system of stochastic differential equations, for $0 \leq \mu \leq d=\mathrm{dim} (\mathcal M)$:
\begin{equation}\label{eqn.flj}
\left \lbrace \begin{array}{l}
\displaystyle{ d \xi^{\mu}_s = \dot{\xi}_s^{\mu} ds}, \\
\displaystyle{ d\xi^{\mu}_s= -\Gamma_{\nu \rho}^{\mu}(\xi_s)\, \xi^{\nu}_s \xi^{\rho}_s ds + d \times \frac{\sigma^2}{2}\, \xi^{\mu}_s ds+ \sigma d M^{\mu}_s},
\end{array}\right.
\end{equation}
where the brakets of the martingales $M^{\mu}_s$ are given by
\[
\langle dM_s^{\mu}, \; dM_s^{\nu}\rangle = (\xi^{\mu}_s \xi^{\nu}_s +g^{\mu \nu}(\xi_s))ds.
\]
In the case of a RW space-time of the form $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$ endowed with its natural global chart, the metric is $g_{\mu \nu} = \mathrm{diag}(-1, \alpha^2(t), \alpha^2(t), \alpha^2(t))$, and the only non vanishing Christoffel symbols are $\Gamma_{i\,i}^0 = \alpha(t) \alpha'(t)$, and $\Gamma_{0\, i}^i = H(t)$ for $i=1, 2, 3$.
Thus, in the case of a spatially flat RW space-time, the system of stochastic differential equations (\ref{eqn.flj}) that defines the relativistic diffusion simply reads:
\begin{equation}\label{eqn.flj.eucli}
\left \lbrace \begin{array}{ll}
\displaystyle{d t_s=\dot{t}_s ds}, & \quad \displaystyle{d \dot{t}_s = - \alpha(t_s) \: \alpha'(t_s) |\dot{x}_s|^2 ds + \frac{3 \sigma^2}{2} \dot{t}_s ds + d M^{\dot{t}}_s,} \\
\displaystyle{d x^i_s = \dot{x}_s^i ds}, & \quad \displaystyle{d \dot{x}^i_s = \left(-2 H(t_s) \dot{t}_s + \frac{3 \sigma^2}{2}\right) \dot{x}^i_s\, ds + dM^{\dot{x}^i}_s},
\end{array}\right.
\end{equation}
where $|\dot{x}_s|$ denote the usual Euclidean norm of $\dot{x}_s$ in $\mathbb R^3$ and the brackets satisfy
\[
\left \lbrace \begin{array}{l}
\displaystyle{d \langle M^{\dot{t}}, \, M^{\dot{t}} \rangle_s = \sigma^2 \left( \dot{t}_s^2-1 \right) ds, \quad d \langle M^{\dot{t}}, \, M^{\dot{x}^i} \rangle_s = \sigma^2 \, \dot{t}_s \dot{x}^i_s ds,} \\
\displaystyle{d \langle M^{\dot{x}^i}, \, M^{\dot{x}^j} \rangle_s = \sigma^2 \left(\dot{x}^i_s \dot{x}^j_s + \frac{\delta_{ij}}{\alpha^2(t_s)} \right) ds}.
\end{array}\right.
\]
Moreover, the parameter $s$ being the arc length, we have the pseudo-norm relation:
\begin{equation} \label{eqn.pseudo.eucli}\dot{t}^2_s -1 = \alpha^2(t_s) \times |\dot{x}_s|^2.\end{equation}
\begin{rmk}\label{rem.pseudo}
The sample paths being future-directed, from the above pseudo-norm relation, we have obviously $\dot{t}_s \geq 1$, in particular as long as it is well defined, the ``time" process $t_s$ is a strictly increasing and we have $t_s>s$.
\end{rmk}
\section{Statement of the results}\label{sec.results}
We can now state our results concerning the asymptotic behavior of the relativistic diffusion and its Poisson boundary in a spatially flat and fast expanding RW space-time.
For the sake of clarity, the proofs of these different results are postponed in Section \ref{sec.proofs}. For the whole section, let us thus fix a spatially flat RW space-time $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$, where $\alpha$ satisfies the hypotheses stated in Sect. \ref{sec.RW}.
\subsection{Existence, uniqueness, reduction of the dimension}
Naturally, the first thing to do is to ensure that the system of stochastic differential equations (\ref{eqn.flj.eucli}) admits a solution, and possibly to exhibit lower dimensional subdiffusions that will facilitate its study. This is the object of the following proposition.
\begin{prop}\label{pro.exiuni}
For any $(\xi_0, \dot{\xi}_0)=(t_0, x_0, \dot{t}_0, \dot{x}_0) \in T^1_+ \mathcal M$, the system of stochastic differential equations (\ref{eqn.flj.eucli})
admits a unique strong solution $(\xi_s, \dot{\xi}_s)=(t_s, x_s, \dot{t}_s, \dot{x}_s)$ starting from $(\xi_0, \dot{\xi}_0)$, which is well defined for all positive proper times $s$. Moreover, this solution admits the two following subdiffusions of dimension two and four respectively: $(t_s, \dot{t}_s)_{s \geq 0}$ and $(t_s, \dot{t}_s, \dot{x}_s/|\dot{x}_s| )_{s \geq 0}$.
\end{prop}
\begin{rmk}
Given a point $(\xi, \dot{\xi})\in T^1_+ \mathcal M$, we will denote by $\mathbb P_{(\xi, \dot{\xi})}$ the law of the relativistic diffusion starting from $(\xi, \dot{\xi})$ and by $\mathbb E_{(\xi, \dot{\xi})}$ the associated expectation. Unless otherwise stated, the word ``almost surely'' will mean $\mathbb P_{(\xi, \dot{\xi})}-$almost surely. The two above subdiffusions will be called the \emph{temporal} and \emph{spherical} diffusions respectively.
\end{rmk}
\subsection{Asymptotics of the relativistic diffusion}\label{sec.asymp}
As conjectured in \cite{flj}, we show that the relativistic diffusion asymptotically behaves like light rays, \emph{i.e.} light-like geodesics. Indeed, from Remark \ref{rem.pseudo}, we know that the first projection $t_s$ of the (non-Markovian) process $\xi_s=(t_s, x_s) \in \mathcal M$ goes almost-surely to infinity with $s$. We shall prove that its spatial part $x_s$ converges almost surely to a random point $x_{\infty}$ in $\mathbb R^3$, so that the diffusion asymptotically follows a line $D_{\infty}$ in $\mathcal M$, see Fig. \ref{fig.asymp} below, which is the typical behavior of a light-like geodesic. Moreover, we shall see that the normalized derivative $\dot{x}_s/|\dot{x}_s|$ is recurrent so that the curve $(\xi_s)_{s \geq 0}$ actually winds along the line $D_{\infty}$ in a recurrent way.
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\caption{Typical path of the relativistic diffusion in $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$.}
\label{fig.asymp}
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To state precise results, let us introduce the following notations.
Given two positive constants $a$ and $b$, let $\nu_{a,b}$ be the probability measure on $(1,+\infty)$ admitting the following density with respect to Lebesgue measure:
\[
\nu_{a, b}(x) :=C_{a,b} \times \sqrt{x^2-1} \times \exp \left(- \frac{2a}{b^2} x \right),
\]
where $C_{a,b}$ is the normalizing constant. If $f \in \mathbb L^1(\nu_{a, b})$, we will write
$\nu_{a, b}(f):=\int f(x) \nu_{a, b}(x)dx.$
The following theorem summarizes the almost sure asymptotics of the relativistic diffusion, its proofs is given in Sect. \ref{sec.proofasymp} below.
\begin{thm}\label{the.asymp}
Let $(\xi_0, \dot{\xi}_0) \in T^1_+ \mathcal M$, and let $(\xi_s, \dot{\xi}_s)=(t_s, x_s, \dot{t}_s, \dot{x}_s)$ be the relativistic diffusion starting from $(\xi_0, \dot{\xi}_0)$. Then as $s$ goes to infinity, we have the following almost sure asymptotics:
\begin{enumerate}
\item the non-Markovian process $\dot{t}_s$ is Harris-recurrent in $(1, +\infty)$. Moreover, if $f$ is a monotone, $\nu_{H_{\infty},\sigma}-$integrable function, or if it is bounded and continuous:
\[
\lim_{s \to +\infty} \frac{1}{s} \: \int_0^s f(\dot{t}_u)du \stackrel{a.s.}{=} \nu_{H_{\infty},\sigma}(f).
\]
In particular, $\displaystyle{\lim_{s \to +\infty} t_s /s \stackrel{a.s.}{=} \nu_{H_{\infty},\sigma}(\mathrm{Id}) \in (0,+\infty)}$.
\item the spatial projection $x_s$ converges almost surely to a random point $x_{\infty}$ in $\mathbb R^3$.
\item the normalized spatial derivative $\dot{x}_s /|\dot{x}_s|$ is a time-changed Brownian motion on the sphere $\mathbb S^2 \subset \mathbb R^3$, in particular it is recurrent.
\end{enumerate}
\end{thm}
As noticed in the introduction, the above asymptotic results can be rephrased concisely thanks to the notion of causal boundary introduced in \cite{geroch}. In fact, in a fast expanding RW space-time $\mathcal M=(0, +\infty) \times_{\alpha} \mathbb R^3$, the causal boundary $\partial \mathcal M_c^+$ identifies with a spacelike copy of $\mathbb R^3$: a causal curve $\xi_u=(t_u, x_u)$ converges to a point $\xi_{\infty} \in \partial \mathcal M_c^+$ iff $t_u \to +\infty$ and $x_u \to x_{\infty} \in \mathbb R^3$, see \cite{flores}.
\subsection{Poisson boundary of the relativistic diffusion}
We now describe the Poisson boundary of the relativistic diffusion, that is we determine its invariant sigma field $\textrm{Inv}((\xi_s, \dot{\xi}_s)_{s \geq 0})$ or equivalently the set of bounded harmonic functions with respect to its infinitesimal generator $\mathcal L$. Owing to Theorem \ref{the.asymp}, the processes $\dot{t}_s$ and $\dot{x}_s/|\dot{x}_s|$ being recurrent, it is tempting to assert that the only non trivial asymptotic variable associated to the relativistic diffusion is the random point $x_{\infty} \in \mathbb R^3$. Nevertheless,
the result is far from being trivial because some extra-information relating the temporal components and the spatial ones could be hidden. For example, in Minkowski space-time i.e. if $\alpha \equiv 1$, we have almost surely $t_s \to +\infty$ and $x_s/|x_s| \to \theta_{\infty} \in \mathbb S^2$, but the Poisson boundary of the relativistic diffusion is not reduced to the sigma algebra $\sigma(\theta_{\infty})$ because the difference $t_s - \langle x_s, \theta_{\infty}\rangle$ converges almost-surely to a random variable that is not measurable with respect to $\sigma(\theta_{\infty})$.
\par
\smallskip
Using shift-coupling techniques, we first prove in Sect. \ref{sec.proofpoisson} (Propositions \ref{pro.poissontemp} and \ref{pro.poissonspatial}) that the invariant sigma fields of the temporal and spherical subdiffusions are indeed trivial. Then, taking into account the regularity of harmonic functions (hypoellipticity), and using the equivariance of infinitesimal generator under Euclidean translations, we construct an abstract conditionning (Proposition \ref{pro.poisson}) to end up with the following result:
\begin{thm}\label{the.poisson}
Let $\mathcal M:=(0, +\infty) \times_{\alpha} \mathbb R^3$ be a RW space-time where $\alpha$ has exponential growth. Let $(\xi_0, \dot{\xi}_0) \in T^1_+ \mathcal M$ and let $(\xi_s, \dot{\xi}_s)=(t_s, x_s, \dot{t}_s, \dot{x}_s)$ be the relativistic diffusion starting from $(\xi_0, \dot{\xi}_0)$. Then, the invariant sigma field $\textrm{Inv}((\xi_s, \dot{\xi}_s)_{s \geq 0})$ of the whole diffusion coincides with the sigma field generated by the single variable $x_{\infty} \in \mathbb R^3$ up to $\mathbb P_{(\xi_0, \dot{\xi}_0)}-$negligeable sets. Equivalently, if $h$ is a bounded $\mathcal L-$harmonic function on $T^1_+ \mathcal M$, there exists a bounded measurable function $\psi$ on $\mathbb R^3$, such that
\[
h(\xi, \dot{\xi}) = \mathbb E_{(\xi, \dot{\xi})}[\psi(x_{\infty})], \; \; \forall (\xi, \dot{\xi}) \in T^1_+ \mathcal M.
\]
\end{thm}
In other words, all the asymptotic information on the relativistic diffusion is encoded in the point $x_{\infty} \in \mathbb R^3$ or equivalently in the point $\xi_{\infty} =(\infty, x_{\infty})$ of the causal boundary $\partial \mathcal M_c^+$. The above theorem is thus very similar to Theorem 1 of \cite{ismael} asserting that the invariant sigma field of the relativistic diffusion in Minkowski space-time is generated by a random point on its causal boundary, which in that case identifies with the product $\mathbb R^+ \times \mathbb S^2$. It is tempting to ask if such a link between the Poisson and causal boundary holds in a more general context. This question is the object of a work in progress of the author and C. Tardif \cite{AT2}.
\section{Proofs of the results}\label{sec.proofs}
This last section is dedicated to the proofs of the different results stated above. Namely, the section \ref{sec.proofuniexi} below is devoted to the proof of Proposition \ref{pro.exiuni}, and in Sections \ref{sec.proofasymp} and \ref{sec.proofpoisson} we give the proofs of Theorems \ref{the.asymp} and \ref{the.poisson} respectively.
\subsection{Existence, uniqueness, reduction of the dimension}\label{sec.proofuniexi}
We first give the proof of Proposition \ref{pro.exiuni} concerning the existence, the uniqueness and the lifetime of the relativistic diffusion. The coefficients in the system of stochastic differential equations (\ref{eqn.flj.eucli}) being smooth, the first assertions follow from classical existence and uniqueness theorems, see for example Theorem (2.3) p. 173 of \cite{ikeda}. Next, the fact that the temporal process $(t_s, \dot{t}_s)$ is a subdiffusion of the whole relativistic diffusion is an immediate consequence of Equation (\ref{eqn.flj.eucli}) and the pseudo-norm relation (\ref{eqn.pseudo.eucli}), which allows to express the norm of the spatial derivative $\dot{x}_s$ in term of the temporal process. Finally, the analogous result concerning the spherical subdiffusion follows from a straightforward computation, namely setting $\Theta_s=(\Theta_s^1, \Theta_s^2, \Theta_s^3)$ where $\Theta^i_s:=\dot{x}_s^i/|\dot{x}_s|$ to lighten the expressions, we have the following lemma:
\begin{lem}\label{cor.eucli.cartesien}
The temporal process $(t_s, \dot{t}_s)$ and the spherical process $(t_s, \dot{t}_s, \Theta_s)$ are solutions of the following system of stochastic differential equations:
\begin{eqnarray}
\displaystyle{d t_s=\dot{t}_s ds,} \quad \displaystyle{d \dot{t}_s = -H(t_s) (\dot{t}_s^2-1)ds+ \frac{3 \sigma^2}{2} \dot{t}_s ds + d M^{\dot{t}}_s,}
\label{eqn.ttpoint} \\
\displaystyle{d \Theta^i_s = - \frac{\sigma^2}{\dot{t}_s^2-1} \times \Theta^i_s ds + d M^{\Theta^i}_s,}
\label{eqn.sphere}
\end{eqnarray}
where the brakets of the martingales $M^{\dot{t}}$ and $M^{\Theta^i}$ are given by
\begin{equation}\label{eqn.crochet}
\left \lbrace \begin{array}{l}
\displaystyle{d \langle M^{\dot{t}}, \, M^{\dot{t}} \rangle_s = \sigma^2 \left( \dot{t}_s^2-1 \right) ds,} \quad
\displaystyle{d \langle M^{\dot{t}}, \, M^{\Theta^i} \rangle_s = 0,} \\
\\
\displaystyle{d \langle M^{\Theta^i}, M^{\Theta^j} \rangle_s = \frac{\sigma^2}{\dot{t}_s^2-1} \left(\delta_{ij} - \Theta^i_s \Theta^j_s \right)ds}.
\end{array}\right.\end{equation}
\end{lem}
\begin{rmk}From Remark \ref{rem.pseudo}, we know that $\dot{t}_s \geq 1$ a.s. for all $s \geq 0$. In fact, the Hubble fuction $H=\alpha'/\alpha$ being non-increasing, using standard comparison techniques, it is easy to see that $\dot{t}_s>1$ a.s for all $s>0$, so that the term $\dot{t}_s^2-1$ in the denominators above never vanishes.
\end{rmk}
\begin{rmk}\label{rem.brownienspherique}
The process $\Theta_s$ is a time-changed spherical Brownian motion. More precisely, introducing the clock
\[
C_s := \sigma^2 \int_0^s \frac{du}{\dot{t}_u^2-1} du,
\]
the time-changed process $\widetilde{\Theta}$ such that $ \widetilde{\Theta}(C_s) := \Theta_s$ is a standard spherical Brownian motion on $\mathbb S^2 \subset \mathbb R^3$ and it is independent of the temporal subdiffusion.
\end{rmk}
\subsection{Asymptotic behavior of the diffusion}\label{sec.proofasymp}
We now prove the results stated in Theorem \ref{the.asymp} concerning the asymptotic behavior of the relativistic diffusion.
We distinguish the cases of the temporal components of the diffusion (Proposition \ref{pro.temp} below) and its spatial components (Proposition \ref{pro.spatial}).
\subsubsection{Asymptotic behavior of the temporal subdiffusion}
In this paragraph, the word ``almost sure'' refers to the law of the temporal subdiffusion solution of Equation (\ref{eqn.ttpoint}). The first point of Theorem \ref{the.asymp}
corresponds to the following proposition.
\begin{prop}\label{pro.temp}
Let $(t_0, \dot{t}_0) \in (0, +\infty) \times [1, +\infty)$ and let $(t_s, \dot{t}_s)$ be the solution of Equation (\ref{eqn.ttpoint}) starting from $(t_0, \dot{t}_0)$.
Then, the process $\dot{t}_s$ is Harris-recurrent in $(1, +\infty)$ and if $f$ is a monotone, $\nu_{H_{\infty},\sigma}-$integrable function, or if it is bounded and continuous:
\[
\lim_{s \to +\infty} \frac{1}{s} \: \int_0^s f(\dot{t}_u)du \stackrel{a.s.}{=} \nu_{H_{\infty},\sigma}(f).
\]
In particular
\[
\lim_{s \to +\infty} \frac{t_s}{s} =\lim_{s \to +\infty} \frac{1}{s} \: \int_0^s \dot{t}_u du \stackrel{a.s.}{=} \nu_{H_{\infty},\sigma}(\textrm{Id}).
\]
\end{prop}
The proof of the proposition, which is given below, is based on standard comparison techniques and on the two following elementary lemmas. Recall that the Hubble function $H$ is supposed to be non-increasing.
\begin{lem}Given a constant $H_0>0$, $\dot{t}_0 \in [1, +\infty)$ and a real standard Brownian motion $B$, the following stochastic differential equation
\[
d \dot{t}_s= -H_0 \times \left( \dot{t}_s^2-1 \right)ds + \frac{3 \sigma^2}{2} \dot{t}_s ds+ \sigma \sqrt{\dot{t}_s^2-1} \,dB_s
\]
has a unique strong solution starting from $\dot{t}_0$, well defined for all times $s \geq 0$. Moreover, $\dot{t}_s$ admits the probability measure $\nu_{H_0,\sigma}$ introduced in Sect. \ref{sec.asymp} as an invariant measure. In particular, it is ergodic.
\label{lem.ergo}\end{lem}
\begin{lem} \label{LEM.COMPARPROJ}
Let $(t_0, \dot{t}_0) \in (0, +\infty) \times [1, +\infty)$ and let $(t_s, \dot{t}_s)$ be the solution of Equation (\ref{eqn.ttpoint}) starting from $(t_0, \dot{t}_0)$, where the martingale $M^{\dot{t}}$ is represented by a real standard Brownian motion $B$, i.e. $d M^{\dot{t}}_s = \sigma ( \dot{t}_s^2-1)^{1/2} d B_s$. Let $u_s$ and $v_s$ be the unique strong solutions, well defined for all $s\geq 0$, and starting from $u_0=v_0=\dot{t}_0$, of the equations:
\[
\begin{array}{l}
\displaystyle{d u_s= -H(t_0)\left(u_s^2-1 \right)ds + \frac{3 \sigma^2}{2} u_s ds+ \sigma \sqrt{u_s^2-1} dB_s}, \\
\displaystyle{d v_s= -H_{\infty}\left(v_s^2-1 \right)ds + \frac{3 \sigma^2}{2} v_s ds+ \sigma \sqrt{v_s^2-1} dB_s}.
\end{array}
\]
Then, almost surely, for all $0 \leq s <+\infty$, one has $\displaystyle{ u_s \leq \dot{t}_s \leq v_s}$.
\end{lem}
\begin{proof}[Proof of Proposition \ref{pro.temp}]
There exists a standard Brownian motion $B$ such that the temporal process $(t_s, \dot{t}_s)$ is the solution of the stochastic differential equations
\[
d t_s = \dot{t}_s ds, \quad d \dot{t}_s= -H(t_s) \times \left( \dot{t}_s^2-1 \right)ds + \frac{3 \sigma^2}{2} \dot{t}_s ds+ \sigma \sqrt{\dot{t}_s^2-1} \,dB_s.
\]
Let $z_s$ be the unique strong solution, starting from $z_0=\dot{t}_0$, of the stochastic differential equation
\[
d z_s= -H_{\infty} \times \left( |z_s|^2-1 \right)ds + \frac{3 \sigma^2}{2} z_s ds+ \sigma \sqrt{|z_s|^2-1} \,dB_s.
\]
For $n \in \mathbb N$, let $z^n_s$ be the process that coincides with $\dot{t}_s$ on $[0,n]$ and is the solution on $[n, +\infty)$ of the stochastic differential equation
\[
d z^n_s= -H(t_0+n) \times \left( |z^n_s|^2-1 \right)ds + \frac{3 \sigma^2}{2} z^n_s ds+ \sigma \sqrt{|z^n_s|^2-1} \,dB_s.
\]
By Lemma \ref{LEM.COMPARPROJ}, for all $n \geq 0$ and $s \geq 0$, one has $z^n_s \leq \dot{t}_s \leq z_s$.
By Lemma \ref{lem.ergo}, both processes $z^0_s$ and $z_s$ are ergodic in $(1, +\infty)$, in particular, they are Harris recurrent and so is $\dot{t}_s$.
Now consider an increasing and $\nu_{H_{\infty},\sigma}-$integrable function $f$, and fix an $\varepsilon>0$.
For all $n \in \mathbb N$, the function $f$ is also integrable against the measure $\nu_{H(t_0+n),\sigma}$ and by dominated convergence theorem, $\nu_{H(t_0+n),\sigma}(f) $ converges to $\nu_{H_{\infty},\sigma}(f)$ when $n$ goes to infinity.
Choose $n$ large enough so that we have $ | \nu_{H(t_0+n),\sigma}(f) - \nu_{H_{\infty},\sigma}(f)| \leq \varepsilon$. As $z^n_s \leq \dot{t}_s \leq z_s$ for $s \geq 0$, one has almost surely:
\[
\displaystyle{\int_0^s f(z^n_u)du \leq \int_0^s f(\dot{t}_u)du \leq \int_0^s f(z_u)du}.
\]
The integer $n$ being fixed, by the ergodic theorem, we have that almost surely:
\[
\nu_{H_{\infty},\sigma}(f)- \varepsilon \leq \nu_{H(t_0+n),\sigma}(f) \leq \liminf_{s \to + \infty} \frac{1}{s} \int_0^s f(\dot{t}_u)du \leq
\limsup_{s \to + \infty}\frac{1}{s} \; \int_0^s f(\dot{t}_u)du \leq \nu_{H_{\infty},\sigma}(f).
\]
Letting $\varepsilon$ goes to zero, we get the desired result.
As any smooth function can be written as the difference of two monotone functions, the above convergence extends to functions in the set $C^{1}_b=\{ f, \; f' \; \textrm{is bounded on} \; (1, +\infty)\}$, and then by regularization, to the set of bounded continuous functions on $(1,+\infty)$.
\end{proof}
\subsubsection{Asymptotic behavior of the spatial components}
The second and third points of Theorem \ref{the.asymp} are the object of the next proposition:
\begin{prop} \label{pro.spatial}
Let $(\xi_s, \dot{\xi}_s)=(t_s, x_s, \dot{t}_s, \dot{x}_s)$ be the relativistic diffusion starting from $(\xi_0, \dot{\xi}_0) \in T^1_+ \mathcal M$. Then, as $s$ goes to infinity, the spatial projection $x_s$ converges almost surely to a random point $x_{\infty} \in \mathbb R^3$, and the process $\Theta_s=\dot{x}_s /|\dot{x}_s|$ is recurrent in $\mathbb S^2 \subset \mathbb R^3$.
\end{prop}
\begin{proof}
From Equation (\ref{eqn.pseudo.eucli}), we have $|\dot{x}_s|=\sqrt{\dot{t}_s^2-1}/\alpha(t_s)$ for all $s \geq 0$. Therefore
\[
|x_s -x_0| \leq \int_0^s |\dot{x}_u |du = \int_0^s \frac{\sqrt{\dot{t}_u^2-1}}{\alpha(t_u)} du \leq \int_0^s \frac{\dot{t}_u}{\alpha(t_u)} du =\int_{t_0}^{t_s} \frac{du}{\alpha(u)}.
\]
The process $t_s$ goes almost surely to infinity with $s$. The expansion function $\alpha$ having exponential growth, the last integral is a.s. convergent, so that the total variation of $x_s$ and the process itself are also convergent, whence the first point in the proposition. According to Remark \ref{rem.brownienspherique}, $\dot{x}_s /|\dot{x}_s|=\Theta_s=\widetilde{\Theta}_{C_s}$ is a time-changed spherical Brownian motion. By Proposition \ref{pro.temp}, we have $\lim_{s \to +\infty} C_s /s \in (0, +\infty)$, in particular the clock $C_s$ goes almost surely to infinity with $s$ and the process $\Theta_s$ is recurrent in $\mathbb S^2$.
\end{proof}
\subsection{Poisson boundary of the relativistic diffusion} \label{sec.proofpoisson}
\noindent
The proof of Theorem \ref{the.poisson} is divided into three parts. We first prove a Liouville theorem for the temporal subdiffusion (Proposition \ref{pro.poissontemp}), then we prove an analogous result for the spherical subdiffusion (Proposition \ref{pro.poissonspatial}). Finally, we deduce the Poisson boundary of the global relativistic diffusion (Proposition \ref{pro.poisson}).
\subsubsection{A Liouville theorem for the temporal subdiffusion}
\noindent
The infinitesimal generator of the temporal subdiffusion $(t_s, \dot{t}_s)$, acting on smooth functions from $(0,+\infty) \times [1, +\infty)$ to $\mathbb R$, is given by
$$
\mathcal L_{H}:= \dot{t} \partial_{t} - H(t) (\dot{t}^2-1) \partial_{\dot{t}} + \frac{\sigma^2}{2} (\dot{t}^2-1) \partial_{\dot{t}}^2.
$$
\begin{prop}
\label{pro.poissontemp}All bounded $\mathcal L_{H}-$harmonic functions are constant.
\end{prop}
\begin{proof}The proof of Proposition \ref{pro.poissontemp} is based on the following fact: there is an automatic shift coupling between two independent solutions of the system (\ref{eqn.ttpoint}). Let $B^1$ and $B^2$ be two independent standard Brownian motions defined on two measured spaces $(\Omega_1, \mathcal F_1)$ and $(\Omega_2, \mathcal F_2)$ as well as the processes $(t_s^1, \dot{t}_s^1)$ and $(t_s^2, \dot{t}_s^2)$, starting from $(t_0^1, \dot{t}_0^1) \neq (t_0^2, \dot{t}_0^2) $ (deterministic) and solution of the following systems, for $ i=1,2$:
\[
dt_s^i= \dot{t}_s^i ds, \;\; d \dot{t}_s^i= \left[-H(t_s^i) \left(|\dot{t}_s^i|^2-1 \right) + \frac{3 \sigma^2}{2} \dot{t}_s^i \right]ds+ \sigma \sqrt{|\dot{t}_s^i|^2-1} d B^i_s.
\]
Define $\tau_0:=\max(t_0^1,t_0^2)$. We denote by $\mathbb P_i$ the law of $(t_s^i, \dot{t}_s^i)$ and by $\mathbb P:=\mathbb P_1 \otimes \mathbb P_2$ the law of the couple.
From Remark \ref{rem.pseudo}, the processes $t^i_s$ are strictly increasing. Denote by $(t^i)^{-1}_s$ their inverse, and define $u^i_s:= \dot{t}^i [ (t^i)^{-1}_s]$. Without loss of generality, one can suppose that $1<u_{\tau_0}^1< u_{\tau_0}^2$. By It\^o's formula, for $s \geq \tau_0$, one has
\begin{equation}\label{eqn.logv}
\frac{1}{2} \log \left(\frac{|u_s^1|^2-1}{|u_s^2|^2-1} \right) =\frac{1}{2} \log \left(\frac{|u_{\tau_0}^1|^2-1}{|u_{\tau_0}^2|^2-1} \right) + Q_s + R_s+M_s,
\end{equation}
where
\[
\begin{array}{ll}
\displaystyle{Q_s:=\sigma^2 \left[(t^1)^{-1}_s -(t^2)^{-1}_s\right] - \sigma^2 \left[(t^1)^{-1}_{\tau_0} -(t^2)^{-1}_{\tau_0}\right], } \\
\\
\displaystyle{R_s:=\frac{\sigma^2}{2}\left(\int_{\tau_0}^{s}\frac{u_r^2 \left(|u_r^2|^2-1\right)-u_r^1 \left(|u_r^1|^2-1\right)}{u_r^1 \left(|u_r^1|^2-1\right) \times u_r^2 \left(|u_r^2|^2-1\right)}dr\right),}\\
\end{array}
\]
and where $M_s$ is a martingale whose bracket is given by:
\begin{equation} \langle M \rangle_s = \sum_{i=1}^2 \int_{(t^i)^{-1}_{\tau_0}}^{(t^i)^{-1}_s}\frac{|\dot{t}_u^i|^2}{|\dot{t}_u^i|^2-1} du \geq (t^1)^{-1}_s- (t^1)^{-1}_{\tau_0}.
\label{eqn.minorcrochet}\end{equation}
Let us show that the coupling time $\tau_c := \inf \{ s > \tau_0, \; u^1_s=u^2_s\}$ is finite $\mathbb P-$almost surely. Consider the set
$A:=\{ \omega \in \Omega_1 \times \Omega_2, \; \tau_c(\omega)=+\infty\}$. By definition, if $\omega\in A$ one has $u^1_s (\omega)< u^2_s(\omega)$ for $s > \tau_0$. We deduce that $R_s(\omega), Q_s(\omega)>0$ for all $s > \tau_0$. Indeed, for $s > \tau_0$, one has :
\[
\int_{\tau_0}^s \frac{dr}{u_r^1} > \int_{\tau_0}^s \frac{dr}{u_r^2}, \;\; \hbox{and} \;\; \int_{\tau_0}^s \frac{dr}{u_r^i}= \int_{\tau_0}^s \frac{dv}{\dot{t}^i[(t^1)^{-1}_v]}=(t^i)^{-1}_s - (t^i)^{-1}_{\tau_0}.
\]
On the set $A$, by Equation (\ref{eqn.logv}), the martingale $M_s$ thus admits the upper bound:
\[
M_s + \frac{1}{2} \log \left(\frac{|u_{\tau_0}^1|^2-1}{|u_{\tau_0}^2|^2-1} \right) \leq \frac{1}{2} \log \left(\frac{|u_s^1|^2-1}{|u_s^2|^2-1} \right) \leq 0,
\]
But by Equation (\ref{eqn.minorcrochet}), as $(t^1)^{-1}_s$ goes to infinity with $s$, we have also $\langle M \rangle_{\infty} = +\infty$ $\mathbb P-$almost surely. Therefore $\mathbb P(A)=0$ and $\tau_c < +\infty \; \mathbb P-$almost surely. In other words, $\mathbb P-$a.s. the two sets $(t_.^1, \dot{t}_.^1)_{\mathbb R^+}$ and $(t_.^2, \dot{t}_.^2)_{\mathbb R^+}$ intersect, where $(t_.^i, \dot{t}_.^i)_{\mathbb R^+}$ denotes the set of points of the curves $(t_s^i, \dot{t}_s^i)_{s \geq 0}$, $i=1,2$. Let us define the random times
\[
\displaystyle{T_1:=\inf \{ s>0, (t_s^1, \dot{t}_s^1) \in (t_.^2, \dot{t}_.^2)_{\mathbb R^+}} \}, \quad
\displaystyle{T_2:=\inf \{ s>0, (t_s^2, \dot{t}_s^2) \in (t_.^1, \dot{t}_.^1)_{\mathbb R^+}} \}.
\]
These variables are not stopping times for the filtration
$\sigma( (t_s^i, \dot{t}_s^i), \; i=1,2, \; s \leq u)_{u \geq 0}$, nevertheless they are finite $\mathbb P-$almost surely. As a consequence, we deduce that both sets
$A_1:= \{ \omega_1 \in \Omega_1, \; T_2 < +\infty \; \mathbb P_2-\hbox{a.s.} \}$ and
$A_2:= \{ \omega_2 \in \Omega_2, \; T_1 < +\infty \; \mathbb P_1-\hbox{a.s.} \}$
verify $\mathbb P_1(A_1)=\mathbb P_2(A_2)=1$.
Moreover, as the processes $t_s^i$ are strictly increasing, one has
\begin{equation} \label{eqn.couplage}(t_{T_1}^1, \dot{t}_{T_1}^1)= (t_{T_2}^2, \dot{t}_{T_2}^2) \quad \mathbb P-\hbox{almost surely}.\end{equation}
Indeed, by definition of $T_1$ and $T_2$, there exists $u, v \in \mathbb R^+$ (random) such that
$(t_{T_1}^1, \dot{t}_{T_1}^1)= (t_u^2, \dot{t}_u^2)$ and $(t_{T_2}^2, \dot{t}_{T_2}^2)= (t_v^1, \dot{t}_v^1)$. If $t_{T_1}^1 =t_u^2 < t_{T_2}^2$, as $t^2_s$ is strictly increasing, we would have $u < T_2$ and $(t_u^2, \dot{t}_u^2) \in (t_.^1, \dot{t}_.^1)_{\mathbb R^+}$ which would contradict the definition of $T_2$ as an infimum. Therefore, we have $t_{T_1}^1\geq t_{T_2}^2$ and $t_{T_1}^1 = t_{T_2}^2$ by symmetry. Finally, using the monotonicity of $t_s^i$ again, we conclude that $u=T_2$ and $v=T_1$, hence the coupling (\ref{eqn.couplage}). Now let $h$ be a bounded $\mathcal L_H-$harmonic function. Fix $\omega_2 \in \Omega_2$. The map
$\omega_1 \in \Omega_1 \mapsto T_1(\omega_1,\omega_2)$ is a stopping time for the filtration $\sigma( (t_s^1, \dot{t}_s^1), \; s \leq t)_{t \geq 0}$, and it is finite $\mathbb P_1-$almost surely. By the optional stopping theorem, one has
\[
h(t_0^1, \dot{t}_0^1) = \mathbb E_1 \left[h (t_{T_1}^1, \dot{t}_{T_1}^1)\right] = \int h (t_{T_1}^1, \dot{t}_{T_1}^1)d \mathbb P_1,
\]
and integrating against $\mathbb P_2$, we get :
\[
h(t_0^1, \dot{t}_0^1) = \int h (t_{T_1}^1, \dot{t}_{T_1}^1)d \mathbb P_1 \otimes d\mathbb P_2 = \int h (t_{T_1}^1, \dot{t}_{T_1}^1)d \mathbb P.
\]
In the same way, we have
\[
h(t_0^2, \dot{t}_0^2) = \int h (t_{T_2}^2, \dot{t}_{T_2}^2)d \mathbb P_1 \otimes d\mathbb P_2 = \int h (t_{T_2}^2, \dot{t}_{T_2}^2)d \mathbb P.
\]
By (\ref{eqn.couplage}), we conclude that $h(t_0^1, \dot{t}_0^1) =h(t_0^2, \dot{t}_0^2)$, \emph{i.e.} the function $h$ is constant.
\end{proof}
\subsubsection{A Liouville theorem for the spherical subdiffusion}
We now extend the above Liouville theorem to the spherical subdiffusion by using a second coupling argument, namely a mirror coupling argument on the sphere. To simplify the expressions in the sequel, we will denote by $(e_s)_{s \geq 0} := (t_s, \dot{t}_s, \Theta_s)_{s \geq 0}$ the spherical subdiffusion with values in the space $E:=(0, +\infty) \times [1, +\infty) \times \mathbb S^2$ and by $\mathcal L_{E}$ its the infinitesimal generator acting on smooth functions from $E$ to $\mathbb R$.
\begin{prop}\label{pro.poissonspatial}All bounded $\mathcal L_{E}-$harmonic functions are constant.
\end{prop}
\begin{proof}
Fix $e^1_0= (t_0^1, \dot{t}_0^1, \Theta_0^1)\neq e^2_0=(t_0^2, \dot{t}_0^2, \Theta_0^2)$ in $E$. As in the proof of Proposition \ref{pro.poissontemp}, consider two independent solutions $(t_s^1, \dot{t}_s^1)$ and $(t_s^2, \dot{t}_s^2)$ of Equation (\ref{eqn.ttpoint}), starting from $(t_0^1, \dot{t}_0^1)$ and $(t_0^2, \dot{t}_0^2)$ respectively, which coincide after the shift-coupling times $T_1$ and $T_2$:
$(t_{T_1+s}^1, \dot{t}_{T_1+s}^1)=(t_{T_2+s}^2, \dot{t}_{T_2+s}^2)$, for $s \geq 0$.
Let us consider two independent spherical Brownian motions $\widetilde{\Theta}^i$ on $\mathbb S^2$, $i=1,2$, which are independent of the two above temporal diffusions and define for $s \geq 0$ and $i=1, 2$:
\[
C^i_s:=\int_{0}^{s} \frac{du}{|\dot{t}_u^i|^2-1}, \quad \Theta_s^i := \widetilde{\Theta}^i \left( C^i_s \right).
\]
By Remark \ref{rem.brownienspherique}, the two diffusions $e^i_s:=(t_s^i, \dot{t}_s^i, \Theta_s^i), \,i=1,2$ are solutions of the stochastic differential equations (\ref{eqn.ttpoint}--\ref{eqn.crochet}), let us denote by $\mathbb P_i$ their law, define $\mathbb P:= \mathbb P_1 \otimes \mathbb P_2$ and denote by $\mathbb E$ the associated expectation. Define a new process $({\Theta'}_{s}^2)_{s \geq 0}$, such that ${\Theta'}_{s}^2$ coincides with ${\Theta}_{s}^2$ on the time interval $[0, T_2]$ and such that the future trajectory $({\Theta'}_{s}^2)_{s \geq T_2}$ is the reflection of $({\Theta}_{s}^1)_{s \geq T_1}$ with respect to the median plan between the points $\Theta_{T^1}^1$ and $\Theta_{T^2}^2$, see figure \ref{fig.couplage} below.
\begin{figure}[ht]
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\caption{Mirror coupling of two independent spherical sub-diffusions.}\label{fig.couplage}
\end{figure}
\if{\begin{figure}[htbp]
\begin{center}
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\caption{Mirror coupling of two independent spherical subdiffusions.}
\label{fig.couplage}
\end{center}
\end{figure}
}\fi
The new process ${e'}^2_s:=(t_s^2, \dot{t}_s^2, {\Theta'}_s^2)$ is again a solution of Equations (\ref{eqn.ttpoint}--\ref{eqn.crochet}). Moreover, the first time $T^*$ when the process $({\Theta}_{s}^1)_{s \geq T_1}$ intersects the median big circle between $\Theta_{T^1}^1$ and $\Theta_{T^2}^2$ is finite $\mathbb P-$almost surely, and one has naturally ${e'}^2_{T_2+T^*}={e}_{T_1+T^*}^1$ $\mathbb P-$almost surely. Now consider $h$ a bounded $\mathcal L_E-$harmonic function, thanks to the above coupling and the optionnal stopping theorem, as in the proof of Prop. \ref{pro.poissontemp}, we have $\mathbb P-$almost surely $h(e_0^2)=h(e_0^1)$ because
\[
\mathbb E [ h({e'}^2_{T_2+T^*}) ] =\mathbb E [ h({e}_{T_1+T^*}^1) ].
\]
Therefore, the function $h$ is constant, hence the result.
\end{proof}
\subsubsection{Poisson boundary of the global relativistic diffusion}
\noindent
In order to describe the Poisson boundary of the whole relativistic diffusion $(\xi_s, \dot{\xi}_s)_{s \geq 0}$ starting from the one of the spherical subdiffusion, we need a few preliminaries. First notice that, thanks to the pseudo-norm relation (\ref{eqn.pseudo.eucli}), the invariant sigma field of the whole diffusion $(\xi_s, \dot{\xi}_s)_{s \geq 0}$ with values in $T^1_+ \mathcal M$ coincides almost surely with the one of the diffusion process $(e_s, x_s)_{s \geq 0}=((t_s, \dot{t}_s, \Theta_s), x_s)_{s \geq 0}$ with values in $E \times \mathbb R^3$ and whose infinitesimal generator $\mathcal G$ is hypoelliptic and reads:
\begin{equation}\label{gene} \mathcal G:= \mathcal L_E+F(e) \, \partial_x, \;\; \hbox{where}\;\; F(e)=F(t,\dot{t}, \Theta):=\Theta \times \frac{\sqrt{\dot{t}^2-1}}{\alpha(t)}.\end{equation}
Without loss of generality, we can suppose that the process $(e_s, x_s)_{s \geq 0}$ is defined on the canonical probability space $(\Omega, \mathcal F)$ where $\Omega:= C(\mathbb R^+, E \times \mathbb R^3)$ and $\mathcal F$ is the standard Borel sigma field. A generic $\omega \in \Omega$ writes $\omega=(\omega^1, \omega^{2})$ where $\omega^1=(\omega^1_s)_{s \geq 0} \in C(\mathbb R^+,E)$ and $\omega^{2}=(\omega^2_s)_{s \geq 0} \in C(\mathbb R^+,\mathbb R^3)$. Without loss of generality again, we can suppose that $(e_s, x_s)_{s \geq 0}$ is the coordinate process, namely $(e_s, x_s) = (\omega^1_s, \omega^2_s)$ for all $s \geq 0$. Given $(e,x)$ in $E \times \mathbb R^3$, we will denote by $\mathbb P_{(e,x)}$ the law of the process $(e_s, x_s)_{s \geq 0}$ starting from $(e,x)$, and by $\mathbb E_{(e,x)}$ the associated expectation.
Let us finally introduce the classical shift operators $(\theta_u)_{u \geq 0}$ acting on $\Omega$ and such that $\theta_u \omega = (\omega_{s+u})_{s \geq 0}$ for all $u \geq 0$. Recall that the tail sigma field $\mathcal F^{\infty}$ of the diffusion process $(e_s,x_s)_{s \geq 0}$ is defined as the intersection
\[
\mathcal F^{\infty}:=\bigcap_{s>0} \sigma( (e_u,x_u), u>s),
\]
and that the invariant sigma field $\textrm{Inv}((e_s,x_s)_{s \geq 0})$ of $(e_s,x_s)_{s \geq 0}$ is the subsigma field of $\mathcal F^{\infty}$ composed of shift invariant events, i.e. events $A$ such that $\theta^{-1}_u A=A$ for all $u \geq 0$.
In this setting, Theorem \ref{the.poisson} is equivalent to the following proposition:
\begin{prop}\label{pro.poisson}
Let $h$ be a bounded $\mathcal G-$harmonic function on $E \times \mathbb R^3$. Then, there exists a bounded mesurable function $\psi$ on $\mathbb R^3$ such that:
\[
h(e, x) = \mathbb E_{(e,x)} [\psi(x_{\infty})], \;\; \forall (e,x) \in E \times \mathbb R^3.
\]
Equivalently, $(e_0,x_0)$ being fixed, the invariant sigma field $\textrm{Inv}((e_s,x_s)_{s \geq 0})$ of the diffusion $(e_s,x_s)_{s \geq 0}$ starting from $(e_0,x_0)$ coincides with $\sigma(x_{\infty})$ up to $\mathbb P_{(e_0,x_0)}-$negligeable sets.
\end{prop}
\begin{proof}
From the second point of Theorem \ref{the.asymp}, for all $(e,x) \in E \times \mathbb R^3$, the process $(x_s)_{s \geq 0}$ converges $\mathbb P_{(e,x)}-$almost surely to a random point $x_{\infty} =x_{\infty}(\omega) \in \mathbb R^3$. With a slight abuse of notation, let us still denote by $x_{\infty}$ the random variable which coincides with $x_{\infty}$ on the subset of $\Omega$ where the convergence occurs and which vanishes elsewhere.
Thanks to the particular form (\ref{gene}) of the infinitesimal generator $\mathcal G$, let us remark the following facts:
\begin{enumerate}
\item for all starting points $(e,x) \in E \times \mathbb R^3$, the law of the process $(e_s, x+ x_s)_{s \geq 0 }$ under $\mathbb P_{(e,0)}$ coincide with the law of $(e_s, x_s)_{s \geq 0 }$ under $\mathbb P_{(e,x)}$, in particular the law of the limit $x_{\infty}$ under $\mathbb P_{(e,x)}$ is the law of $x + x_{\infty}$ under $\mathbb P_{(e,0)}$;
\item the push-forward measures of both measures $\mathbb P_{(e, 0)}$ and $\mathbb P_{(e,x)}$ under the following mesurable map
$\omega=(\omega^1, \omega^2) \mapsto (\omega^1, \omega^2-x_{\infty}(\omega))$ coincide.
\end{enumerate}
\noindent
Let $h$ be a bounded $\mathcal G-$harmonic function on $E \times \mathbb R^3$. From the classical duality between harmonic functions and invariant events, there exists a bounded variable map $Z : \Omega \to \mathbb R$, such that $Z$ is $\mathcal F^{\infty}-$measurable and satisfies $Z( \theta_u \omega) = Z(\omega)$ for all $\omega \in \Omega$, and such that
\[
h(e,x) = \mathbb{E}_{(e,x)} [ Z ], \quad \hbox{for all} \;\; (e,x) \in E \times \mathbb R^3.\]
Moreover, $(e, x) \in E \times \mathbb R^3$ being fixed, for $\mathbb P_{(e,x)}-$almost all paths $\omega$, we have:
\[
Z(\omega)= \lim_{s \to +\infty} h(e_s(\omega), x_s(\omega)).
\]
For $y \in \mathbb R^3$, consider the new random variable
\[
Z^y(\omega) := Z(( \omega^1, \omega^2-x_{\infty}(\omega) + y) ).
\]
The variable $Z^y $ is again $\textrm{Inv}((e_s,x_s)_{s \geq 0})-$measurable. Indeed, since the constant function equal to $y$ and the random variable $Z$ are shift-invariant, for all $u \geq 0$ we have
\[
Z( (\omega_{.+u}^1, \omega^2_{.+u}-x_{\infty}(\omega_{.+u}) + y) )= Z (\theta_u ( \omega^1, \omega^2-x_{\infty}(\omega) + y ))= Z( ( \omega^1, \omega^2-x_{\infty}(\omega) + y )).
\]
Since $Z^y$ is bounded and $\textrm{Inv}((e_s,x_s)_{s \geq 0})-$measurable, the function $(e,x) \mapsto \mathbb E_{(e,x)} [Z^{y}]$ is also a bounded $\mathcal G-$harmonic function. But from the point 2 of the beginning of the proof, for all starting points $(e,x, x') \in E \times \mathbb R^3 \times \mathbb R^3$, we have
$\mathbb{E}_{(e,x)} [ Z^{y} ]= \mathbb{E}_{(e,x')} [ Z^{y} ]$.
In other words, the harmonic function $(e, x) \mapsto \mathbb{E}_{(e,x)} [ Z^{y} ]$ is constant in $x$ and its restriction to $E$ is $\mathcal L_E-$harmonic. From Proposition \ref{pro.poissonspatial}, we deduce that the function $(e, x) \mapsto \mathbb{E}_{(e,x)} [ Z^{y} ]$ is constant. In the sequel, we will denote by $\psi(y)$ the value of this constant. Note that $y \mapsto \psi(y)$ is a bounded measurable function since $y \mapsto Z^y$ is.
Let us now introduce an approximate unity $(\rho_n)_{n \geq 0}$ on $\mathbb R^3$, fix $\mathbf{x} \in \mathbb R^3$, $n \in \mathbb N$ and consider the ``conditionned and regularized'' version $Z$, namely:
$$
Z^{ \mathbf{x},n}(\omega):= \int_{\mathbb R^3} Z^y(\omega) \rho_n( \mathbf{x}-y)dy.
$$
The same reasoning as above shows that $Z^{ \mathbf{x},n}$ is a bounded $\textrm{Inv}((e_s,x_s)_{s \geq 0})-$measurable variable so that the function $(e, x) \mapsto \mathbb{E}_{(e,x)} [ Z^{\mathbf{x},n} ] $ is constant. Hence, for all $\mathbf{x} \in \mathbb R^3$, $n \in \mathbb N$ and $(e,x) \in E \times \mathbb R^3$, there exists a set $\Omega^{\mathbf{x},n,(e,x)} \subset \Omega$ such that $\mathbb P_{(e,x)}(\Omega^{\mathbf{x},n,(e,x)} ) =1$ and such that for all paths $\omega$ in $\Omega^{\mathbf{x},n,(e,x)}$, we have:
\[
Z^{\mathbf{x},n} (\omega) = \lim_{s \to \infty} \mathbb{E}_{(e_s(\omega),x_s(\omega))} [ Z^{\mathbf{x},n} ]
= \mathbb{E}_{(e_0(\omega),x_0(\omega))} [ Z^{\mathbf{x},n} ]
= \mathbb{E}_{(e,x)} [Z^{\mathbf{x},n} ].
\]
Let $D$ be a countable dense set in $\mathbb R^3$ and consider the intersection
\[
\Omega^{(e,x)} := \underset{\mathbf{x} \in D, n \in \mathbb N}{\bigcap } \Omega^{\mathbf{x},n, (e,x)}.
\]
We have naturally $\mathbb P_{(e,x)} ( \Omega^{(e,x)} )=1$ and for all $\omega \in \Omega^{(e,x)}$, $\mathbf{x} \in D$, $ n \in \mathbb N$, we have
\[
Z^{\mathbf{x},n} (\omega) = \mathbb{E}_{(e,x)} [ Z^{\mathbf{x},n} ].
\]
Since the above expressions are continuous in $\mathbf{x}$, we deduce that the last inequality is true for all $\mathbf{x} \in \mathbb R^3$. In other words, we have shown that for all $\mathbf{x} \in \mathbb R^3$ and $\omega $ in $\Omega^{(e,x)}$:
\[
Z^{\mathbf{x},n} (\omega) =\mathbb{E}_{(e,x)} [ Z^{\mathbf{x},n} ]= \int_{\mathbb R^3} \psi(y ) \rho_n(\mathbf{x} -y)dy.
\]
In particular, taking $\mathbf{x}= x_\infty (\omega)$, we obtain that for all $\omega \in \Omega^{(e,x)}$ and for all $n \in \mathbb N$:
\[
Z^{x_{\infty}(\omega),n} (\omega) = \displaystyle{\int_{\mathbb R^3} Z((\omega^1, \omega^2+y) ) \rho_n(-y)dy }= \displaystyle{\int_{\mathbb R^3} \psi(y+ x_{\infty}(\omega)) \rho_n(-y)dy}.
\]
Taking the integral in $\omega$ with respect to $\mathbb P_{(e,x)}$ on $\Omega^{(e,x)}$, we deduce that for all $ n \in \mathbb N$:
\[
\displaystyle{\mathbb E_{(e,x)} \left[ Z^{x_{\infty},n} \right] } = \displaystyle{\int_{\mathbb R^3} \mathbb{E}_{(e,x)} [ \psi(y+x_{\infty}) ] \rho_n(-y)dy },
\]
which, from the first point at the beginning of the proof yields
\[
\displaystyle{\int_{\mathbb R^3} h(e, x+y ) \rho_n(-y)dy } = \displaystyle{\int_{\mathbb R^3} \mathbb{E}_{(e,x+y)} [ \psi(x_\infty ) ] \rho_n(-y)dy.}
\]
To conclude, recall that the infinitesimal generator of the diffusion is hypoelliptic so that $\mathcal G-$harmonic functions are continuous, hence we can let $n$ go to infinity in the above expressions to get the desired result, namely $h(e,x) = \mathbb{E}_{(e,x)} [\psi(x_\infty)]$.
\end{proof}
\begin{rmk}
As already noticed at the end of the introduction, the proof of the last proposition is the starting point of the very recent work \cite{AT1} in collaboration with C. Tardif, where our main motivation is to exhibit a general setting and some natural conditions that allow to compute the Poisson boundary of a diffusion starting from the Poisson boundary of a subdiffusion of the original one. Indeed, the main ingredients of the proof above are that the infinitesimal generator $\mathcal G$ acting on $E \times \mathbb R^3$ is equivariant under the action of Euclidean translations and that it is hypoelliptic so that harmonic functions are continuous. The devissage method introduced in \cite{AT1} actually shows that, under similar equivariance and regularity conditions, the scheme of the proof of Proposition \ref{pro.poisson} can be generalized to an abstract setting where $E$ is replaced by any differentiable manifold and $\mathbb R^3$ is replaced by a finite dimensional Lie group or a co-compact homogeneous space.
\end{rmk}
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Pittsburg State senior third baseman Katelyn Birchfield helped deliver another Saturday split in the Mid-America Intercollegiate Athletics Association, driving home four runs on four hits in a 9-6 win over the No. 14-ranked Fort Hays State Tigers.
The Gorillas came back for their win after dropping a 1-0 heartbreaker in 10 innings, as Pitt State freshman Kyndel Shelburn matched low numbers with Fort Hays senior Maggie Holub, one of the premier players in the nation fresh off perfect games against Northwest Missouri and Missouri Southern.
Shelburn allowed one run on one hit but that run did not score on a hit and she fanned 12 Fort Hays batters. Holub improved to 20-1, striking out 13 and surrendering five hits. Shelburn and Holub combined for 11 walks.
Shelburn, who dropped to 6-7, pitched the PSU softball program’s first-ever one-hit extra-inning game. The last Pitt State one-hitter was tossed by Melissa Slayden against Northeastern State all the way back in 2010.
Tori Beltz drove in Amanda Vaupel on a sacrifice fly for the game’s lone run.
The game was decided by the good old international tiebreaker. The Gorillas were cursed by 14 runners left behind.
Pitt State answered the bell in the top of the first of the second game with two runs. Tiffany Brown drove in Brenna George with a sacrifice fly and Birchfield picked up her first RBI with a single plating Catie Cummins. Both runs were unearned.
Fort Hays tied it up in the bottom half of the first with a two-run single by Danie Brinkmann.
Birchfield drove in a run in all four innings the Gorillas scored and she again pushed across Cummins with a RBI single to spark a four-run third. McKenzie Rynard pulled off the nifty feat of both reaching on a bunt single and driving in Brown with that bunt single. Kayla Sears delivered a two-run double.
After the Tigers halved the Gorilla lead to 6-4, Birchfield singled home fellow senior Kreslee Ketcham in the fourth.
Fort Hays came back with two runs in the bottom of the fourth.
With a 7-6 lead in the sixth, Pitt State — victims of so many one-run losses this season, nine in fact — closed out the scoring with Brown and Birchfield RBI. Birchfield doubled home Ketcham and conversely, all four Birchfield RBI involved Neosho (Mo.) products Cummins and Ketcham.
Pitt State (19-19, 7-11 MIAA) closes out its road schedule today against Nebraska-Kearney and the Gorillas play their final 12 regular season games at the PSU Softball Complex.
Game One
No. 14 Fort Hays State 1, Pittsburg State 0 (10 innings)
PSU 000 000 000 0 — 0 5 0
FHSU 000 000 000 1 — 1 1 1
Kyndel Shelburn and Kayla Sears. Maddie Holub and Callie Wright. WP: Holub (20-1). LP: Shelburn (6-7). E: Danie Brinkmann. SF: Tori Beltz. SH: Kreslee Ketcham; Brinkmann, Paxton Duran. SB: Brenna George 2; Bianca Adame, Holub. CS: Holub. HBP: Katelyn Birchfield, Sears, Riley Campbell.
Page 2 of 2 - Game Two
Pittsburg State 9, Fort Hays State 6
PSU 204 102 0 — 9 10 3
FHSU 202 200 0 — 6 10 2
Haleigh Sills, Kylie Harpst (3) and McKenzie Rynard. Paxton Duran, Jordan Jones (4), Maggie Holub (5) and Amy Dunn. WP: Harpst (4-3). LP: Duran (13-6). E: Rynard, Jordan Bradshaw, Riley Campbell; Holub, Dunn. 2B: Katelyn Birchfield, Kayla Sears. SF: Tiffany Brown. SH: Catie Cummins 2, Kreslee Ketcham 2. SB: Bianca Adame 4, Courtney Dobson, Adara Erickson. HBP: Jordan Bradshaw. LOB: PSU 6, FHSU 11.
| 146,572
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What is nothing 13
Company & Friends 13 - Nothing (2013)
A dance piece by Linda Weißig with the company of the Leipziger Tanztheater under the direction of Alessio Trevisani
(Text: Linda Weißig)
Inspired by the novel by Janne Teller “Nothing. What is important in life ”. Translated from Danish by Sigrid Engeler. Carl Hanser Verlag Munich, 2010.
"Nothing means anything, so it's not worth doing anything."
In a world full of possibilities, adolescence is no less difficult. The search for meaning appears to be made more difficult by the infinite and global possibilities that the world offers us today. The search for meaning in our own life often accompanies us for a lifetime. The question of “What is?” And “What is nothing?” Affects classes and cultures across generations. But what happens when someone really leaves, questions the meaning of life critically and thereby calls on the masses to face these thoughts. Conforming mass against the individual, who are we, where do we want to go? What has meaning and what if nothing has meaning? “Nothing” is a danced journey of meaning around the individual thought of meaning.
Premiere: April 23, 2013, Plant 2, Leipzig
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| 280,778
|
The Victorian Economy is forecast to grow by 3.25 per cent in 2006-07, continuing the State’s strong record of growth.
This budget forecasts an operating surplus of $317 million in 2006-07 and surpluses averaging $316 million over the following three years, maintaining Victoria’s triple-A credit rating.
Infrastructure
This budget includes a $4.9 billion capital works program, including;
- $200 million to renew the Regional Infrastructure Development Fund.
- $737 million over four years to improve the Monash-West Gate corridor
- more than $850 million to extend and improve bus, train and tram services, improve rail safety and upgrade rail stations.
- $345 million to upgrade arterial roads across the State.
Tax
In the interests of further reducing the cost of business, this budget outlines measures worth $1.4 billion over the next four years, including;
- payroll tax cut from 5.25 per cent to 5 per cent over three years
- land tax relief worth $167 million over four years
- reducing the middle rates of land tax by 20 per cent
- reducing the top land tax rate to 3 per cent from 2006-07
- increases in land tax liabilities capped for a further year
- indexation factors eliminated
- WorkCover premiums reduced by 10 per cent, saving business $170 million a year.
Tourism
The budget allocates $73 million in funding to boost the Victorian tourism and events industry. This includes;
- $52 million to attract new events to Victoria
- $8 million to attract major business events to the new Melbourne Convention Centre
- $12 million to market Melbourne and Victoria in interstate and international markets.
Schools and Skills
This budget announces an investment of more than $1 billion in schools and skills. This includes;
- a new $300 School Start Bonus for every Victorian child starting Prep or Year 7
- an additional $448 million in education and training infrastructure, including nine new and replacement schools and $58 million to buy land for 11 new schools in Melbourne’s growing outer suburbs
- $20 million to commence major regeneration projects across 25 school sites
- $50 million to fast track school maintenance
- a new $500 Trades Bonus to encourage young people to finish their apprenticeships
Water
This budget provides an additional $160 million for vital water projects, including;
- $30 million towards building a new pipeline to secure Bendigo’s water supply and the future of surrounding irrigators
- $50 million in contributions to the Gippsland Water Factory
- $50 million to the Wimmera Mallee Pipeline
- an extra $25 million for the Murray Darling Basin Commission.
Health
Funding for health in this budget include;
- $87 million to fight obesity, promote health and fitness, and tackle chronic conditions, such as diabetes.
- funding for the redevelopment of the Royal Children’s Hospital.
- $498 million to treat an additional 37 000 hospital patients in 2006-07 with a focus on areas under the greatest strain, such as intensive care, maternity services and neonatal care.
- $10 million to reduce waiting lists
- $114 million to extend diversion programs to reduce hospitalisations for people with chronic and complex conditions.
A Fairer Victoria
The 2006-07 budget provides a further $818 million for A Fairer Victoria – with a strong emphasis on giving Victorian children the best start in life. This includes;
- $268 million to protect vulnerable children, improve the wellbeing of children in care, deliver more early intervention services for families and employ more than 100 extra child protection workers.
- $10 million to provide greater support for disengaged young people
- $25 million to boost services for children in Melbourne’s fastest growing areas
- $170 million to improve mental health services
- $67 million for additional disability support services
- $62 million to continue to work with Indigenous Victorians to tackle disadvantage and strengthen local communities.
Community safety
The budget’s community safety program includes;
- $53 million to continue the upgrade of police stations and courts
- $109 million to fight terrorism and organised crime
- more than $520 million for the next phase of the road safety strategy.
Provincial Victoria
The budget provides more than $800 million for provincial Victoria, including new investment in schools, roads and health and community services and to support regional industries. This includes;
- $100 million for a new Provincial Victoria Growth Fund
- an $11 million package of support for the dairy industry
- $27 million for the energy and resource sector
- a $27 million boost for regional tourism
- an extra $1.5 million to support Victoria’s horticulture industry
Other
- $230 million to support growth in Victorian medical research and life sciences through the Healthy Futures statement (released in April)
- $8.9 million in increased funding to Victoria’s film, TV and digital media industries
- $15 million to boost research in information and communications technologies
- $1.4 million for the APEC Regional Finance centre
| 283,397
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\begin{document}
\title{Pseudo-parabolic category over \\quaternionic projective plane}
\author{
Gareth Jones and Andrey Mudrov\vspace{20pt}
\vspace{10pt}\\
\small Department of Mathematics,\\ \small University of Leicester, \\
\small University Road,
LE1 7RH Leicester, UK\\
}
\date{ }
\maketitle
\begin{abstract}
Quaternionic projective plane $\Hbb P^2$ is the next simplest conjugacy class
of the symplectic group $SP(6)$ with pseudo-Levi
stabilizer subgroup after the sphere
$\Sbb^4\simeq \Hbb P^1$.
Its quantization gives rise to a module category $\O_t\bigl(\Hbb P^2\bigr)$ over finite-dimensional representations of $U_q\bigl(\s\p(6)\bigr)$,
a full subcategory
in the category $\O$. We prove that $\O_t\bigl(\Hbb P^2\bigr)$ is semi-simple
and equaivalent to the category of
quantized equivariant vector bundles on $\Hbb P^2$.
\end{abstract}
{\small \underline{Key words}: quaternionic Grassmannians, quantum symplectic group, module category, cotravariant form.}
\\
{\small \underline{AMS classification codes}: 17B10, 17B37, 53D55.}
\section{Introduction}
With every point $t$ of a maximal torus $T$ of a simple complex algebraic group $G$ one can associate a subcategory $\O_t$
in the category $\O$ of the corresponding quantum group, $U_q(\g)$. This subcategory is
stable under the tensor product with the category $\Fin_q(\g)$ of finite-dimensional (quasi-classical) $U_q(\g)$-modules. As a $\Fin_q(\g)$-module category,
it is generated by a base module $M_t$, whose locally finite part of $\End(M)$, in generic situation, is
an equivariant quantization $\Ac$ of the coordinate ring of $C_t=\Ad_G(t)$, the conjugacy class of $t$.
If the category $\O_t$ is semi-simple, then its objects can be regarded as "representations" of quantum equivariant vector
bundles on $\Ad_G(t)$. According to the famous Serre-Swan theorem \cite{S,Sw}, global sections of vector bundles on an affine variety
are finitely generated projective modules over its coordinate ring. Finitely generated projective right $\Ac$-modules
equivariant with respect to $U_q(\g)$ can be viewed as quantum equivariant vector bundles. They constitute a $\Fin_q(\g)$-module
category, $\Proj_q(\Ac,\g)$.
The equvalence of $\Fin_q(\g)$-module categories $\O_t$ and $\Proj_q(\Ac,\g)$ is established via functors
acting on objects as $\Proj_q(\A,\g)\ni \Gamma \mapsto \Gamma\tp_\Ac M \in \O_t$
and $\O_t\ni M'\mapsto \Hom^\circ_\C(M,M')\in \Proj_q(\Ac,\g)$, where the circle designates the locally finite part.
The module $M$ is absent in the classical picture as there is no faithful irreducible representation of the classical coordinate ring.
Vector bundles on non-commutative spaces are of interest in the non-commutative geometry \cite{C} and its applications to mathematical physics \cite{DN}.
There is one more area of their applications in connection with the theory of symmetric pairs and universal K-matrices,
\cite{Let,Kolb}.
If the class $C_t$ is a symmetric space, there is a solution to the reflection equation
\cite{KS} defining a coideal subalgebra $\Bc\subset U_q(\g)$
and realizing $\Ac$ as the subalgebra of $\Bc$-invariants in the Hopf algebra $\C_q[G]$ dual to $U_q(\g)$.
In the classical limit, $\Bc$ turns into the centralizer $U(\k')$ of a point $t'\in C_t$,
which is isomorphic to the centralizer $U(\k)$ of the point $t$.
The representation theory
of $\Bc$ is a hard topic since $t'\not \in T$ and the triangular decomposition of $\Bc$ is not compatible with that of $U_q(\g)$.
The category $\O_t$, if semi-simple, plays the role of a bridge between $\Proj_q(\A,\g)$ and the category of finite-dimensional
$U(\k)$-modules via a chain of equivalences, see e.g. \cite{M4}.
In the present paper we study the category $\O_t$ for $G=SP(6)$ and $t\in T$ one of $6$ points with
the stabilizer $\simeq SP(4)\times SP(4)$. In this case, $\Ad_G(t)$ is the quaternionic plane $\Hbb P^2$
which enters one of the two infinite series $\Hbb P^n$ of rank one non-Hermitian symmetric conjugacy classes.
The other series comprises even spheres and has been studied in \cite{M4}.
We prove that the base module $M_t$ is irreducible and explicitly construct an orthonormal basis with respect
to the contravariant form on it. Our approach is based on viewing $M_t$ as a module over $U_q(\l)\subset U_q(\g)$,
where $\l \simeq \g\l(2)\op \s\p(2)$ is the maximal reductive Lie subalgebra of $\k$ that $U(\l)$ is quantized as a Hopf
subalgebra in $U_q(\g)$. This is the content of Section 2.
In Section 3, we prove semi-simplicity of the category $\O_t$. It is an illustration of
the complete reducibility criterion for the tensor products
based on a contravariant form and Zhelobenko extremal cocycle \cite{M4,M5,Zh}.
We show that for every finite-dimensional quasi-classical $U_q(\g)$-module $V$ the tensor product
$V\tp M_t$ is completly reducible and its simple submodules are in bijection with simple $\k$-submodules in
the classical $\g$-module $V$. This way we establish equivalence of $\O_t$ and $\Fin(\k)$
as Abelian categories.
\subsection{Quantum group $U_q\bigl(\s\p(6)\bigr)$ and basic conventions}
We fix the notation $\g=\s\p(6)$, $\k= \s\p(4)\op \s\p(2)$ and $\l=\g\l(2)\op \s\p(2)$.
There are inclusions $\g\supset \k\supset \l$ of Lie algebras,
which we describe by inclusions of their root bases as follows.
Both $\k$ and $\l$ are reductive subalgebras of maximal rank, i. e. they contain the Cartan subalgebra $\h$
of $\g$. Fix the inner product on $\h$ such that the long root has length $2$.
There is an orthornormal basis of weights $\{\ve_i\}_{i=1}^{3}\in \h^*$ such that $\Rm^+_\g=\{\ve_i\pm \ve_j\}_{i<j}\cup \{2\ve_i\}_{i=1}^{3}$.
The roots $\al_i=\ve_i-\ve_{i+1}$, $i=1,2$, and $\al_3=2\ve_{3}$ form a basis of simple roots $\Pi_\g$.
The basis of simple roots of $\k$ is $\Pi_\k=\{\al_{1}, 2\al_2+\al_3,\al_{3}\}$.
Note that the root $2\al_2+\al_3$ is not simple,
so $\k$ is not a Levi subalgebra in $\g$. On the contrary, $\l$ is the maximal subalgebra in $\k$ that is Levi in $\g$.
Its basis of simple roots is $\Pi_\l=\{\al_{1}, \al_{3}\}$.
The quantum group $U_q(\g)$ is a $\C$-algebra generated by the simple root vectors (Chevalley generators)
$e_{i}$, $f_{i}$,
and invertible Cartan generators $q^{h_{i}}$, $i=1,2,3$.
The elements $q^{\pm h_{i}}$ generate a commutative
subalgebra $U_\hbar(\h)$ in $U_\hbar(\g)$ isomorphic to the polynomial algebra on a torus.
They obey the following commutation
relations with $e_{i}$, $f_{i}$:
$$
q^{h_i} e_{j}= q^{(\al_i,\al_j)} e_{j}q^{h_i}
\quad
q^{h_i} f_{j}= q^{-(\al_i,\al_j)} f_{j}q^{h_i}
\quad i,j=1,2,3.
$$
Furthermore,
$
[e_{i},f_{j}]=\delta_{ij} \frac{q^{h_{i}}-q^{-h_{j}}}{q-q^{-1}}
$ for all $i,j=1,2, 3$.
The non-adjacent positive Chevalley generators commute while the adjacent generators satisfy the Serre relations
$$
[e_i,[e_i,e_j]_q]_{\bar q}=0, \quad i,j=1,2, \>i\not =j
,
\quad [e_3,[e_3,[e_3,e_2]_{q^2}]]_{\bar q^2}=0,
\quad
[e_2,[e_2,e_3]_q]_{\bar q}=0,
$$
where $[x,y]_a=xy-ayx$, $x,y\in U_q(\g)$, $a\in \C$, and $\bar q= q^{-1}$.
Similar relations hold for the negative Chevalley generators $f_i$ on replacement $f_i\to e_i$,
which extends to an involutive
automorphism of $U_\hbar(\g)$ with $\si(h_i)=-h_i$. Its composition $\omega=\si\circ \gm$
with the antipode $\gm$ preserves multiplication and flips the comultiplication
$$\Delta(f_i)= f_i\tp 1+q^{-h_i}\tp f_i,\quad\Delta(q^{\pm h_i})=q^{\pm h_i}\tp q^{\pm h_i},\quad\Delta(e_i)= e_i\tp q^{h_i}+1\tp e_i.$$
The Serre relations are homogeneous with respect to the $U_q(\h)$-grading via its adjoint action on $U_q(\g)$.
They are determined by the corresponding weight, so we refer to a relation by its weight in what follows.
We remind that a total ordering on the set of positive roots is called normal
if any $\al\in \Rm^+$ presentable in a sum $\al=\mu+\nu$ with $\mu,\nu\in \Rm^+$ lies between $\mu$ and $\nu$.
Recall that a reductive Lie subalgebra $\l\subset \g$ of maximal rank is called Levi if it has a basis $\Pi_\l$ of
simple roots which is a part of $\Pi$.
Then there is an ordering such that every element of $\Rm^+_{\g/\l}$ is preceding all elements of $\Rm_\l$.
In this paper, $\l$ designates the sublagebra $\g\l(2)\op \s\p(2)$.
With a normal ordering one can associate a system $\tilde f_\al\in U_q(\g_-)$ of elements such that normally ordered monomials
in $\tilde f_\al$ form a PBW-like basis in $U_q(\g_-)$. In particular, the algebra $U_q(\g_-)$ is freely generated
over $U_q(\l_-)$ by the ordered monomials in $f_\al$ with $\al\in \Rm^+_{\g/\l}$.
In the classical limit, the elements $\tilde f_\al$ form a basis of root vectors in $\g_-$.
By $\La_\g$ we denote the root lattice of $\g$, i.e. the free Abelian group generated by the fundamental weights.
The semi-group of integral dominant weights is denoted by $\La^+_\g$. All $U_q(\g)$-modules are assumed
diagonalizable over $U_q(\h)$.
A non-zero vector $v$ of $U_q(\h)$-module $V$ its is said to be of weight $\mu\in \h^*$ if
$q^{h_\al}v=q^{(\al,\la)}v$ for all $\al\in \Pi^+$. Vectors of weight $\mu$ span a subspace in $V$ denoted by $V[\mu]$. The set of weights of $V$ is denoted by $\La(V)$.
Character of a $U_q(\h)$-module is defined as a formal sum
$
\sum_{\mu\in \La(V)} \dim V[\mu]_\mu q^{\mu}.
$
We write $\Char(V)\leqslant \Char(W)$ if $\dim V[\mu]\leqslant \dim W[\mu]$ for all $\mu$
and $\Char(V)< \Char(W)$ if this inequality is strict for some $\mu$.
We say that a property holds for all $q$ meaning $q$ not a root of unity. We say it is true for generic $q$ if
it holds upon extension of scalars to the local ring of rational functions in $q$ regular at $q=1$.
\section{Base module for $\Hbb P^2$}
In this section we study a $U_q(\g)$-module $M$ that generates the category of our interest.
We prove its irreducibility and construct an orthonormal basis with respect
to a contraviant form on it.
Set $\dt=2\al_2+ \al_{3}$ and $f_{\dt}=f_2^2 f_{3}-(q^2+\bar q^2)f_2f_{3} f_2+ f_{3} f_2^2$.
It is easy to check that $f_\dt$ commutes with $f_{3}$ and $e_{3}$.
Let $\hat M_\la$ denote the Verma module with highest weight $\la$ such that $q^{2(\la,\ve_{3})}=1$,
$q^{2(\la,\ve_1)}= q^{2(\la,\ve_2)}=-q^{-2}=q^{(\la,\ve_1+\ve_2})$. Define $M$ as the quotient of $\hat M_\la$ by its proper submodules
generated by singular vectors $f_11_\la$, $f_{3}1_\la$, and $f_\dt1_\la$.
It is isomorphic to $U_q(\g_-)/J$ as a $U_q(\g_-)$-module, where $J\subset U_q(\g_-)$ is the left ideal generated by $f_1$, $f_{3}$, $f_\dt$.
The module $M$ supports quantization of the conjugacy class $\Hbb P^2$ in the sense that
an equivariant quantization $\C_q[\Hbb P^2]$ of its affine coordinate ring can be realized as a
$U_q(\g)$-invariant subalgebra in $\End(M)$. An explicit formulation in terms of generators and relations
can be found in \cite{M2}. We do not use it in this presentation.
As $\l$ is a Levi subalgebra in $\g$, its universal enveloping algebra is quantized to a Hopf subalgebra $U_q(\l)\subset U_q(\g)$.
The module $M$ is a quotient of the parabolic Verma module of the same weight, by the submodule generated
by $f_\dt 1_\la$.
It follows that $M$ is locally finite over $U_q(\l)$, \cite{M5}.
\subsection{$U_q(\l)$-module structure of $M$}
We identify a subalgebra $U_q\bigl(\s\l(3)\bigr)\subset U_q(\g)$ that plays a role in this presentation.
Set $\xi=\al_1+\al_2+\al_3$ and $\theta=\al_1+2\al_2+\al_3$ and define
$$ f_\xi=[[f_1,f_2]_{\bar q},f_{3}]_{\bar q^2},\quad f_\theta=[f_2,f_\xi]_{q}
,\quad
e_\xi=[e_{3},[e_2,e_1]_q]_{q^2}, \quad e_\theta=[e_\xi,e_2]_{\bar q}.
$$
Remark that $e_{\phi}$ is proportional to $\si(f_\phi)$ for $\phi=\xi,\theta$.
The set $\{f_\xi, e_\xi, q^{\pm h_\xi}\}$ forms a quantum $\s\l(2)$-triple with
$
[e_\xi,f_\xi]=[2]_q[h_\xi]_q
$.
\begin{propn}
The elements $e_2,f_2,q^{\pm h_2}, e_\xi,f_\xi, q^{\pm h_\xi}$ generate
a subalgebra $U_q(\m)\subset U_q(\g)$ isomorphic to $U_q\bigl(\s\l(3)\bigr)$, with the
set of simple roots $\{\al_2, \xi\}$.
\label{RAsl3}
\end{propn}
\begin{proof}
Observe that the set $\Rm_\m=\{\pm \al_2,\pm \xi, \pm \theta\}\subset \h^*$ is a root system of the $\s\l(3)$-type with
$$
(\xi,\xi)=2, \quad (\al_2,\al_2)=2, \quad (\xi,\al_2)=-1,
$$
so the commutation relations between the Cartan and simple root generators are correct.
Furthermore, it is straightforward to check that $[e_2,f_\xi]=0$ and $[e_\xi,f_2]=0$.
Finally, so long $f_\theta=[f_2,f_\xi]_q$, the Serre relations $[f_\theta,f_2]_{q}=0=[f_\xi,f_\theta]_{q}$ hold by
Propositions \ref{Ap-Serre_step} and \ref{Ap-xi-theta}. This also yields the Serre relations $[e_\theta,e_2]_{q}=0=[e_\xi,e_\theta]_{q}$ via the
involution $\si$.
\end{proof}
\begin{propn}
Vectors $\{f_2^k f_\theta^l 1_\la\}_{k,l\in \Z_+}\subset M$ are $U_q(\l_+)$-invariant.
\label{l-singular}
\end{propn}
\begin{proof}
Both $e_{1}$ and $e_{3}$ commute with $f_2$, so we check their interaction with $f_\theta$.
An easy calculation gives $[e_3,f_\theta]=0$ and $[e_1,f_\theta]=f_\dt q^{h_{1}}$.
Hence $f_2^kf_\theta^l 1_\la$ is $e_{1}$- and $e_3$-invariant, by Corollary \ref{Ap-theta delta}.
\end{proof}
\begin{corollary}
The vector $f_\theta$ belongs to the normalizer of the left ideal $J$.
\end{corollary}
\begin{proof}
Indeed, $ f_\dt f_\theta\in J$ by Corollary \ref{Ap-theta delta}. Furthermore, $f_\theta1_\la$ generates
a finite-dimensional $U_q(\l)$-submodule in $M$. Since $(\la-\theta,\al_{i})=0$ for $i=1,3$, this submodule is trivial,
hence $f_{1}f_\theta$ and $f_{3} f_\theta$ are in $J$.
\end{proof}
We denote by $B$ the set $\{f_2^k f_\theta^l 1_\la\}_{k,l\in \Z_+}\subset M$.
Let $L_{l,k}\subset M$ be the $U_q(\l)$-submodule generated by $f_2^k f_\theta^l 1_\la$
and $L=\op_{l,k=0}^\infty L_{l,k}\subset M$.
Introduce notation $f_{ij}$ for $i\leqslant j$
by setting $f_{ii}=f_i$ and recursively $f_{i,j+1}=[f_{i,j},f_{i+1}]_a$, where $a=q^{(\al_i+\ldots +\al_j,\al_{j+1})}$.
The Serre relations imply
\be
f_{1}f_2^{k}=
[k]_qf_2^{k-1} f_{12} + q^{-k} f_2^{k} f_{1},
\quad
f_{3}f_2^{k}=
-q^2[k]_{q^2}f_2^{k-1} f_{23} + q^{2k} f_2^{k} f_{3} \mod J
\label{aux_comm_rel}
\ee
since $f_\dt$ is central in $U_q(\g_-)$.
\begin{lemma}
For all $k\geqslant 2$,
$
f_{1}f_{3}f_2^{k}=
[k]_{q^2} f_2^{k-2}\Bigl([k]_qf_2f_3 f_1f_2 -\frac{[k-1]_q[2]_q}{(1-\bar q^2)}f_\theta\Bigr).
$
\label{fullness}
\end{lemma}
\begin{proof}
Pushing $f_{3}$ and then $f_{1}$ to the right in $f_{1}f_{3}f_2^{k}$
we find it equal to
$$
-q^2f_{1} [k]_{q^2}f_2^{k-1} f_{23}\mod J=
-q^2[k]_{q^2} [k-1]_q f_2^{k-2} f_{12} f_{23} -q^2q^{-k+1} [k]_{q^2}f_2^{k-1}f_{1} f_{23}\mod J.
$$
Expressing $f_{12} f_{23}$ and $f_2f_{1} f_{23}$ on the right
through $f_2f_\xi$ and $f_\theta$ modulo $J$
we prove the lemma.
\end{proof}
\begin{propn}
$L$ exhausts all of $M$.
\end{propn}
\begin{proof}
It is sufficient to check that the $U_q(\l)$-submodule $L$ is invariant under $U_q(\g_-)$ as it contains $1_\la$.
That is so if and only it is $f_{2}$-invariant.
The vectors $f_{ij}$ quasi-commute with $f_k$, $k=1,3$, unless $k=i-1$ or $k=j+1$.
Therefore
$$
f_2L\subset U_q(\l_-)f_{12}B+U_q(\l_-)f_{23}B+U_q(\l_-)f_{\xi}B+L.
$$
We have $f_{23}B\subset L$ by the right equality in (\ref{aux_comm_rel}). Furthermore,
$U_q(\l_-)f_{12}B \subset L
$
by the left equality in (\ref{aux_comm_rel}).
Finally, since $[f_2,f_\xi]_q=f_\theta$ and $f_\theta$ $q$-commutes with $f_2$ (cf. Proposition \ref{RAsl3}),
Lemma \ref{fullness} implies that $f_\xi B\subset L$.
Then $f_2L\subset \sum_{i=1}^{1}U_q(\l_-)f_{\xi}B+L\subset L$, as required.
\end{proof}
If follows from Proposition \ref{RAsl3} that
\be
[e_{2},f_\theta^k]=[k]_qf_\xi f_\theta^{k-1} q^{-h_2}, \quad [e_\theta^k,f_\xi]=-q^{-(k-1)}[2]_q[k]_qe_\theta^{k-1}e_2q^{-h_\xi}.
\label{U(m)}
\ee
Setting $\la_i=(\al_i,\la)$ we get as a corollary the following identity:
\be
e_{2}f_2^lf_\theta^k 1_\la=[l]_q[\la_3-l-k]_qf_2^{l-1} f_\theta^k 1_\la +[k]_q q^{-\la_2}f_2^l f_\xi f_\theta^{k-1}1_\la.
\label{e2-action}
\ee
\begin{propn}
The module $M$ is irreducible.
\end{propn}
\begin{proof}
It is sufficient to check that none of the $U_q(\l_+)$-singular vectors $f_2^l f_\theta^k1_\la$
with $l+k>0$ is killed by $e_2$.
The operator $e_{1}e_{3}$ annihilates the first term in (\ref{e2-action}) and returns $f_2^{l+1} f_\theta^{k-1}1_\la$, up to a non-zero scalar multiplier, on the second.
Proceeding this way we obtain
$
(e_{1}e_{3}e_2)^kf_2^lf_\theta^k1_\la\varpropto f_2^{k+l}1_\la\not =0.
$
Therefore $f_2^lf_\theta^k1_\la\not =0$ and $f_2^lf_\theta^k1_\la\not \in \ker(e_2)$ unless $l+k=0$. Hence these vectors
are highest for different $U_q(\l)$-submodules in $M$ and none of them is killed by $e_2$.
\end{proof}
In summary, $M$ isomorphic to the natural $U_q\bigl(\g\l(2)\bigr)-U_q\bigr(\s\l(2)\bigl)$-bimodule $\C_q[\End(\C^2)]$.
It is semi-simple and multiplicity free and its character equals $\prod_{\al\in \Rm^+\backslash\Rm^+_\k}(1-q^{-\al})^{-1} q^\la$.
In the classical limit, the subalgebra of $U(\g_+)$-invariants in $\C[\C^2\tp \C^2]\simeq \C[\End(\C^2)]$ is a polynomial algebra of two variables
generated by the principal minors of the coordinate matrix.
\subsection{Orthonormal basis in $M$}
Recall that every highest weight module over a reductive quantum group has a unique contravariant form
with respect to the involution $\omega$ such that the squared norm of the highest vector is $1$.
In this section we construct an orthonormal basis in $M$, with the help of the subalgebra $U_q(\m)$.
\begin{propn}
\label{recurrence for matrix elements}
The assignment $(l,k)\mapsto\tilde c_{l,k}=\langle 1_\la,e_\theta^k e_2^l f_2^l f_\theta^k1_\la\rangle$
is a unique function $\Z_+^2\to \C$ satisfying
$$
\tilde c_{l,k}=-\tilde c_{l,k-1} [2]_q[k]_q^2 q^{-\la_\theta+l+1}+q^{-k} [l]_q[\la_2-l+1]_q\tilde c_{l-1,k}, \quad lk\not =0,
$$
$$
\hspace{-1.4in}\mbox{and }\quad\quad \quad \quad \quad \quad\quad c_{l,0}=[l]_q!\prod_{i=0}^{l-1}[\la_2-i]_q, \quad
\tilde c_{0,k}=[k]_q![2]_q^k\prod_{i=0}^{k-1}[\la_\theta-i]_q, \quad
$$
\end{propn}
\begin{proof}
The boundary conditions easily follow from the basic relations of $U_q(\m)$. Uniqueness is proved by the obvious
induction on $l+k$. To prove the recurrence relation permute $f_\theta^k$ and $f_2^l$.
In the resulting matrix element $q^{-lk}\langle 1_\la,e_\theta^k e_2^l f_\theta^kf_2^l1_\la \rangle$ push one
copy of $e_2$ to the right:
$$
\tilde c_{l,k}=q^{-kl}\langle 1_\la, e_\theta^kf_\xi e_2^{l-1}f_\theta^{k-1} f_2^l 1_\la\rangle [k]_qq^{-\la_2+2l}
+q^{-k}\langle 1_\la, e_\theta^k e_2^{l-1}f_2^{l-1} f_\theta^k 1_\la\rangle [l]_q[\la_2-l+1]_q
$$
$$
=-\tilde c_{l,k-1} [2]_q[k]_q^2 q^{-\la_\theta+l-1}+q^{-k} [l]_q[\la_2-l+1]_q\tilde c_{l-1,k}.
$$
This calculation is actually done in $U_q(\m)$. In particular, we used (\ref{U(m)}) and $[f_2,f_\theta]_{\bar q}=0$.
\end{proof}
\begin{propn}
\label{matrix elements}
The matrix element $c_{l,k}=\langle f_2^l f_\theta^k1_\la,f_2^l f_\theta^k1_\la\rangle$ equals $(-1)^{l+k} q^{k(k-5)+ lk+l(l-1)} \times q^{-l(\la,\al_2)}\tilde c_{l,k}$,
where
$$
\tilde c_{l,k}=[l]_q![k]_q![2]_q^k\prod_{i=0}^{l-1}[\la_2-i]_q\frac{\prod_{i=0}^{l+k-1}[\la_\theta-i]_q}{\prod_{i=0}^{l-1}[\la_\theta-i]_q}.
$$
\end{propn}
\begin{proof}
Let $\bar f_\theta$ be the vector obtained from $f_\theta$ by the replacement $q\to q^{-1}$. One can check that
$q f_\theta+q^{-1} q \bar f_\theta\in J$ and replace $f_\theta$ with $q^{-2} \bar f_\theta$ in the left argument.
Then $c_{l,k}$ equals
$$
\langle f_2^l f_\theta^k 1_\la,f_2^lf_\theta^k 1_\la \rangle=(-1)^kq^{-2k}\langle f_2^lf_\theta^k 1_\la,f_\theta^kf_2^l1_\la \rangle
=(-1)^{l} q^{-2k} \langle 1_\la,(q^{-h_\theta-4}e_\theta)^k(q^{-h_2}e_2)^lf_2^lf_\theta^k1_\la \rangle
$$
since $\omega(\bar f_\theta)=-q^{-h_\theta-4}e_\theta$.
One can express the right hand side through $\tilde c_{l,k}=\langle 1_\la,e_\theta^k e_2^l f_2^l f_\theta^k1_\la\rangle$ and check that $\tilde c_{l,k}$ defined as above satisfies the conditions of
Proposition (\ref{recurrence for matrix elements}).
\end{proof}
\begin{corollary}
The system
$
y^{l,k}_{i,j}=\frac{1}{[i]_q[j]_q\sqrt{c_{l,k}}}f_1^i f_3^j f_2^l f_\theta^k 1_\la$, where $l,k\in \Z_+$ and
$i,j\leqslant l$,
is an orthonormal basis with
respect to the contravariant form on $M_\la$.
\end{corollary}
\section{Category $\O_t(\Hbb P^2_q)$}
While the base module $M$ supports a representation of the function algebra on quantized $\Hbb P^2$,
it generates a family of modules which may be regarded as "representations" of more general vector bundles.
This interpretation is only possible if all such modules are completely reducible - then they give rise
to projective modules over $\C_q[\Hbb P^2]$. They appear as submodules in tensor products $V\tp M$ (corresponding to the trivial vector bundle),
where $V$ is a $U_q(\g)$-module from $\Fin_q(\g)$. Therefore the key issue is complete reducibility of tensor products $V\tp M$.
We solve this problem in the present section using a technique developed in $\cite{M4,M5}$.
\subsection{Complete reducibility of tensor products}
Suppose that $V$ and $Z$ are irreducible modules of highest weight. Each of them has a unique, upon normalization,
nondegenerate contravariant symmetric bilinear form, with respect to the involution $\omega\colon U_q(\g)\to U_q(\g)$.
Define the contravariant form on $V\tp Z$ as the product of the forms on the factors.
Then the module $V\tp Z$ is completely reducible if and only if the form on $V\tp Z$ is non-degenerate when
restricted to the span of singular vectors $(V\tp Z)^+$. Equivalently, if and only if every submodule of highest
weight in $V\tp Z$ is irreducible, \cite{M4}.
For practical calculations, it is convenient to deal with the pullback of the form under
an isomorphism of $(V\tp Z)^+$ with a certain vector subspace in $V$ (alternatively, in $Z$)
which is defined as follows. Let $I^-_Z\subset U_q(\g_-)$ be the left
ideal annihilating the highest vector $1_\zt\in Z$, and $I^+_Z=\si(I^-_Z)$ a left ideal in $U_q(\g_+)$.
Denote by $V^+_Z\subset V$ the kernel of $I^+_Z$, i.e. the subspace of vectors killed by $I^+_Z$.
There is a linear isomorphism between $V^+_Z$ and $(V\tp Z)^+$
assigning a singular vector $u=v\tp 1_\zt+\ldots $ to any weight vector $v\in V^+_Z$. Here we suppressed
the terms whose tensor $Z$-factors have lower weights than the highest weight $\zt$.
The pullback of the contravariant form under the map $V^+_Z\to (V\tp Z)^+$
can be expressed through the contravariant form $\langle -, -\rangle$ on $V$
as $\langle \theta(v), w\rangle$, for a certain operator $\theta$ with values in the dual space to $V^+_Z$.
In this paper, the contravariant form on $V$ is always non-degenerate when restricted to $V^+_Z$,
so we can write $\theta \in \End(V^+_Z)$.
This operator is related with the extremal projector $p_\g$, which is an
element of a certain extension $\hat U_q(\g)$ of $U_q(\g)$, \cite{KT}.
It is constructed as follows. A normal order on $\Rm^+$ defines an
embedding $\iota_\al\colon U_q\bigl(\s\l(2)\bigr)\to U_q(\g)$ for each $\al \in \Rm^+$, \cite{ChP}.
Set $p_\g(\zt)$, $\zt \in \h^*$, to be the ordered product $\prod_{\al\in \Rm^+}^<p_\al\bigl((\zt+\rho, \al^\vee )\bigr)$,
where $p_\al(z)$ is the image of
\be
\label{translationed_proj}
p(z)=\sum_{k=0}^\infty f^k e^k \frac{(-1)^{k}q^{k(t-1)}}{[k]_q!\prod_{i=1}^{k}[h+z+i]_q}
\in \hat U_q\bigl(\s\l(2)\bigr), \quad z\in \C,
\ee
under $\iota_\al$ (sending $q$ to $q_\al=q^{\frac{(\al,\al)}{2}}$ and $q^h$ to $q^{h_\al}$).
For generic $\zt$, the operator $p_\g(\zt)$ is well defined and invertible on every finite-dimensional $U_q(\g)$-module. The specialization $p_\g=p_\g(0)$ is an idempotent satisfying
$e_\al p_\g=0=p_\g f_\al$ for all $\al\in \Pi$.
\begin{thm}[\cite{M5}]
\label{com_red_crit}
Suppose that the maps $p_\g\colon V^+_M\tp 1_{\zt}\to (V\tp M)^+$ is well defined
and $p_\g(\zt)\in \End(V_{Z}^+)$ is invertible. Then $\theta = p^{-1}(\zt)$.
\end{thm}
In the case of our concern, $p_\g$ is well defined, cf. Proposition \ref{ext_proj_reg}.
Furthermore, the operator $p_\g(\zt)$ is rational trigonometric in $\zt$, so it may have poles.
The theorem assumes that such poles can be regularized.
In the special case when all weights in $V^+_Z$ are multiplicity free,
$\det(\theta)\propto \prod_{\al\in \Rm^+}\prod_{\mu\in \La(V)}\theta^\al_\mu$ up to a non-zero factor, with
\be
\theta^\al_\mu=\prod_{k=1}^{l_{\mu,\al}}\frac{[(\zt+\rho+\mu,\al^\vee)+k]_{q_\al}}{[(\zt+\rho,\al^\vee)-k]_{q_\al}}.
\label{theta igenvalues}
\ee
Here $l_{\mu,\al}$ is the maximal integer $k$ such that $\tilde e_{\al}^kV^+[\mu]\not =\{0\}$ for
$\tilde e_\al=\iota_\al(e)$. We compute $\theta$ in
the next section.
\subsection{Regularization of extremal projector}
Denote the positive simple roots of the Lie subalgebra $\k$ by $\bt_1=\al_1, \bt_2=\dt,\bt_3=\al_3$.
The correspondng fundamental weights of $\k$ are
$
\mu_1=\ve_1$, $\mu_2=\ve_1+\ve_2$, $\mu_{3}=\ve_{3}$.
Pick up an integral dominant (with respect to $\k$) weight $\xi=\sum_{s=1}^{3}i_s\mu_s$ with $\vec i= (i_s)_{s=1}^{3} \in \Z^{3}_+$
and set $\zt=\xi+\la$.
The Verma module $\hat M_{\zt}$ of highest weight $\zt$ and highest vector $1_\zt$ has a submodule generated by
singular vectors
$\hat F_s^{i+1}1_{\zt}$, where $\hat F_s=f_s$, $s=1, 3$, and
$$
\hat F_2=\bar q^2\Bigl( f_2^2 f_{3} \frac{[h_2]_p}{[h_2]_q}-f_2 f_{3} f_2[2]_q\frac{[h_2]_q}{[h_2]_q}+ f_{3} f_2^2\Bigr)\in \hat U_q(\b_-),
$$
cf. \cite{M4}, Proposition 2.7. Define also $\hat E_s=\si(\hat F_s)\in \hat U_q(\b_+)$ for $s=1,2,3$.
Denote by $\tilde M_{\vec i}$ the quotient of $\hat M_{\zt}$ by the submodule generated by $\Span\{\hat F_s^{i+1}1_{\zt}\}_{s=1}^{3}$.
The projection $\hat M_\zt\to \tilde M_{\vec i}$ factors through a parabolic Verma module relative to $U_q(\l)$.
Therefore $\tilde M_{\vec i}$ is locally finite over $U_q(\l)$, \cite{M4}.
We use the same notation $1_\zt$ for the highest vector in $\tilde M_{\vec i}$.
Let $\tilde I^-_{\vec i}\subset U_q(\g_-)$ denote the left ideal annihilating the highest vector in $\tilde M_{\vec i}$.
and put $\tilde I^+_{\vec i}=\si(\tilde I^-_{\vec i})\subset U_q(\g_+)$.
They are generated by $\{F^{i_s+1}_s\}_{s=1}^3$ and $\{E^{i_s+1}_s\}_{s=1}^3$, respectively,
where $F^{i_s+1}_s$ the lift of $\hat F^{i_s+1}_s1_{\zt}$ to $U_q(\g_-)$, and $E^{i_s+1}_s=\si(F^{i_s+1}_s)$.
These elements are simply powers of the Chevalley generators for $i=1,3$.
From now to the end of the paper we fix $V=\C^{6}$, the smallest fundamental module of $U_q(\g)$.
Up to non-zero scalar factors, the action of $U_q(\g_+)$ is described by a graph
\be
v_{-1}\stackrel{e_1}{\longrightarrow} v_{-2}\stackrel{e_2}{\longrightarrow} v_{-3}\stackrel{e_3}{\longrightarrow} v_{3}\stackrel{e_2}{\longrightarrow} v_{2}\stackrel{e_1}{\longrightarrow} v_{1}
\label{graph}
\ee
where the vectors $v_{\pm i}$ of weights $\pm \ve_i$, $i=1,2,3$, form
an orthonormal basis with respect to the contravariant form.
The $U_q(\g_-)$-action is obtained by reversing the arrows. Observe that all $e^2_k$ vanish on $V$.
Thus we readily find from the diagram that $\ker(E_s)$ equals
\be
V\ominus\Span\{v_{-1},v_{2}\},\> s=1,\quad
V\ominus \Span\{v_{-2}\}, \> s=2,\quad
V\ominus \Span\{v_{-{3}}\}, \> s=3.
\label{ker_E}
\ee
Furthermore, $\ker(E_s^i)$ is entire $V$ if $i>1$. Similar results for
$\ker(F_s^i)$, $s=1,2,3$, $i\in \Z_+$, are obtained by replacing $v_{i}\to v_{-i}$.
Denote by $\tilde V^+_{\vec i}=\cap_{s=1}^{3} \ker(E^{i_s+1}_s)$ the
kernel of $\tilde I^+_{\vec i}$ in $V$.
All weights in $\tilde V^+_{\vec i}$ are dominant with respect to $\l$ (in fact, with respect to $\k$).
\begin{propn}
\label{ext_proj_reg}
The extremal projector $p_\g\colon \tilde V^+_{\vec i}\tp 1_\zt\to (V\tp \tilde M_{\vec i})^+$ is well defined.
\end{propn}
\begin{proof}
It is argued in \cite{M5} that the factors $p_\al(t)$ for $\al\in \Rm^+_\l$ are regular on $\tilde V^+_{\vec i}\tp 1_{\zt}$
at $t=(\rho,\al^\vee)$.
Suppose that $\al \in \Rm^+_\g-\Rm^+_\l$.
The denominators in $p_\al(t)$ specialized at weight $\eta=\zt+\mu$, $\mu\in \La(\tilde V^+_{\vec i})$, contain
$[t+(\eta,\al^\vee)+k]_{q_\al}$ with $k\in \N$. For $\al\in \Rm^+_\g-\Rm^+_\k$,
this factor is proportional to $q^{m+t}_\al+q^{-t-m}_\al$ for some $m\in \Z$ and does not vanish. Therefore
all such $p_\al(t)$ are regular at $t=(\rho,\al^\vee)$, and the extremal projector of
the subalgebra $U_q(\g^{\al_2})$ is well defined on $V\tp 1_{\zt}$.
Suppose that $\al\in \Rm^+_\k-\Rm^+_\l$. With $\xi=0$, $[(\eta+\rho,\al^\vee)+k]_q$ equals
$$
[(\mu,\al^\vee)+2+k]_{q^2},\quad [(\mu,\al^\vee)+1+k]_{q^2}, \quad [(\mu,\al^\vee)+3+k]_{q},
$$
for $\al=2\ve_1,2\ve_2,\ve_1+\ve_2$, respectively. They are not zero since as $k>0$ and $(\mu,\al^\vee)\in \{-1,0,1\}$
for $\mu\in \La(\tilde V^+_{\vec i})$.
That is {\em a fortiori} true when $\xi\not =0$ because $(\xi,\al^\vee)\in \Z_+$. Therefore such $p_\al(t)$ are
regular on $\tilde V^+_{\vec i}\tp 1_\la$ at $t=(\rho,\al^\vee)$.
It follows that all root factors in $p_\g(\psi)$ are regular on $V\tp 1_{\zt}$ at $\psi=0$, so $p_\g(0)$ is independent of normal order.
For a simple root $\al$ choose an order with $\al$ on the left. Then $e_\al p_\g(0)=0$ on $\tilde V^+_{\vec i}\tp 1_{\zt}$.
We already checked that for $\al=\al_2$, while for $\al=\al_1,\al_3$
this is true because all weights in $\tilde V^+_{\vec i}\tp 1_{\zt}$ are dominant with respect to $\l$, cf. \cite{M5}.
\end{proof}
\noindent
Thus the first condition of Theorem \ref{com_red_crit} is satisfied. The second condition is secured by the following calculation.
\begin{propn}
\label{complete_red}
For all $\xi= \sum_{s=1}^{3}i_s\mu_s$ with $\vec i\in \Z^3_+$, the operator $p_\g(\xi+\la)$ is invertible on $\tilde V^+_{\vec i}$.
\end{propn}
\begin{proof}
Let us calculate $\theta^\al_\mu$, which are inverse eigenvalues
of the root factors constituting $p_\g(\zt)$.
From (\ref{graph}) we conclude that all integers $l_{\mu,\al}$ in (\ref{theta igenvalues}) are at most $1$.
Put $\zt=\la+\xi$. Then (\ref{theta igenvalues}) reduces to $\theta^\al_\mu=1$ for $l_{\al,\mu}=0$ and $\theta^\al_\mu=\frac{[(\zt+\rho+\mu,\al^\vee)+1]_{q_\al}}{[(\zt+\rho,\al^\vee)-1]_{q_\al}}$
for $l_{\al,\mu}=1$.
Observe
that
$$
\theta^{\ve_1-\ve_3}_{-\ve_1},\quad \theta^{\ve_1-\ve_3}_{\ve_3},
\quad
\theta^{\ve_2-\ve_3}_{-\ve_2}, \quad \theta^{\ve_2-\ve_3}_{\ve_3},
\quad
\theta^{\ve_2+\ve_3}_{-\ve_2}, \quad \theta^{\ve_2+\ve_3}_{-\ve_3},\quad
\theta^{\ve_1+\ve_3}_{-\ve_1},\quad\theta^{\ve_1+\ve_3}_{-\ve_3}.
$$
are of the form $\frac{\{m_1\}_q}{\{m_2\}_q}$ for some integers $m_1,m_2$, where
$\{x\}_{q}=\frac{q^{x}+q^{-x}}{q+q^{-1}}$. They cannot turn zero as $q$ is not a root of unity.
The remaining non-trivial factors $\theta^\al_\mu$ are
$$
\theta^{2\ve_1}_{-\ve_1}=\frac{[i_1+i_2+2]_{q^2}}{[i_1+i_2+1]_{q^2}}, \quad
\theta^{2\ve_2}_{-\ve_2}=\frac{[i_2+1]_{q^2}}{[i_2]_{q^2}},
\quad
\theta^{2\ve_3}_{-\ve_3}= \frac{[i_{3}+1]_{q^2}}{[i_{3}]_{q^2}},
$$
$$
\theta^{\ve_1-\ve_2}_{-\ve_1}=
\frac{[i_1+1]_{q}}{[i_1]_{q}}= \theta^{\ve_1-\ve_2}_{\ve_2},\quad
\theta^{\ve_1+\ve_2}_{-\ve_1}=
\frac{[i_1+2i_2+3]_{q}}{[i_1+2i_2+2]_{q}}=\theta^{\ve_1+\ve_2}_{-\ve_3}.
\quad
$$
Observe that the denominator in
$\theta^\al_\mu$ may turn zero only
for $\al\in \Pi_\k$. That happens if $i_s=0$, $s=1,2,3$. However, such $\mu$ do not belong to $\La(\tilde V^{+}_{\vec i})$,
as seen from (\ref{ker_E}).
Since $q$ is not a root of unity, all $\theta^\al_\mu$ never turn zero. Therefore, $p_\g(\zt)$ is invertible, and $\theta=p_\g(\zt)^{-1}$.
\end{proof}
\subsection{Semi-simplicity of $\O_t(\Hbb P^2)$}
Denote by $M_{\vec i}$ the irreducible quotient of $\tilde M_{\vec i}$ (we conjecture that they coincide, at least they do for generic $q$).
We define $V_{\vec i}^+$ as the kernel of the left ideal $I^+_{\vec i}=\si(I^-_{\vec i})$,
where $I^-_{\vec i}$ is the annihilator of the highest vector in $M_{\vec i}$. Obviously $V^+_{\vec i} \subseteq \tilde V^+_{\vec i}$
because $\tilde I_{\vec i}^+\subseteq I_{\vec i}^+$.
The subspace $V^+_{\vec i}$ is isomorphic to the span of singular vectors in $V\tp M_{\vec i}$.
In principle, $\tilde V^+_{\vec i}$ might be bigger than $V^+_{\vec i}$ but we shall see that they coincide.
\begin{propn}
For all $\vec i\in \Z^3_+$, the tensor product $V\tp M_{\vec i}$ is completely reducible.
\label{VMi-comp-red}
\end{propn}
\begin{proof}
Since $V^+_{\vec i}\subseteq \tilde V^+_{\vec i}$ and $M_{\vec i}$ is a quotient of $\tilde M_{\vec i}$,
the operator $p_\g\colon V^+_{\vec i}\tp 1_\zt\to (V\tp M_{\vec i})^+$ is well defined,
by Proposition \ref{ext_proj_reg}. The operator $p_\g(\zt)$ is invertible on $V^+_{\vec i}$
by Proposition \ref{complete_red}. This proves the assertion thanks to Theorem \ref{com_red_crit}.
\end{proof}
Our next goal is to describe the irreducible summands in such tensor products.
Identify $V$ with the classical $\k$-module and denote by
$f_{\beta_s},e_{\beta_s}\in \k$ its simple root vectors.
\begin{lemma}
For all $\zt\in \La^+_\k+\la$, $\ker (F_s^i)\simeq \ker (e_{\beta_s}^i)$ and $\ker (E_s^i)\simeq \ker (e_{\beta_s}^i)$,
where $s=1,2,3$, and
$i\in \Z_+$.
\label{class-quant}
\end{lemma}
\begin{proof}
Elementary calculation. Remark that $F_s^i$ and $E_s^i$ are regular at the specified weights.
\end{proof}
\noindent
Let $X_{\vec i}$ denote the classical $\k$-module of highest weight $\xi= \sum_{s=1}^{3}i_s \mu_s$.
\begin{corollary}
\label{classical_ext_space}
\begin{enumerate}
\item The vector space $\tilde V^+_{\vec i}$ is isomorphic to $(V\tp X_{\vec i})^{\k_+}$.
\item For every $\vec i\in \Z_+^3$ and $v\in V^+_{\vec i}[\mu]$ there is a singular
vector $u=v\tp 1_{\la+\xi}+\ldots $,
and a module homomorphism $\tilde M_{\vec i'}\to V\tp M_{\vec i}$, where $i'_s=(\xi+\mu,\bt_s^\vee)$,
extending $1_{\la+\xi+\mu}\mapsto u$.
\end{enumerate}
\end{corollary}
\begin{proof}
The first statement is due to the isomorphism $\tilde V_{\vec i}^+\simeq \cap_{s=1}^3\ker (e_{\bt_s}^{i_s+1})$ because the
right-hand side is in bijection with the span of singular vectors in the $\k$-module $V\tp X_{\vec i}$.
The singular vector $u=v\tp 1_{\la+\xi}+\ldots $ exists due to irreducibility of $M_{\vec i}$.
So there is a homomorphism $\hat M_{\la+\xi+\mu}\to V\tp M_{\vec i}$
of the Verma module assigning $u$ to $1_{\la+\xi+\mu}$. The vector $v$ belongs to $\bigl(\cap_{s=1}^3\ker(e_s^{i_s+1})\bigr)\cap \bigl(\cap_{s=1}^3\ker(f_s^{i'_s+1})\bigr)$ and thus to $\bigl(\cap_{s=1}^3\ker(E_s^{i_s+1})\bigr)\cap \bigl(\cap_{s=1}^3\ker(F_s^{i'_s+1})\bigr)$, by Lemma \ref{class-quant}.
Hence the homomorphism $\hat M_{\la+\xi+\mu}\to V\tp M_{\vec i}$ factors through $\tilde M_{\vec i'}\to V\tp M_{\vec i}$.
\end{proof}
For each $\vec i\in \Z_+^3$, introduce a set of triples $\tilde I(\vec i)\subset \Z_+^3$ labelling weights in $\tilde V_{\vec i}$.
We put
\be
\tilde I(\vec i)=\bigl\{(i_1\pm 1,i_2, i_{3}), \> (i_1, i_2, i_{3}\pm 1), \> (i_{1}\pm 1,i_{2}\mp 1,i_3)\bigr\},
\label{checked summands}
\ee
where
the triples with negative coordinates are excluded.
Since $M_{\vec i}$ is a quotient of $\tilde M_{\vec i}$,
singular vectors in $V\tp M_{\vec i}$ may have only weights $\sum_{s=1}^{3} i'_s\mu_s+\la$
with $\vec i'\in \tilde I(\vec i)$, by Corollary \ref{classical_ext_space}, 2).
Let $I(\vec i)\subseteq \tilde I(\vec i)$ denote the subset of such triples.
\begin{lemma}
If $\Char (M_{\vec i})=\Char (M)\Char (X_{\vec i})$ for all $q$,
then $V^+_{\vec i}=\tilde V^+_{\vec i}$ and $\Char (M_{\vec i'})=\Char (M)\Char (X_{\vec i'})$ for all $\vec i'\in I(\vec i)=\tilde I(\vec i)$. Furthermore,
$\tilde M_{\vec i'}\simeq M_{\vec i'}$ for generic $q$.
\label{check-to-unckeck}
\end{lemma}
\begin{proof}
Remind that all $q$ means but roots of unity.
By Lemma \ref{complete_red}, $V\tp M_{\vec i}$ is completely reducible. Therefore all submodules in $V\tp M_{\vec i}$ are $M_{\vec i'}$ with $\vec i' \in I(\vec i)$.
By deformation arguments (see e. g. \cite{M4}), we have
$
\Char(\tilde M_{\vec i})\leqslant \Char (X_{\vec i'})\Char(M)
$
for generic $q$. Then
$$
\Char(V)\Char(M_{\vec i})=\sum_{\vec i'\in I(\vec i)} \Char(M_{\vec i'})\leqslant \sum_{\vec i'\in I(\vec i)} \Char( \tilde M_{\vec i'})
\leqslant \sum_{\vec i'\in \tilde I(\vec i)} \Char( X_{\vec i'})\Char(M)
$$
for generic $q$.
But the leftmost term is equal to the rightmost term for all $q$, by Corollary \ref{classical_ext_space}, 1).
This is possible if and only if $\tilde I(\vec i)=I(\vec i)$
and $\Char(\tilde M_{\vec i'})=\Char(M_{\vec i'})$ for all $\vec i'\in I(\vec i)$. This proves the statement
for generic $q$. The module $M_{\vec i'}$ is rational in $q$ and it is a submodule
in a rational module $V\tp M_{\vec i}$ that is flat at all $q$ by the hypothesis. Therefore
$\Char(M_{\vec i'})\leqslant \Char( X_{\vec i'})\Char(M)$ for all $q$.
But the strict inequality or $\tilde I(\vec i)\not =I(\vec i)$ lead to
$\Char(V)\Char(X_{\vec i})<\sum_{\vec i'\in \tilde I(\vec i)} \Char( X_{\vec i'})$, which is absurd.
\end{proof}
Denote by $\O_t(\Hbb P^2)$ the full subcategory in the category $\O$ whose objects are submodules in $W\tp M$,
where $W$ is a quasiclassical finite-dimensional module over $U_q(\g)$. By construction,
it is a module category over quasi-classical finite-dimensional representations of $U_q(\g)$.
Denote by $\mathrm{Fin}(\k)$ the category of finite-dimensional $\k$-modules. It is a module category
over $\mathrm{Fin}(\g)$ via the restriction functor.
\begin{propn}
The modules $M_{\vec i}$ are in $\O_t(\Hbb P^2)$ for all $\vec i\in \Z_+^3$.
\label{all Verma in PPO}
\end{propn}
\begin{proof}
Essentially it is sufficient to prove that every irreducible finite-dimensional $\k$-module can be realized
as a submodule in a tensor power of $V$.
We do induction on $|\vec i|=i_1+i_2+ i_{3}$ applying
Lemma \ref{check-to-unckeck}.
For $|\vec i|=0$, $M_{\vec i}$ is the base module $M$, which satisfies the hypothesis of Lemma \ref{check-to-unckeck}.
Suppose that the statement is proved for all $M_{\vec i}$ with $m=|\vec i|\geqslant 0$.
Fix an index $\vec i$ with $|\vec i|=m+1$ and let $\ell$ be the minimal $s$ such that $i_s>0$.
We consider separately the following two cases depending on the value of $\ell$.
If $\ell=3$ then $\vec i \in \tilde I(\vec i^{3})$, where the multi-index $\vec i^{3}$ has zero
coordinates but $i^{3}_{3}=i_{3}-1$.
Since $|\vec i^{3}|=m$, $M_{\vec i}$ is in $\O_t(\Hbb P^2)$.
If $\ell\leqslant 2$, define a sequence $\vec i^l\in \Z^{3}_+$ for $l=0,\ldots, \ell$ as follows.
Set $i^0_s=i_s-\dt_{1 s}$ and $i^l_s=i_s+\dt_{ls}-\dt_{\ell s}$ for $l=1,\ldots, \ell$.
Since $|\vec i^0|=m$, $M_{\vec i^0}$ is in $\O_t(\Hbb P^2)$ by the induction assumption.
Now observe that $\vec i^{l+1}\in \tilde I(\vec i^{l})$ for $0\leqslant l\leqslant \ell-1$ and $\vec i^\ell=\vec i$.
Ascending induction on $l$ proves that $ M_{\vec i^l}$ are in $\O_t(\Hbb P^2)$, for all $l=1,\ldots ,\ell$.
This completes the proof.
\end{proof}
Now we summarize the main result of the paper.
\begin{thm}
\begin{enumerate}
\item $\O_t(\Hbb P^2)$ is semi-simple.
\item $\O_t(\Hbb P^2)$ is equivalent to the category $\mathrm{Fin}(\k)$.
\item $\O_t(\Hbb P^2)$ is equivalent to the category of equivariant finitely generated projective modules
over the quantized function algebra $\C_q[\Hbb P^2]$, for generic $q$.
\end{enumerate}
\end{thm}
\begin{proof}
The category $\O_t(\Hbb P^2)$ is clearly additive.
To prove the first statement, observe that a module $V$ from $\Fin_q(\g)$ can be realized as a submodule
in a tensor power of $\C^{6}$. Then apply Propositions \ref{VMi-comp-red} and \ref{all Verma in PPO}.
Equivalence $\O_t(\Hbb P^2)\sim \mathrm{Fin}(\k)$ as Abelian categories can be proved similarly to \cite{M4}, Proposition 3.8.
Finally, for generic $q$, $\C_q[\Hbb P^2]$ is the locally finite part of $\End(M)$, hence invariant idempotents
in $\End(W)\tp \C_q[\Hbb P^2]$ are exactly those in $\End(W\tp M)$. Equivariant projective $\C_q[\Hbb P^2]$-modules are then in a
natural correspondence with $U_q(\g)$-modules in $\O_t(\Hbb P^2)$. This correspondence respects tensor multiplication
by modules from $\Fin_q(\g)$.
\end{proof}
Concerning an explicit description of pseudoparabolic Verma modules, we have proved that $\tilde M_{\vec i}$ and $M_{\vec i}$ are isomorphic upon extension of scalars to the local ring of rational functions in $q$
regular at $q=1$. It is natural to expect that they are isomorphic at all $q$.
\appendix
\section{Generalized Jacobi identity}
In this technical section, we establish some identities in the subalgebra $U_q(\g_-)$, which
are crucial for this exposition. This material is also an illustration of modified Jacobi identity,
which may be of some interest.
\subsection{}
In the algebra $U_q(\g)$, the usual commutator $[x,y]$ has no preference over
$[x,y]_a$ due to the lack of the underlying Lie structure. In this situation, a modified Jacobi identity
\be
[x,[y,z]_a]_b=[[x,y]_c,z]_{\frac{ab}{c}}+c[y,[x,z]_{\frac{b}{c}}]_{\frac{a}{c}}
\label{Jacobi}
\ee
appears to be useful. It holds true for any elements $x,y,z$ of an associative algebra and any scalars $a,b,c$ with invertible $c$.
Here is an example of its application.
\begin{lemma}
\label{Ap-Great Auxiliary}
Suppose elements $y,z,x$ of an associative algebra satisfy the identities
\be
[x,[x,y]_r]_{\bar r}=0, \quad [y,[y,z]_s]_{\bar s}=0,\quad
[x,z]=0.
\label{auxiliary}
\ee
for some invertible scalars $r,s$ such that $[2]_r\not=0$.
Then
$
[[x,y]_{r},[[x,y]_{r},z]_{s}]_{\bar s}=0.
$
\end{lemma}
\begin{proof}
We apply $[x,[x,-]_{r^2}]$ to first identity. The commutator $[x,-]_{r^2}$ gives
$$
0=[x,[y,[y,z]_s]_{\bar s}]_{r^2}=
[[x,y]_r,[y,z]_s]_{\bar s r}+r[y,[[x,y]_{r},z]_{s}]_{\bar s \bar r}.
$$
Application of $[x,-]$ to the first term in the right hand side gives
$$
[x[[x,y]_r,[y,z]_s]_{\bar s r}]=\bar r[[x,y]_r,[[x,y]_{r},z]_{s}]_{\bar s r^2},
$$
where (\ref{Jacobi}) with $c=\bar r$ and the left and right identities in (\ref{auxiliary}) were used.
Commutator of $x$ with the second term (without factor $r$) gives
$$
[x,[y,[[x,y]_{r},z]_{s}]_{\bar s\bar r}]=[[x,y]_r,[[x,y]_{r},z]_{s}]_{\bar s\bar r^2}+
r[y,[x,[[x,y]_{r},z]_{s}]_{\bar r}]_{\bar s \bar r^2}=[[x,y]_r,[[[x,y]_{r},z]_{s}]_{\bar s\bar r^2}
$$
via (\ref{Jacobi}) with $c=a$ and the right and left identities in (\ref{auxiliary}).
Collecting the results we get
$$
0=\bar r[[x,y]_r,[x,y]_{r},z]_{s}]_{\bar s r^2}+r[[x,y]_r,[[x,y]_{r},z]_{s}]_{\bar s \bar r^2}=
(\bar r+r)[[x,y]_r,[x,y]_{r},z]_{s}]_{\bar s }
$$
as required.
\end{proof}
Remark that the hypothesis of the lemma is symmetric with respect to replacement
of $a$ by $a^{-1}$, as well as $b$ by $b^{-1}$. These replacements
can be made arbitrarily.
\subsection{}
Define $\bar f_\theta$ obtained from $f_\theta$ by replacement $q\to \bar q$.
One has
\be
f_\theta&=&[f_2,[[f_{1},f_2]_{\bar q},f_{3}]_{\bar q^2}]_q=\bar q[[f_{1},f_2]_{\bar q},[f_2,f_{3}]_{q^2}]_{\bar q},
\\
\Bar f_\theta&=&[f_2,[[f_{1},f_2]_{q},f_{3}]_{q^2}]_{\bar q}=q[[f_{1},f_2]_{q},[f_2,f_{3}]_{\bar q^2}]_{q}.
\ee
The equalities in the right-hand side follow from (\ref{Jacobi}), with $c=\bar q$ in (\ref{Jacobi}), and the Serre relation of weight $-(2\al_2+\al_{1})$.
Remark that
$qf_\theta+\bar q\bar f_\theta=[f_1,f_\dt]\in J$, which can be proved via (\ref{Jacobi}).
\begin{lemma}
\label{Ap-theta-beta}
The vectors $f_\theta$ and $\Bar f_\theta$ commute with $f_{3}$.
\end{lemma}
\begin{proof}
Applying (\ref{Jacobi}) to $[f_{3},[f_2,f_\xi]]$ with $c=q^2$, we kill the second term
due to the Serre relation of the weight $-2\al_{3}-\al_2$. The result is
$$
[f_{3},[f_2,f_\xi]_{\bar q}]_q]=[[f_{3},f_2]_{q^2},[f_{1},[f_2,f_{3}]_{\bar q^2}]_{\bar q}]_{\bar q}
=\bar q^{3}[[f_{3},f_2]_{q^2},[[f_{3} ,f_2]_{q^2},f_{1}]_{q}]_{\bar q}=0.
$$
The last equality is a specialization of Lemma \ref{Ap-Great Auxiliary} with $x=f_{3}$, $y=f_2$, and $z=f_{1}$.
\end{proof}
\begin{propn}
\label{Ap-Serre_step}
Suppose that $[{3}]_q\not =0$. Then the elements $f_2$, $f_\theta$ and $\Bar f_\theta$ satisfy the relations
$$
f_2 f_\theta=\bar q f_\theta f_2, \quad f_2\Bar f_\theta=q\Bar f_\theta f_2.
$$
\end{propn}
\begin{proof}
Observe that these equalities are flipped under the replacement $q\to q^{-1}$,
so we will prove only left one.
Let us introduce the shortcut $w=[f_2,f_{3}]_{q^2}$.
Starting with the Serre relation of weigth $-(3\al_n+\al_{3})$ we get
$$
0=[f_{1},[f_2,[f_2,w]_{\bar q^2}]]_{\bar q}=[[f_{1},f_2]_{\bar q},[f_2,w]_{\bar q^2}]+\bar q[f_2,[f_{1},[f_2,w]_{\bar q^2}]]_{q}.
$$
We used (\ref{Jacobi}) for $x=f_{1}$ and $c=\bar q$. Further by Serre relation we mean
the one of weight $-(2\al_2+\al_{1})$.
Applying (\ref{Jacobi}) with $c=\bar q$ to $[f_{1},[f_2,w]_{\bar q^2}]$ in the second term
we get
$$
0=[[f_{1},f_2]_{\bar q},[f_2,w]_{\bar q^2}]+
\bar q[f_2,[[f_{1},f_2]_{\bar q},w]_{\bar q}]_{q}
+
\bar q^2[f_2,[f_2,[f_{1},w]_{q}]_{\bar q}]_{q}.
$$
The second term equals $[f_2,f_\theta]_{q}$.
In the third term, present $[f_{1},w]_{q}$ as $[[f_{1},f_2]_{q},f_{3}]_{q^2}$ and
apply the Jacobi identity with $x=f_2$ and $c=q$ to $[f_2,[[f_{1},f_2]_{q},f_{3}]_{q^2}]_{\bar q}$.
Thanks to the Serre relation, the third term turns to $\bar q[f_2,[[f_{1},f_2]_{q},[f_2,f_{3}]_{\bar q^2}]_{q}]_{q}=\bar q^2[f_2,\Bar f_\theta]_{q}$.
Finally, apply the Jacobi identity with $x=[f_{1},f_2]_{\bar q}$, $c=q$ to the first term and using the Serre relation transform
it to $q[f_2,[[f_{1},f_2]_{\bar q},w]_{\bar q}]_{\bar q^3}=q^2[f_2,f_\theta]_{\bar q^3}$.
Thus we arrive at the equation
$$
0=q^2[f_2,f_\theta]_{\bar q^3}+
[f_2,f_\theta]_{q}
+
\bar q^2[f_2,\Bar f_\theta]_{q}.
$$
This leads to the first line of the following system
\be
f_2\bigl((q^2+1)f_\theta+\bar q^2 \Bar f_\theta\bigr)
&=&
\bigl((\bar q+q)f_\theta +\bar q\Bar f_\theta\bigr)f_2
\label{Delta'}
\\
f_2\bigl(q^2 f_\theta+(\bar q^2+1)\Bar f_\theta\bigr)
&=&
\bigl(qf_\theta+(\bar q+q)\Bar f_\theta\bigr)f_2
\label{Delta''}
\ee
The second line is obtained by the replacement $q\to q^{-1}$.
Multiply (\ref{Delta''}) by $\bar q$ and (\ref{Delta''}) by $q+\bar q$, then subtract one from another
and get
$[3]_qf_2 f_\theta=\bar q [3]_qf_\theta f_2$, which implies the statement.
\end{proof}
\begin{corollary}
The vectors $f_\theta$ and $f_\dt$ satisfy $f_\dt f_\theta= \bar q^2 f_\theta f_\dt$.
\label{Ap-theta delta}
\end{corollary}
\begin{lemma}
\label{Ap-nu-theta}
Suppose that $[2]_q\not =0$ and put $f_\nu=[f_1,f_2]_{\bar q}$. Then
$
f_\nu f_\theta=qf_\theta f_\nu.
$
\end{lemma}
\begin{proof}
We will use relations $[f_2,f_\theta]_{\bar q}=0$, and
$[f_2,f_\nu]_{\bar q}=0=[f_1,f_\nu]_q$.
We start with the equality
\be
[f_1,f_\theta]=[[f_1,f_2]_{\bar q},f_\xi]_{q^2}
\label{f1Delta}
\ee
obtained via (\ref{Jacobi}) with $c=\bar q$ using the presentation $f_\theta=[f_\nu,f_\xi]_q$
and the equality $[f_1,f_\xi]_q=0$. Taking commutator of $f_2$ with the lef-hand side we get
$$
[f_2,[f_1,f_\theta]] =[[f_2,f_1]_{q},f_\theta]_{\bar q},
$$
again using (\ref{Jacobi}) with $c=q$, where the second term is gone due to $[f_2,f_\theta]_{\bar q}=0$.
Taking commutator of $f_2$ with the right-hand side of (\ref{f1Delta}) we get
$$
[f_2,[[f_1,f_2]_{\bar q},f_\xi]_{q^2}]
=\bar q[[f_1,f_2]_{\bar q},[f_2,f_\xi]_{q}]_{q^3},
$$
using (\ref{Jacobi}) with $c=z\bar q$; the first term is killed by the Serre relation of weight $\al_1+2\al_2$.
We arrive at the equality
$$
-q[[f_1,f_2]_{\bar q},f_\theta]_{\bar q}
=\bar q[[f_1,f_2]_{\bar q},f_\theta]_{q^3}
\quad\mbox{or}\quad
-f_\nu f_\theta (q+\bar q)
=-f_\theta f_\nu q(\bar q+q),
$$
as required.
\end{proof}
Since $f_\xi=[f_\nu,f_3]_{\bar q^2}$ and in view of Lemma \ref{Ap-theta-beta}, we come up with the following.
\begin{propn}
\label{Ap-xi-theta}
If $[2]_q\not =0$, then the identity $[f_\xi,f_\theta]_q=0$ holds true.
\end{propn}
| 132,857
|
TITLE: How to know there is a subgroup of order $p^{n(n-1)/2}$ in Aut$\left({\mathbb{Z}\over{p\mathbb{Z}}}\right)^n$ without using Sylow theorem?
QUESTION [3 upvotes]: Suppose we have the $V=\left({\mathbb{Z}\over{p\mathbb{Z}}}\right)^n$, $p$ is a prime number. Then we can prove the order of the group $\operatorname{Aut}(V)$ is $p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)...(p-1)$ ($\operatorname{Aut}(V)$ consists of all the automorphisms in $V$).
To prove this, the basic idea is that we can suppose $e_{1}=({1,0, \ldots, 0}), e_{2}=(0,1,0, \ldots, 0), \ldots e_{n}=(0,0, \ldots, 1)$. If $u \in \operatorname{Aut}(V)$, then $u(e_{2}) \notin \langle u(e_{1}) \rangle, u(e_{3})\notin \langle u(e_{2}),u(e_{1}) \rangle$ and so on. And the order of $\langle u(e_{1}),u(e_{2})...u(e_{i}) \rangle = p^i$, then the $u(e_{i+1})$ can be $p^n-p^i$ possibilities. Based on this fact, we can finally prove the order of the group $\operatorname{Aut}(V)$ is $p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)...(p-1)$.
Then, how can we prove that there is subgroup whose order is $p^{n(n-1)\over 2}$ in $\operatorname{Aut}(V)$ without using Sylow theorem?
Can someone help me prove it?
REPLY [2 votes]: Write $N$ for the upper triangular matrices, with coefficients in $\mathbb F_p = \mathbb Z/p\mathbb Z$, and with $1$ along the diagonal. $N$ is a sub-group of the $\mathop{\rm Aut}(\mathbb F^n_p)$ - closed under multiplication, inversion, and contains the identity matrix - check this! Above the diagonal, there are $d= (n-1)n/2$ entries (slots) for arbitrary elements of $\mathbb F_p$, so the cardinality of $N$ is $p^d$.
| 69,798
|
You will need
- calculator.
Instruction
1
To calculate root of the third degree, take a calculator designed for engineering calculations. To calculate root of the third degree, use the equivalent function of raising to the power 1/3.
2
To raise a number to the power 1/3, enter that number, then click on exponentiation and enter the approximate value of the number 1/3 - 0,333. Such accuracy is sufficient for most calculations. However, the accuracy of calculation is very easy to improve – just add as many triples, how many will fit on the display of the calculator (for example, 0,3333333333333333). Then click "=".
3
To calculate root of the third degree using the computer, run the Windows calculator. The procedure for calculating root of third degree fully similar to that described above. The only difference is in the design of the buttons exponentiation. On the virtual keyboard of the calculator it is labelled as "x^y".
4
The root of the third degree can be calculated in MS Excel. To do this, enter in any cell of the symbol "=" and select the "insert function" (fx). Select menu option "DEGREE" and click "OK". In appeared window enter the number for which you want to calculate root of the third degree. In the window, "Degree", enter the number "1/3". Dial the number 1/3 exactly in this form – as an ordinary fraction. After that, click "OK". In that cell of the table where the created formula will appear in the cube root of a given number.
5
If the root of the third degree has to calculate constantly, then slightly improve the above-described method. As the number from which you want to extract the root, specify not the number itself, and the cell of the table. After that, each time you enter in that cell of the original number in the cell with the formula will appear its cube root.
Note
Conclusion. In this paper we have discussed various methods of calculating values of the cube root. It turned out that the values of the cube root can be found with the help of the method of iterations, it is also possible to approximate the cube root, raise a number to the power of 1/3 to find the values of the root of the third degree with Microsoft Office Ecxel, setting formulas in cells.
Useful advice
The roots of the second and third degree are used especially often and thus have special names. Square root: In this case the exponent is usually omitted, and the term "root" without indicating the extent often implies the square root. Practical computation of the roots of the algorithm for finding the root of n-th degree. Square roots and cube roots are usually included in all the calculators.
| 294,371
|
- KO
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We have embraced the Kaizen philosophy for quality manufacturing and continuous improvement since 1996, operating in a cellular manufacturing environment. The combination of cellular manufacturing and Kaizen events has resulted in our ability to flex production to meet changing customer requirements, while leading to greater manufacturing efficiencies, superior quality, and effective throughput.
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Becoming A Christian
Have you ever wondered any of the following:
"What do I have to do to go to heaven?"
"What steps must I take to become a Christian?"
According to the Bible, the answer to these questions contains some bad news and some good news.
The BAD NEWS.
The bad news is, you can never do enough or be good enough to earn your salvation. No amount of religious practice can make you deserving of heaven. Why? Because ALL of us are sinners, unfit to dwell in the presence of a perfect, pure and holy God. The Bible says,
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The wages of sin - what we deserve - is death. We all face physical death, which is a result of sin. But a worse death is spiritual death that alienates us from God for all of eternity. The Bible plainly teaches that there is a place called Hell where people who are not saved while on earth will be in torment forever. Now THAT'S bad news!
But THANK GOD, there is some GOOD NEWS!
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"But God demonstrates his own love for us in this:
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so that your sins may be forgiven. And you will receive the gift of the Holy Spirit" (Acts 2:38).
Notice that along with repentance, Peter also instructed them to be baptized. Baptism means to immerse under water. It is an outward demonstration of what God does inwardly to your heart. When you surrender your life to Christ, you are putting to death your old sinful nature, and your record is wiped clean as all your sins are washed away. You then begin a new life, living for Jesus instead of for self. The sin that used to be in your heart is replaced by God's presence (the Holy Spirit). Going under the water is a clear outward expression of burying the old sinful nature. Rising out of the watery grave, just as Jesus arose from the tomb, demonstrates the new and eternal life you now enjoy because of God's amazing grace. Once you begin a relationship with Jesus, your life will never be the same. You will do "good works" and obey His teachings as found in Scripture, not because you HAVE to in order to merit salvation, but because you WANT to, out of love, appreciation, and devotion to your Creator and Savior. The Bible).
You will never follow the teachings and commands of the Bible perfectly, for temptation and evil will continue to entice you. But the more you grow in your love and devotion to Jesus, the more you will grow to be LIKE Jesus. And THAT'S GOOD NEWS!
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\section{Conclusion and future work} \label{sec:conc}
In order to account for the heterogeneity of real-world power grid networks, we designed a generative random graph model that distinguishes between nodes and edges of different type, according to their voltage level. We found that same-voltage subgraphs in real power grid graph data exhibited an unusual combination of structural properties which make modeling them accurately difficult. We proposed a two-phase model which generates both the same-voltage subgraphs as well as the transformer edges that connect them, according to tunable user-specified input. We found that this model either matched or outperformed the Chung-Lu model in nearly all categories tested, particularly with regard to the large diameter and average distance observed in the data.
The close match of these structural properties were also seemingly reflected in visualizations that, even at small scales, bore resemblance to the original graph data. Lastly, our model also produced graphs with comparable resiliency to single edge failures.
{Ultimately, our model may be used to generate synthetic graph data, test hypotheses and algorithms at different scales, and serve as a baseline model on top of which further electrical network properties, device models, and interdependencies to other networks, may be appended. In particular, given that our model only requires desired degrees and diameter, users may easily generate power grid graphs using artificially generated inputs, under guidelines identical or similar to those we provided above. This feature is attractive in light of the limited availability of real power grid data: given that these guidelines don't require user-input beyond specifying the desired number of vertices of a given voltage level, our model may still be used in the complete absence of real data.}
{Many questions remain for future work. First, we note possible areas for improvement in our model and refinement in our analyses. Perhaps most notably, in Section \ref{sec:agg}, we observed the accuracy of the model in matching diameter declined for the aggregate graphs, which can likely be attributed to choosing leaf vertices of $k$-stars uniformly at random in the stars algorithm. Accordingly, one possible avenue to improving our model would be to devise a weighted sampling scheme for leaf selection, which we speculate might also produce more accurate structural match in the interconnection graphs. Beyond visualizations, we didn't investigate the structural properties of these interconnection networks; indeed, taking into account both vertex and edge-weights makes this analysis more complicated. With regard to resiliency, while we examined single edge failures, we didn't consider scenarios of multiple simultaneous or cascading edge failures, which would provide a more extensive perspective. And lastly, while our model generates static graphs, real-world physical transmission systems are evolving in time, and it would be advantageous to capture such changes. }
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TITLE: Probabilities of no ace and exactly not one ace in throwing 10 dies
QUESTION [1 upvotes]: The probabilities i am looking for are used for a problem in Feller's introduction to probability that asks for the probability $p$ of finding $2$ or more aces in throwing $10$ dies. For this problem i considered that:
$p = 1-x-y$
Where $x$ is the probability of finding no ace at all, and $y$ being the probability of finding exactly one ace. Of course $x = \frac{5^{10}}{6^{10}}$... i think. But i am a little more confused about finding $y$, more because the answer to this exercise is that: $p = 1- 10\times\frac{5^9}{6^{10}-5^{10}} $ but i don't really understand where this result can come from, more considering the result i got for $x$
REPLY [1 votes]: The book is wrong. And your factorization is slightly off. (You need a plus sign inside the parenthesis).$1-\frac{5^{10}}{6^{10}}-10*\frac{5^9}{6^{10}}=1-\frac{5^9}{6^{10}}\left(5+10\right)\approx 0.5154833$ whereas the answer in the book is $1-10*\frac{5^9}{6^{10}-5^{10}}\approx 0.6147724$.
This can be computed also without using complementary events as ${10\choose2}(1/6)^2(5/6)^8+...+{10\choose 10}(1/6)^{10}(5/6)^0$. Using code, this is
> pbinom(1, 10, 1/6, lower.tail=FALSE)
[1] 0.5154833
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Talk:Southampton, United Kingdom/Archive 1
[edit] Update for weekend of 19th July 2008:
Both days are mid-Solent.
Going north we find:
Saturday:
Sunday:
Both in fields near Salisbury. Saturday's looks a bit close to a farmhouse.
--Dave 14:07, 18 July 2008 (UTC)
[edit] Saturday 6/9/08
Looks very possible, for the first time in ages. The spot itself is certainly physically accessible from the street; to me, the building it's associated with looks like a pub (see the umbrellas in the garden). Geohash landing actually on a pub - couldn't really be more convenient. There's a good chance I'll be there, especially if anyone else makes their plans known here. --Pete
[edit] May/June 2008 meets
When I want to go to a meet, I probably shan't sign up here or check that anyone else is going. The mystery about how many people are going to be there is part of the fun, does noone agree? 131.111.202.214 18:19, 21 May 2008 (UTC)
- Definitely a purist :-)
- Some people feel the other way - I'd be pretty disappointed to travel up to 50 miles only to stand in an empty field. If just 3 people sign the wiki page, then there might be 10 or more people that didn't sign the page, making it a good event Hello1024 18:52, 21 May 2008 (UTC)
I'm more concerned about the amount of "our" area that is sea - it seems like it's going to be rare for the location to be on land. 81.187.153.189 18:39, 21 May 2008 (UTC)
- Yep - cross your fingers for Saturday! Hello1024 18:52, 21 May 2008 (UTC)
- Provided this keeps going long enough (and why shouldn't it) I think at least one sea meetup needs to happen one day. We live on the coast, we ought to be able to beg, borrow or hire boats, right? 81.187.153.189 23:01, 21 May 2008 (UTC)
It's good to see some fellow xkcd'ers around in this area!!
guys just a heads up that i'm TOTALLY interested in a meet with cool people that like XKCD but, alas, this weekend my sister is coming down and i've gotta stay in soton :( My apologies, maybe next time yes? :) SinJax
- no worries - remember since so much of our area is on the sea there'll probably only be a meetup one in 3 weeks anyway - remember to "watchlist" the main page so you can find out a week you can come! Hello1024 19:33, 21 May 2008 (UTC)
Sorry guys not this week, too much to do. Incidently, when location falls in the sea, but close to coast, we should hire a boat and go, inorder to get our Pirate badges. See Water Geohash AX
Would love to come have a meetup one day but the cost of getting off/on the IOW these days is ridiculous. Will have to find a Saturday when it hits land and I'm in civilisation, suppose. Coincidently, is everyone here from Southampton Uni? Rvsmortimer 21:51, 10 June 2008 (UTC)
[edit] We've got a position for saturday!
BUT... it looks like it's in the water... :(
is anyone going to an alternate location instead? We could use sundays location just north east of southampton instead, or give it a miss this week. 192.102.214.6 14:15, 23 May 2008 (UTC)
People are going north if the location is wet. AX
Dave: This week's Saturday location is water. The alternate (north one graticule) location is certainly private property. As per the main page "Disclaimer: When any coordinates generated by the Geohashing algorithm fall within a dangerous area, are inaccessible, or would require illegal trespass, DO NOT attempt to reach them." I suggest we bail on this and see how things go in the future. Sunday's is also private land. Monday is just off the Needles on the IoW if anyone wants their pirate badge.
- Although it is on private land the farm house looks to be very close by, and from experience if you ask i am sure the farmer would grant access (and if the field is growing crop you can use the tram lines). If however everyone does bail due to it being on private land i fear there will never be many meets as most of the Swindon/Oxford graticule is agricultural. Sadly i cannot make this meet but i will come next week with Sinjax --Incanus 10:46, 24 May 2008 (UTC)
The location at [1] looks to be a field or paddock intimately associated with the building in its eastern corner, so I agree that it's "too private" to just turn up at. However, I don't think we should be scared off a typical agricultural field - as Incanus says, if we did that we'd never get anywhere unless the position happened to land right in Southampton High Street. Anywhere else is going to be residential property, industrial estates, etc that are less accessible than a field. It's unlikely that anyone would even notice a dozen people standing in their field one day, and if they did I'd be happy to explain that we're part of a strange club that goes to random locations, and we haven't caused any damage, and thanks, and we'll go now if that's what they'd like. I can't really see even the most unfavourable response to that extending beyond "grumpy". Separately, it might also be worth looking up the Right to Roam stuff that came in a year or two ago - maybe we're even allowed to visit random fields these days? 81.187.153.189 13:09, 25 May 2008 (UTC)
- If i remember correctly as long as you don't hop a fence and that there is no crop growing in the filed you are free to roam it. Also in this country trespass is only a civil offense, so even if you do hop a fence or go into a field with crop the most that will happen if you a caught is that you will be told to leave as long as you didnt cause any damage (by using tram lines) and move along.--Incanus 15:09, 25 May 2008 (UTC)
- "This new legal right - or right to roam - provided by The Countryside and Rights of Way Act 2000 (CRoW), applies only to mapped areas of uncultivated, open countryside namely mountain, moor, heath, down and registered common land." So no, you still have no right to go on private property. I'm all for getting permission where possible, but actually know your rights before claiming them! Annoyingly, Saturday's meet was a few hundred meters from Danesbury down, which would have made a good meet. Except very few people could make it. We'll get there eventually! --Dave 09:47, 27 May 2008 (UTC)
[edit] Thursday 12/6/08
It's almost a shame that this location didn't come up as a Saturday meetup - it's near as dammit on Calshot nudist beach :-)
[edit] Saturday 2009-07-18
Near Hambledon and Clanfield. Could be pleasant walk/ride/drive if the weather holds off. Who's up for it? Apologies if I've done this wrong, I'm a bit new to this. I added some text to the main page because it said to add names there. --macronencer 17:12, 17 July 2009 (UTC)
[edit] Inaccurate?
My apologies if I've offended you. But I'm curious as to your choice of words. What, exactly, did you feel was "inaccurate"? -- Benjw 21:03, 16 January 2009 (UTC)
We're active here, but suffer from being in a graticule that is mostly water. I think there were a few other points, but as there was a lot changed and most of it was rewording, I knee-jerked a bit and decided to undo all of it. I'll probably take a more detailed look and put some bits back when I have time. --d7415 21:16, 16 January 2009 (UTC)
Fair enough! Glad there's someone active after all, and I shall leave it well alone in future. Just that nothing seemed to have been changed since September, and the only active person I could find was you, and that was in a different graticule. Hope all goes well. -- Benjw 21:21, 16 January 2009 (UTC)
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Prof. (Dr.) Niranjan Mohanty.
National President,Indian Institute of Homoeopathic Physicians
Introduction
With the beginning of trans-border migration of birds from November, chances are there that the Avian Flu virus too flies into new areas. Is this the beginning of a pandemic? Nations are in a state of panic and scare as the virus threatens to melt borders and spread the wings of death.
With the bird flu outbreak in Asia spreading to Europe, alarm bells have started ringing with the UN also asking for action. Thousands of ducks and chickens have been killed to curb the disease’s spread. As human death toll continues to grow, many are concerned that the virus will mutate and trigger a human pandemic.
But the subtle homoeopathic philosophy as well as vivo and vitro studies conducted in the past with clinical evidences indicate the efficacy in viral infectious diseases. However, it is felt imperative to establish scientifically that homoeopathic medicament do act curatively in combating this Avian Flu.
Methodology (For mathematical model)
Aim – To ascertain the preventive and curative medicine for Avian Flu.
Methods – A diagnostic criterion was determined by taking clinical features such as: fever, sore throat, rhinitis, dry cough, headache, bodyache, malaise, redness of conjunctiva, nausea, vomiting and otitis media.
Results of mathematical model
By using RADAR & HOMPATH Software above symptoms were repertorised. Drugs evolved as closer remedy for Avian Flu are as follows:
RADAR HOMPATH
Belladonna – 24/10 Belladonna – 27/10
Pulsatilla – 24/10 Ars. alb. – 26/10
Mercurius – 21/10 Nux vom. – 26 /10
Rhus tox. – 21/10 Sulphur – 26/10
Causticum – 18/10 Pulsatilla – 25/10
Silicea – 23/9 Bry. alb. – 23/10
Sulphur – 23/9 Calc. carb. – 23/10
Ars. alb. – 22/9 Mercurius – 23/10
Calc. carb – 21/9 Nat. mur. – 23/10
Proposed Methodology
Aim – To determine the efficacy of the drugs evolved through mathematical model in providing prophylactic and curative role in Avian Flu.
Methods – Target groups
Domesticated birds
Human being of globe
The experiment will be double blind control trial with different determinants with various dosages and repetition schedule
The exclusion and inclusion criteria will be as per our earlier discussed clinical feature and diagnostic tests (viral culture / serology / antigen test / PCR and immunofluroscent assay). Route of administration of medicine will be oral. Follow up and monitoring will be done periodically for improvement of clinical feature and diagnostic tests. A standard protocol will be developed to document the data for its positive and negative responses. Finally statistical evaluation will be done by using EPINFO soft ware.
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- Articles
- Reviews
- Boxoffice
- The Sked
NOVITIATE (Sony Classics): It’s not clear how much of an audience there can be for a dark drama set amid the physical and psychological hardships of a pre-Vatican II midwestern abbey, but Margaret Betts’s Novitiate provides an utterly convincing insight into that world. (Betts won a “breakthrough” directing award at the festival.) The story […]
CALL ME BY YOUR NAME (Sony Classics): Luca Guadagnino’s sumptuous gay romance has been anointed as the Sundance entry most likely to figure into next year’s Oscar race, and it’s easy to see why. It combines the appeal of traditional prestige drama (James Ivory, who practically invented the modern version of that genre, […]
SID […]
REBEL IN THE RYE (no distrib): Danny Strong’s first film as a director is a biography of J. D. Salinger (Nicholas Hoult), and it hits all the Salinger bullet points: his early struggles to get published, his spectacularly doomed romance with legendary playwright’s daughter Oona O’Neill (he lost her to Charlie Chaplin), his difficult […] […]
When […] […]
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TITLE: Equivalence of Weierstrass and Ramification points of a Riemann Surface
QUESTION [0 upvotes]: It is known that on a hyperelliptic surface the set of Weierstrass points and the set of ramification points of the extension of the projection map $(x,y)\mapsto x$ to $\mathbb{P}^1$ coincide.
However, I am not sure if this is the case for a general Riemann surface. Maybe $p$ a Weierstrass point iff there is a ramified covering of $\mathbb{P}^1$ with $p$ as a ramification point.
It seems like I can't do much in either direction to prove this, but it also seems like something similar to this proposition would be nice to have since ramification provides a nice geometric intuition.
REPLY [1 votes]: There is no relationship like the one you proposed:
If $p$ is an arbitrary point on a regular projective curve $C$, then by Riemann-Roch there is a function $f$ having a zero of order $>1$ at $p$. The covering induced by $f$ is ramified at $p$.
Of course one could require $p$ to be the only ramification point. But then:
By the Riemann-Hurwitz formula a covering $C\rightarrow\mathbb{P}^1$ alwways has more than $1$ ramification point provided that $C$ has genus $>0$ (maybe plus some additional requirements in the case of characteristic $>0$ and in the case $K$ is not algebraically closed).
H
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Surprising Benefits of Psychic Reading
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TITLE: Finite disjoint unions of intervals?
QUESTION [0 upvotes]: Let $O = [0, \infty)$ and $F_1$ the class of all intervals of the type $[a, b)$ or $[a, \infty)$, where $0 \le a < b < \infty$. Let $F_2$ be the class of all finite disjoint unions of intervals of $F_1$. Show that $F_1$ is not a field and $F_2$ is a field but not a sigma field.
What does "finite disjoint unions of intervals" mean in this context ? Does that mean $F_2$ is empty should the word disjoint be in there ?
REPLY [2 votes]: $F_2$ is the set of all finite disjoint unions of intervals, that is
$$[a_1,b_1) \cup \cdots \cup [a_n,b_n),$$
where $b_i \leq a_{i+1}$ and $b_n$ could be $\infty$. The restriction $b_i \leq a_{i+1}$ ensures that the intervals are disjoint. Some examples:
$$ [1,2) \cup [3,4), \quad [5,6), \quad \emptyset, \quad [7,8) \cup [8,9) \cup [9,\infty). $$
But in fact, $F_2$ is also the set of all finite (unrestricted) unions of intervals. They don't have to be disjoint. Do you see why?
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TITLE: Find the number of functions.
QUESTION [1 upvotes]: Let $A = \{1,2,3,4\}$ and $B = \{a,b,c,d,e\}$. How many functions from $A$ to $B$ are either one-to-one or map the element $1$ to $c$? (you need not simplify your answer)
First. the number of functions which are one-to-one : $5\cdot4\cdot3\cdot2$
Second. the number of functions that map the $1$ to $c$ : $5^3$
answer : $5\cdot4\cdot3\cdot2 + 5^3$
is it right? please help me..
REPLY [4 votes]: Your sub-answers are correct, but the final answer is not: you forgot that some functions from $A$ to $B$ are in both categories $-$ they’re one-to-one and take $1$ to $c$. You’ve counted these functions twice in your answer of $5\cdot4\cdot3\cdot2+5^3$. To correct for this, you need to figure out how many functions are simultaneously in both categories and subtract this number from your current answer. How many injections from $A$ to $B$ take $1$ to $c$? If you use the kind of reasoning that you used to get your sub-answers, you should have little difficulty with this.
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TITLE: The difference of two normal random variables
QUESTION [0 upvotes]: Below is a problem that I did. I would like somebody to check it for me.
Thanks,
Bob
Problem:
Suppose that $z_1$ and $z_2$ are two independent random variables that are normally distributed with
mean $2$ and standard deviation of $4$. Now if $z = |z_1 - z_2|$, what is the probability that $z$ will be greater than 10?
Answer:
Let $u_z$ by the mean of the variable $z$. Let $p$ be the probability we seek.
\begin{align*}
u_z &= 0 \\
p &= 2P(z_1 - z_2 \geq 10) \\
\text{Let } z_3 &= z_1 - z_2 \\
p &= 2P(z_3 \geq 10) \\
\end{align*}
The variance of $z_3$ is $4^2 + 4^2 = 32$ and the standard deviation fo $z_3$ is $\sqrt{32} = 4\sqrt{2}$. Now we need to ask,
how many standard deviations does $10$ represent. The number $10$ represents $\frac{10}{4\sqrt{2}} = 1.7678$ standard deviation. The Z-score of $1.7678$ is $0.9614528$.
\begin{align*}
p &= 2( 1 - 0.9614528) \\
p &= 0.0770944 \\
\end{align*}
REPLY [0 votes]: Your approach is correct. The variance of the $z_3$ is indeed equal to the sum of the variances of $z_1$ and $z_2$, and you correctly computed both tails of the distribution corresponding to $z_1-z_2<-10$ and $z_1-z_2>10$.
| 107,836
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Fidelity Investments is the first fund management company to establish a range of tax-free child trust funds (CTFs), ready for when the Government's CTF scheme goes live next year.
Richard Wastcoat, the managing director, pictured above, said the company would offer a "stakeholder" product, with charges capped at 1.5 per cent, which will invest in a range of Fidelity funds.
There will also be a "self-invested" CTF offering a choice of 900 funds through Fidelity's fund supermarket. But this product will have standard unit trust fees - an initial charge of three to five per cent and an ongoing 1.5 per cent fee.
The Government will donate £250 into a child trust fund for every baby born after September 1 2002.
The money must stay invested until the child is 18, and returns will be free of income and capital gains tax.
Parents and grandparents can donate a further £1,200 a year.
Andrew Oxlade
Time to panic? No, follow the investment rulebook
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TITLE: Are some numbers more likely to count conjugacy classes than others?
QUESTION [18 upvotes]: Is there some reason for there to be more groups with 16 conjugacy classes than with 15 or 17?
It is a well-known exercise to show that a group with one conjugacy class has only one element, a finite group with two conjugacy classes must be cyclic of order two, and a finite group with three conjugacy classes must be cyclic of order three or non-abelian of order six. At some point I (along with everyone and their brother) classified finite groups with four conjugacy classes, but I never really looked beyond that until today.
My results are only partial, but I couldn't help noticing local maximums in my census at 10 classes (gentle), 14 classes (medium), 16 classes (sharp), and 18 classes (medium), with corresponding dips at 11 (gentle), 15 (sharp), and 17 (sharp).
Off hand I can't think of why a group might be more likely to have an even number of classes than an odd number, but perhaps this is well known. Four and five classes are relatively well known, as in one of my favorite papers, Miller (1919).
Miller, G. A.
"Groups possessing a small number of sets of conjugate operators."
Trans. Amer. Math. Soc. 20 (1919), no. 3, 260–270.
MR1501126
JFM47.0094.04 DOI:10.2307/1988867
REPLY [13 votes]: Your number $16$ reminds me of the beautiful theorem that for a group $G$ of odd order, $k(G) \equiv |G|$ (mod $16$). Here $k(G)$ is the number of conjugacy classes of $G$.
See also this post and here.
And I would like to add that in 1903 Edmund Landau proved that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes. However, his proof does not say anything about the frequencies with which conjugacy class numbers arise.
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Freeport boys finish fourth at WPIAL golf championship
Freeport's first trip to the WPIAL boys team golf finals in 18 years didn't come with a victory, but it gave next year's team a goal.
“We want to come back here,” junior Cole Hepler said.
Freeport shot a 447 to tie Shenango for fourth place out of six teams at the WPIAL Class AA team championship Thursday at Cedarbrook Golf Course's Gold Course in Rostraver.
“It's an awesome accomplishment to get here,” Freeport coach Joe Sprumont said. “You have to try and look at it both ways. Like when a team makes the Super Bowl, you try and soak it all in. But at the same time, you're here to take care of business.”
Longtime WPIAL power Burgettstown shot 411 to win, giving it five titles. It will be the first WPIAL team to compete for a PIAA title in the lower classification this month. Junior Owen Miller led the Blue Devils with a 74.
Sewickley Academy was second at 415, and Neshannock shot 430.
“We could have done better than we did,” said Hepler, who led Freeport with an 85. “I think we kind of clenched up a little bit. Overall, it was a good time and a great experience.”
Senior Robbie Miller shot 86, and freshman Audrey Clawson had 88.
“I know at least I had a slow start,” Miller said. “I had some trouble on the first two holes and never really got going.
“This is a great accomplishment to make it this far. A lot of people compare us to the team I was on as a freshman. They went 17-1 but lost in the first round.”
Central Catholic won the boys Class AAA title, its first in team history. The team collected on a bet with their coach, Corey O'Connor, who jumped into a lake near the 16th tee to celebrate.
“I was thinking to myself as the day went on, I wonder if I have any extra clothes in my car just in case?” O'Connor said. “The water was a lot deeper than I thought.”
But not as deep as his Vikings, who were led by four rounds in the 70s. Senior Kris Wright, who joined his coach for a swim, matched medalist honors with Peters Township's Tommy Nettles with a 2-over-par 74. Senior Hogan Cuny added a 76, senior Collin Haag 77 and junior Brent Rodgers 79.
Central finished with 387 to edge Peters Township (395), which was seeking its 14th title and first since 2008. Latrobe took third at 406.
The long-hitting Rodgers won the Class AAA individual title last week at Fox Chapel Golf Club.
“It's awesome to win this,” said Rodgers, the youngest competitor to qualify for the ReMax World Long-Drive Championships, which are the same day as the PIAA individual championships. “We have wanted this for so long.”
Rodgers again showed he can cave in a club face. He hit his tee shot on No. 9 over the green. Why hit driver on a short, tight par 4?
“Because it's fun,” Rodgers said.
The Penn-Trafford girls cruised to their first Class AAA championship, as senior Haley Borkovich shot a tournament-low 79 on the Red Course.
The girls Class AA title went to Central Valley, which won by 78 shots.
All four champions advance to the PIAA finals Oct. 22-24 at Heritage Hills Resort in York.
Bill Beckner Jr. is a staff writer for Trib Total Media. He can be reached at bbeckner@tribweb.com.
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| 272,684
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Updated at 11 a.m. ET's friends and foes react to his death.
."
CBSNews.com political reporter Lucy Madison and CBS News/ National Journal reporter Rebecca Kaplan contributed to this report.
Watch CBSNews.com's extensive interview with Breitbart from May, 2011, in which he discussed his disdain for the mainstream media and offered his perspective on the Republican presidential race:
| 103,327
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TITLE: Which power of an integer matrix is identity modulo $p^\alpha$?
QUESTION [3 upvotes]: I've read this question about identity power of an integer matrix.
But how about power of a matrix modulo $p^\alpha$.
$$A^m \equiv I \pmod{p^\alpha} $$
How can I find the minimal $m$ that the above equation hold?
Or how to prove that such $m$ does not exist?
REPLY [1 votes]: We assume that $p$ is a prime. If $\alpha >1$, then $\mathbb{Z}/p^{\alpha}\mathbb{Z}$ is a ring and not a field. Then the probem is difficult. For instance consider the equation $A^3=I_4$ over $\mathbb{Z}/4\mathbb{Z}$; I think that the problem is feasible but I do not know how to do.
If $\alpha=1$, then $K=\mathbb{Z}/p\mathbb{Z}$ is a field and it's easier...
Case 1. $p$ does not divide $m$, then $x^m-1$ has only simple roots. We decompose $x^m-1=p_1(x)\cdots p_k(x)$ in irreducible over $K$. The minimal polynomial $m(x)$ of $A$ is a product of some of the $p_i$, for instance $m(x)=p_1(x)\cdots p_r(x)$. Finally, $A$ is similar over $K$ to $diag(C_1,\cdots,C_1,\cdots ,C_r,\cdots C_r)$, where $C_i$ is the companion matrix of $p_i$.
Case 2. $m=pq$. Then $x^m-1=(x^q-1)^p$ and it suffices to decompose $x^q-1$.
Method. STEP 1.Calculate the minimal polynomial $m(x)$ of the matrix $A$ and $d=\det(A)$. If $d=0$, then there is no $m$. Otherwise, let $s$ be the minimum of $t>0$ s.t. $d^t=1$; if $m$ exists, then $s$ divides $m$. Find $s$.
STEP 2. for $u=1,2,\cdots$, calculate the remainder of the division of $x^{us}-1$ by $m(x)$. You stop when $m(x)$ divides $x^{us}-1$.
Example: $p=7$,$A=\begin{pmatrix}1&2&2&2&2\\1&1&0&1&0\\2&3&4&0&2\\2&2&3&2&5\\2&3&6&2&1\end{pmatrix}$, $\det(A)=3$, $s=6$. We obtain $m=300$.
It remains to obtain a stopping test.
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TITLE: Is $0.248163264128…$ a transcendental number?
QUESTION [18 upvotes]: My question is in the title:
Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$).
It is possible to express $a$ as a series :
$$a = \sum\limits_{n=1}^{\infty} 2^n \cdot
10^{ -\sum\limits_{k=1}^{n} (\lfloor{ k \cdot \log_{\,10}\,(2) }\rfloor + 1) } \tag{*}$$
I know that $a$ is irrational.
I know that if I consider the powers of $10$ instead the powers of $2$, i.e. if I consider $b=0.10100100010000...$, this number is transcendental.
Looking at the series (*), it seems very difficult to establish the transcendence of $a$. However, it is known (thanks to Kurt Mahler) that numbers as:
$$c = 0.149162536… =
\sum\limits_{n=1}^{\infty} n^2 \cdot
10^{ -\sum\limits_{k=1}^{n} (\lfloor{ 2 \cdot \log_{\,10}\,(k) }\rfloor + 1) } \tag{**} $$
are transcendental ($c$ is the concatenation of the square numbers in base $10$ ; the same holds for third powers and so on).
I am aware that this could be a difficult problem. Similar numbers, as Copeland-Erdős constant, are not known to be transcendental. I would really appreciate if anyone had a reference about this number $a$, because I didn't find anything that could help me to determine whether $a$ is transcendental, or whether it is still unknown.
Thank you very much!
REPLY [14 votes]: Yann Bugeaud "Distribution Modulo One and Diophantine Approximation", page 221:
For integers $b \geq 2$ and $c \geq 2$, let $(c)_b$ denote the sequence of digits of $c$ in its representation in base $b$. Mahler [471] proved that the real number $0 (c)_{10}(c^2)_{10} \dots$ is irrational. This was subsequently reproved and extended to every base $b \geq 2$ by Bundschuh [170] and Niederreiter [539]; see also [69, 172, 647, 652].
Problem 10.48. With the above notation, prove that, for arbitrary integers $b \geq 2$ and $c \geq 2$, Mahler's number $0 (c)_{b}(c^2)_{b} \dots$ is transcendental and normal to base $b$.
The question of normality of $0.248163264 \dots$ to base $10$ was already posed by Pillai [561].
Some of the links I was able to recover:
[172] P.Bundschuh, P. J.-S. Shiue and X.Y. Yu, Transcendence and
algebraic independence connected with Mahler type numbers, Publ. Math.
Debrecen 56 (2000), 121-130.
[647] Z.Shan, A note on irrationality of some numbers, J. Number
Theory 25 (1987), 211-212.
[652] Irrationality criteria for numbers of Mahler's type. In:
Analytic Number Theory (Kyoto, 1996), 343-351, London Math. Soc.
Lecture Note Ser., 247, Cambridge University Press, Cambridge, 1997.
Some data on this number from me. More digits:
$$0.2481632641282565121024204840968192163843276865536\dots$$
Simple continued fraction:
$$[0; 4, 33, 1, 3, 2, 565, 3, 5, 1, 10, 1, 43, 1, 1, 1, 1, 3, 1, 4, 1, 1, 3, 2, 3, 3, 2, 1, 1, 3, 5, 1, 16, 1, 15, 1, 2, 1, 3, 1, 3, 3, 327, \dots]$$
Euler type continued fraction:
$$\cfrac{1}{5-\cfrac{5}{6-\cfrac{5}{6-\cfrac{5}{51-\cfrac{50}{51-\cfrac{50}{51-\cfrac{50}{501-...}}}}}}}$$
The probability of a bigger partial quotent to occur after a smaller one in this fraction is equal to:
$$\frac{\ln 2}{\ln 10}=0.30103 \dots$$
Note that this fraction always approaches the number from below, fot example this truncation is exactly equal to $0.248163264128$
Unfortunately, general continued fractions do not afford any insight in the trancendentality of a number, as far as I know.
| 141,042
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Lea Valley – Paths Less Trodden
Sunday, 22 January 2017 - 8:30am
Sunday 22 January
Lea Valley – Paths Less Trodden
Hopefully we will find at least some parts of the Lea Valley Park that you have not been to – I certainly will.
About 13½ miles, on mostly good paths. Meet at Sollershott Green at 8.30am for car-sharing to the car park at the Old Mill and Meadows at Broxbourne. Please let me know if you will be going direct. Bring packed lunch and drink for the day.
Leader: Tony Maynard-Smith 07800 632879.
Contact Details:
See Group Contact.
| 407,634
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Sealand - A Maersk Company, the Intra-Americas regional ocean carrier of the Maersk Group, this week introduced Hispaniola, a new direct, all-water service that connects South Florida's Port Everglades with the Dominican Republic and Haiti. This service provides another connection to the entire Sealand network..
About Sealand - A Maersk Company
Sealand - A Maersk Company is a regional container logistics company that combines passionate local teams and agile-thinking with a global network powered by the larger Maersk family. We move our customers' cargo quickly and efficiently across the Americas, Asia, Europe, North Africa and the Middle East. Through the close connection to Maersk we ensure our customers the benefits of logistics expertise and cutting-edge technology. As the global leader in shipping services, A.P. Moller Maersk operates in 130 countries, employs roughly 76,000 people and works to connect and simplify its customers' supply chains.
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| 83,172
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.11 Six Days, Part 1 |
Version 20 -
view current page
◄ Back to
Grey's Home
"Six Days, Part 1"
Episode Number
: 11
Season
: 3
Original Air Date
: January 11, 2007
Writer
: Krista Vernoff
Director
: Greg Yaitanes
Special Guests
: Steven W. Bailey, Sarah Utterback, Jeff Perry, Greg Pitts, Debra Monk, George Dzunzda
Synopsis
Hookups
George's father has surgery for his cancer; Thatcher Grey arrives to see his new granddaughter; Meredith finds that Derek has trouble sleeping.
Recap of Events:-us..
Best Moments
Best Quotes
When Dr. Torres stares down Dr. Shepherd
Patient: "Sorry I was a ***** to you earlier."
Izzy: "That's okay, you're in pain. Sometimes I'm a ***** for no reason at all."
Opening Statement/Quote:
Scene opens to Derek, smiling and watching Meredith snore in her sleep and cuts to George berating Izzie about not depositing her check from Denny.
Ending Statement/Quote:
Scene ends with George & Izzie holding hands after his father's surgery and cuts to Derek unhappily listening to Meredith snore
Music:
Title Reference:
Six Days, Part 1
"Lonely Hearts Still Beat The Same" by The Research
"Love Will Come Through" by Travis
"Passion Play" by William Fitzsimmons
"Rest Of My Life" by Michelle Featherstone
"Beggars Prayer" by Emiliana Torrini
This episode's title refers to a song by DJ Shadow.
Goofs:
Aha! Moments
Six Days, Part I
Six Days, Part 1 Recap
| 266,598
|
Tara Pearls
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- Total ruby carat weight is 4.00
- Total sapphire carat weight is 2.72
- Total emerald carat weight is 2.46
- Total diamond carat weight is 0.08
- Diamond color is G-H
- Diamond clarity is SI1-SI2
- 18 inches long
- 0.55 inches at widest point
- Push button closure
Material: 18K white gold, pearl, ruby, sapphire, emerald, and diamond
Brand: Tara Pearls
Origin: United States
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MTV
Learn more about MTV
- This article is about the original U.S. music television network. For all of MTV's sister channels in the U.S., and all international MTV channels, see List of MTV channels. For other uses, see MTV (disambiguation). culture, and reality television shows aimed at adolescents and young adults.
Since its premiere, MTV.
The launch of MTV
Previous concepts
MTV's pre-history began in 1977, when Warner Cable (a division of Warner Communications and an ancestor of WASEC, Warner Satellite Entertainment Company); with the interactive Qube service, viewers could vote for their favorite songs and artists.
MTV's programming format was created by the visionary media executive, Bob Pittman, who later became president and chief executive officer of MTV Networks.<ref></ref> Pittman had test driven the music format by producing and hosting a 15 minute show, Album Tracks, on WNBC, New York, in the late 1970s. And Pittman's boss, WASEC COO John Lack, had sheparded a TV series called PopClips, created by former Monkee-turned solo artist Michael Nesmith, the latter of whom by the late 1970s was turning his attention to the music video format.<ref></ref>
Music Television debuts
- Further information: First music videos aired on MTV
On August 1, 1981, at 12:01 a.m., MTV: Music Television launched with the words "Ladies and gentlemen, rock and roll!" (by original COO John Lack) and the original MTV theme song, a crunching guitar riff written by Jonathan Elias and John Petersen, playing over a montage of the Apollo 11 moon landing. MTV producers used this footage because it was in the public domain.<ref></ref>
Appropriately, the first music video shown on MTV was "Video Killed the Radio Star" by The Buggles. The second video shown was Pat Benatar's "You Better Run". Sporadically, the screen would go black when someone at MTV inserted a tape into a VCR.<ref></ref>
At launch time, the official subscriber count across America was 3,000,000 (the actual number was 500,000), but the immediate impact would have argued that every young adult's television in the country was tuned to MTV.
MTV's early days
Personalities and format
- Further information: List of MTV VJs
The early format of MTV.<ref></ref>
The early music videos that made up the bulk of MTV's programming in the 1980, including Spike Jonze, Michel Gondry, and David Fincher.
A large number of rock bands and performers of the 1980s were made into household names by MTV. Some 1980s acts immediately identifiable with MTV include Van Halen, The Police, The Cars, Eurythmics, RATT, Culture Club, Def Leppard, Duran Duran, Bon Jovi, and "Weird Al" Yankovic, who made a career out of parodying other artists videos.
The hard rock band KISS publicly appeared without their trademark makeup for the first time on MTV in 1983. Michael Jackson launched the second wave of his career as an MTV staple. Madonna rose to fame on MTV in the 1980s. Madonna is the most successful video performer in MTV history, and to this day she uses MTV to market her music.
Station IDs and slogans
- Further information: List of MTV slogans
MTV's innovative station IDs were created by independent animation studios like Colossal Pictures (San Francisco) [1], Broadcast Arts (Washington, D.C.), and Buzzco (New York) [2]. The radical MTV logo was designed by tiny New York design firm Manhattan Design (Pat Gorman, Frank Olinsky, and Patty Rogoff). Alan Goodman and Fred Seibert developed the identity and promotional strategies that put MTV in the forefront of the world media industry by establishing the idea that television channels were brands.
Many of the more successful musicians featured on MTV could frequently be seen doing station identification spots for the network, exclaiming the signature line, I want my MTV!, and other phrases. Over the years, MTV would gather a large list of slogans.
Award shows
- Further information: List of MTV award shows.
Initial criticism
As early as 1985, because of its visibility as a promotional tool for the recording industry, MTV became criticized as overly commercial. It was accused of denigrating the importance of music in the music industry, replacing it with a purely visual aesthetic), and putting equally popular but less image-centric or single-based acts at a distinct disadvantage. One musician that criticized MTV for these reasons was Dead Kennedys, with the song "MTV - Get off the Air," from the album Frankenchrist. Although it could be said that MTV simply gave airtime to the most popular acts in a given country, it is also possible that these acts became popular simply because of the exposure that MTV gave them.
MTV comes of age
Format evolution
- Further information: List of MTV shows
Before 1987, MTV featured almost exclusively music videos, but as time passed they introduced a variety of other shows. Some of these new shows, such as 120 Minutes, still featured music videos. However, many of these shows were originally intended for other channels.
This non-music video programming began in the late 1980s with the introduction of a music news show The Week in Rock, which was also the beginning of MTV's news division, MTV News. Around this time, MTV also introduced a.
Animated shows
In the early 1990s, MTV introduced animated shows to its line-up. MTV has a history of cartoons with mature themes, notably Beavis and Butt-head, and its spin-off, Daria. The channel would go on to debut any other animated shows. Few of MTV's other cartoons have been renewed for additional seasons, regardless of their reception.
Variety of programming
By the second half of the 1990s, MTV's programming consisted primarily of non-music shows. In 1997, MTV was being heavily criticized for not playing as many music videos as it had in the past. In response, MTV created four shows that centered around the channel's unofficial flagship program. In 1999, MTV shifted its focus to prank/comedic shows such as The Tom Green Show, Jackass, and Punk'd; and soap operas such as Undressed.
MTV in recent years
Reality shows
- Further information: List of MTV shows
In the early 2000s, MTV put a stronger focus on reality shows, building on the success of The Real World and Road Rules in the 1990s. MTV continued to play music videos (albeit rarely) instead of exclusively relegating them to their genre channels; however, the music videos aired either in the early morning hours or in a condensed form on Total Request Live.
In 2000, MTV's Fear became the first 'scary' reality show in which contestants filmed themselves. The show ran for three seasons and spawned numerous imitations, including. television.
In 2003, Newlyweds, another popular reality TV show that follows the lives of Jessica Simpson and Nick Lachey, a music celebrity couple, began airing. It ran for four seasons and ended in early 2005 and they later divorced. The success of Newlyweds was followed in June 2004 by The Ashlee Simpson Show, which documented the beginnings of the music career of Ashlee Simpson, Jessica Simpson's younger sister. In the fall of 2004, Ozzy Osbourne's reality show Battle for Ozzfest aired.
Controversies
In 2004, MTV faced criticism in the wake of the Super Bowl XXXVIII half time show, which it produced. This infamous halftime show, which was shown on live television, featured the partial exposure of one of Janet Jackson's breasts. Afterwards, the NFL indicated that MTV would not produce future Super Bowl halftime shows or any NFL-sponsored public event. In 2006, fans of Janet Jackson started a petition against MTV for blacklisting the video for her single "Call on Me," possibly as a result of the 2004 controversy.
In July 2005, MTV drew heavy criticism for their coverage of Live 8. The network cut to commercials while bands were still performing, specifically legendary rock acts Pink Floyd (during the legendary guitar solo for "Comfortably Numb") and The Who. Criticism was also aimed at MTV and VH1 for focusing too much on ill-informed VJs and not enough on the music. In some instances, VJs referred to the event at "Live 8 2005" or even "Live Aid 8," demonstrating that they had little or no knowledge of the cause going into the event.
25th anniversary
On August 1, 2006, MTV celebrated its 25th anniversary. On their web site, MTV.com, visitors could watch the very first hour of MTV, including airing the original promos and commercials from Mountain Dew, Atari, Chewels gum, and Jovan. Videos were also shown from The Buggles, Pat Benatar, Rod Stewart, and more. The introduction of the first five VJs was also shown.
Additionally, MTV.com put together a "yearbook" consisting of the greatest videos of each year from 1981 to 2006. Along with that, music.mtv.com offered a special online viewing of the top music video of each year since 1981.
MTV itself only mentioned the anniversary once on TRL. The main highlight of the day on the channel was The Real World, the show that has divided viewers into two sides: those who think it made the network more of a pop culture force and created the reality television genre, and those who think it caused the network to "jump the shark" by reducing its focus on music videos.
Current trends
- Further information: List of MTV shows
In 2005 and 2006, MTV continued its focus on reality shows, with the debuts of popular shows such as Laguna Beach: The Real Orange County, NEXT, Two-A-Days, My Super Sweet 16, and Parental Control.
Today, MTV's main source of music video programming is still Total Request Live, airing four times per week. A hip-hop music video show, Sucker Free, also airs regularly. On most days, music video rotation continues in the late night and early morning hours.
Moral influence of MTV
Since its inception, critics of MTV have claimed that the channel's programming promotes bad behavior, including violence and recreational drug use, to the youth of America by embracing the behaviors of certain celebrities who are not good role models. Some critics have even claimed that MTV is "pornography for children."<ref></ref>
In response to this initial criticism, since the early 1990s, MTV restructured its programming to incorporate moral behaviors that might influence their audience. Personalities on the channel began to support environmental issues and emphasize being "socially responsible," encouraging young people to take part in volunteer work in their community.
More recently, in the summer of 2005, MTV began to examine the depiction of women in their programming after women's rights groups criticized MTV for allowing misogyny in images and music videos.
Censorship
On the other side of the moral influence debate, MTV has also come under criticism for being too politically correct and sensitive, censoring too much of their programming. Many of MTV's shows were altered or removed from the channel's schedule. Additionally, many music videos aired on the channel were censored, moved to late-night rotation, or banned entirely from the channel.
Social activism
MTV has a long history of promoting social, political, and environmental activism in young people.
In 1992, MTV started a pro-democracy campaign called Choose or Lose, to encourage up to 20 million people to register to vote, and hosted a town hall forum for Bill Clinton.<ref>MTV's traveling "Choose or Lose" vehicle brings politics. Salon.</ref> Shepherd, a gay college student.
MTV also aired a popular band's Sum 41 trip to the Democratic Republic of Congo, documenting the conflict there. The group ended up being caught in the midst of an attack outside of the hotel and were subsequently flown out of the country.<ref>"Rocked: Sum 41 in Congo" War Child Canada. 2001-2006.</ref>.<ref>Sherman, Tom, "The Real Story of the Youth Vote in the 2004 Election." Underscorebleach.net, 2004-11-04. Retrieved on 2006-04-14.</ref> MTV worked with P. Diddy's "Vote or Die" campaign, designed to encourage young people to vote.<ref>Vargas, Jose Antonio, "Vote or Die? Well, They Did Vote." Washingtonpost.com, 2004-11-09. Retrieved on 2006-04-14.</ref>
MTV's most recent activism campaign is "think MTV," which discusses current political issues such as gay marriage, U.S. elections, and war in other countries. The slogan of the program is "Reflect. Decide. Do." As part of think MTV, the channel also airs a series of pro-environmental ad spots entitled "Break The Addiction", as a way of encouraging their viewers to find ways to use less fossil fuels and energy.
MTV in popular culture
MTV has been referenced countless times in popular culture. Other TV channels, TV shows, musicians, films, and books have made reference to MTV in their works. An incomplete list of these references can be found at MTV in popular culture.
Beyond MTV
Sister channels in the U.S.
- Further information: List of MTV channels
The advent of satellite television and digital cable brought MTV greater channel diversity, including its current sister channel MTV2, which initially played 24/7 music videos and now focuses on other music-related programming. Two additional channels, MTV Hits and MTV Jams, play music videos exclusively. MTV also broadcasts mtvU, a college-oriented channel on campus at various universities.
In 1985, MTV saw the introduction of its first true sister channel, VH1, short for Video Hits One. Today, MTV Networks still operates VH1, which is aimed at celebrity and popular culture programming, as well as CMT, which targets the country music market. Recently, MTV Networks launched MHD (Music: High Definition), a high definition channel that features programming from all three music-themed channels owned by MTV Networks: MTV, VH1, and CMT.
In 2005 and 2006, MTV launched a series of channels for Asian Americans. The first channel was MTV Desi, launched in July 2005, dedicated towards South-Asian Americans. Next was MTV Chi, in December 2005, which catered to Chinese Americans. The third installment was MTV K, targeted toward Korean Americans, which was launched on June 27, 2006. Each of these channels feature music videos and shows from MTV's international affiliates as well as original U.S. programming, promos, and packaging.
The Internet
MTV.com, the official website of MTV, expands on the channel's broadcasts by bringing additional content to its viewers. The site's notable features include an online version of MTV News, podcasts, and a video streaming service supported by commercials. There are also movie features, profiles and interviews with recording artists and even clips from MTV television programs. In 2006, MTV.com went through a massive change, transforming the entire site into a video-based entity, in the style of the former MTV Overdrive service.
MTV around the world
- Further information: List of MTV channels
MTV Networks and Viacom have launched numerous native-language MTV-branded music channels to countries worldwide. These channels include, but are not limited Latin America, MTV Puerto Rico, MTV Brasil, MTV Australia, MTV New Zealand, MTV Russia, MTV Ukraine, MTV Türkiye, and MTV Base in Africa.
See also
- List of MTV channels
- List of MTV shows
- List of MTV animated shows
- List of MTV award shows
- List of MTV VJs
- List of MTV slogans
- First music videos aired on MTV
- Censorship on MTV
- MTV in popular culture
- MTV News
- MTV Generation
- MTV Buzz Bin
- MuchMusic
- Fuse TV
- Coolhunting
References
<references />
External links
fr:Modèle:Viacom, Inccs:MTV da:MTV de:MTV es:MTV fr:Music Television gd:MTV hr:MTV id:MTV it:MTV he:MTV la:MTV hu:MTV nl:MTV ja:MTV no:MTV pl:MTV pt:MTV ru:MTV Россия sq:MTV sr:МТВ fi:Music Television sv:MTV th:เอ็มทีวี tr:MTV zh:音樂電視網
Categories: Semi-protected | MTV Networks | Companies based in New York City | Music video networks | TV channels with British versions | Viacom subsidiaries | 1981 establishments
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News and Press Releases
Archives: 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | 2007 | 2006 | 2005 | 2004 | 2003 | 2002
Former Crestwood Water Officials Sentenced for Concealing
Village’s Use of Well in Drinking Water Supply
CHICAGO — Two former water department officials for the southwest suburban Village of Crestwood were each sentenced today to two years’ probation for lying repeatedly to environmental regulators for more than 20 years about using a water well to supplement the village’s drinking water supply. The defendants, FRANK SCACCIA, a retired certified water operator, and THERESA NEUBAUER, former water department clerk and supervisor and, later, Crestwood’s police chief, effectively thwarted the government from implementing the federal Safe Drinking Water Act’s notice and testing requirements designed to ensure the safety of municipal water supplies.
In addition to probation, Scaccia, 61, of Crestwood, was ordered to serve the first six months in home confinement. He pleaded guilty on April 11 this year to making false statements. Neubauer, 56, of Crestwood, was fined $2,000 and ordered to perform 200 hours of community service. She was convicted by a jury on April 29 of 11 counts of making false statements after a week-long trial.
U.S. District Judge Joan Gottschall cited Scaccia’s serious health condition in imposing his sentence. She said the case involved a “breach of the public trust for years,” which had as its purpose “the perpetual re-election of the mayor.”
Both defendants concealed the village’s use of its well from the government and the citizens of Crestwood to save money. By doing so, the village didn’t properly monitor for contaminants that could have been introduced to Crestwood’s water supply, avoided having to fix its leaking water distribution system, or paying the neighboring Village of Alsip more money for water drawn from Lake Michigan.
“Providing safe drinking water is one of the most fundamental and important functions of local government. Those who operate municipal water systems are now on notice that defeating the Safe Drinking Water Act in exchange for selfish political and personal objectives is an extremely serious crime that will be dealt with through vigorous federal prosecution,” said Zachary T. Fardon, United States Attorney for the Northern District of Illinois.
“Public servants swear an oath to protect the citizens of their community,” said Randall Ashe, Special Agent-in-Charge of the U.S. Environmental Protection Agency’s Criminal Enforcement Program in Illinois. “Rather than protecting the citizens of Crestwood, Scaccia and Neubauer engaged in a very lengthy scheme to deny Crestwood citizens their basic right to know the source of their drinking water, and to deceive them into thinking that their drinking water was properly tested for dangerous contaminants. As a result, Crestwood residents will never fully know what contaminants from the well they ingested. This case demonstrates that anyone who violates the public trust to assure the distribution of safe, potable and properly tested drinking water will face the consequences in court.”
According to court records, Scaccia, Neubauer were among of a small circle of trusted village employees ― directed by Crestwood’s longtime former mayor, Chester Stranczek, who was not charged ― who concealed that Crestwood was supplementing its Lake Michigan water with water drawn from Well #1. Scaccia was responsible for ensuring that water distributed by Crestwood met all federal and state regulations, including filing annual Consumer Confidence Reports (CCRs); obtaining the raw data that was used to complete the Monthly Operation and Chemical Analysis Reports . Neubauer also helped prepare and submit various false reports stating that Well #1 was on standby status and that the sole source of Crestwood’s drinking water was Lake Michigan water purchased from Alsip.
Under the federal Safe Drinking Water Act of 1974, the U.S. EPA created regulations to ensure the safety of drinking water distributed by public water systems by requiring testing and establishing maximum contaminant levels for various contaminants. The EPA delegated the primary responsibility for enforcement to the Illinois EPA,, an unmonitored and unreported water source, the village should have periodically tested.
The government was represented by Assistant U.S. Attorneys Erika Csicsila and Timothy Chapman, and Special Assistant U.S. Attorney Crissy Pellegrin, criminal enforcement counsel for the U.S. EPA Region V.
Direct: (312) 353-5318, Cell: (312) 613-6700
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Drilling Site Camps, Homes, Living & Office Containers
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\begin{document}
\maketitle \thispagestyle{empty}
\maketitle
\begin{abstract}
We investigate the geometrical and combinatorial structures of multipartite quantum systems based on conifold and toric variety. In particular, we study the relations between resolution of conifold, toric variety, a separable state, and a quantum entangled state for bipartite and multi-qubit states. For example we show that the resolved or deformed conifold is equivalent with the space of a pure entangled two-qubit state. We also generalize this result into multi-qubit states.
\end{abstract}
\section{Introduction}
Pure quantum states are usually defined on complex Hilbert spaces which are very complicated to visualize. The simplest case, namely, the space of a single qubit state can be visualized with Block or Riemann sphere. Beyond that there have been little progresses to visualize quantum state. Recently, we have established a relation between quantum states and toric varieties. Based on such a construction or mapping it is possible to visualize the complex Hilbert space by lattice polytop.
In algebraic
geometry \cite{Griff78}, a conifold is a generalization of the notion of a
manifold. But, a conifold can contain conical
singularities, e.g., points whose neighborhood look like a cone
with a certain base. The base is usually a five-dimensional
manifold. However, the base of a complex conifold is a product of one dimensional complex projective space. Conifold are interesting space in string theory, e.g., in
the process of compactification of Calabi-Yau manifolds. A
Calabi-Yau manifold is a compact K\"{a}hler manifold with a
vanishing first Chern class. A Calabi-Yau manifold can also be
defined as a compact Ricci-flat K\"{a}hler manifold.
During recent decade toric varieties have been constructed in different
contexts in mathematics \cite{Ewald,GKZ,Fulton}. A toric variety $\mathbb{X}$ is a complex
variety that contains an algebraic torus $T=(\mathbb{C}^{*})^{n}$
as a dense open set and with action of $T$ on $\mathbb{X}$ whose restriction to
$T\subset\mathbb{X}$ is the usual multiplication on $T$.
In this paper, we establish relations between toric varieties and space of entangled states of bipartite and multipartite quantum systems. In particular, we discuss resolving the singularity and deformation of conifold and toric variety of the conifold. We show that by removing the singularity of conifold variety we get a space which not anymore toric variety but is the space of an entangled two-qubit state. We also investigate the combinatorial structure of multi-qubit systems based on deformation of each faces of cube (hypercube) which is equivalent to deformation of conifold variety.
Through this paper we will use the following notation
\begin{equation}\label{qubit}
\ket{\Psi}=\sum^{1}_{x_{m}=0}\sum^{1}_{x_{m-1}=0}\cdots
\sum^{1}_{
x_{1}=0}\alpha_{x_{m}x_{m-1}\cdots x_{1}}\ket{x_{m}x_{m-1}\cdots
x_{1}},
\end{equation}
with $\ket{x_{m}x_{m-1}\cdots
x_{1}}=\ket{x_{m}}\otimes\ket{x_{m-1}}\otimes\cdots\otimes\ket{x_{1}}\in
\mathcal{H}_{\mathcal{Q}}=\mathcal{H}_{\mathcal{Q}_{1}}\otimes
\mathcal{H}_{\mathcal{Q}_{2}}\otimes\cdots\otimes\mathcal{H}_{\mathcal{Q}_{m}}
$ for a pure multi-qubit state.
\section{Conifold}
In this section we will give a short review of conifold.
Let $\mathbb{C}$ be a complex algebraic field. Then, an affine
$n$-space over $\mathbb{C}$ denoted $\mathbb{C}^{n}$ is the set of
all $n$-tuples of elements of $\mathbb{C}$. An element
$P\in\mathbb{C}^{n}$ is called a point of $\mathbb{C}^{n}$ and if
$P=(a_{1},a_{2},\ldots,a_{n})$ with $a_{j}\in\mathbb{C}$, then
$a_{j}$ is called the coordinates of $P$.
A complex projective space $\mathbb{P}_{\mathbb{C}}^{n}$ is
defined to be the set of lines through the origin in
$\mathbb{C}^{n+1}$, that is,
\begin{equation}
\mathbb{P}_{\mathbb{C}}^{n}=\frac{\mathbb{C}^{n+1}-\{0\}}{
(x_{0},\ldots,x_{n})\sim(y_{0},\ldots,y_{n})},~\lambda\in
\mathbb{C}-0,~y_{i}=\lambda x_{i}
\end{equation}
for all $0\leq i\leq n $.
An
example of real (complex) affine variety is conifold which is
defined by
\begin{equation}
\mathcal{V}_{\mathbb{C}}(z)=\{(z_{1},z_{2},z_{3},z_{4})
\in\mathbb{C}^{4}: \sum^{4}_{i=1}z^{2}_{i}=0\}.
\end{equation}
and conifold as a real affine variety is define by
\begin{equation}
\mathcal{V}_{\mathbb{R}}(f_{1},f_{2})=\{(x_{1},\ldots,x_{4},y_{1},\ldots,y_{4})\in\mathbb{R}^{8}:
\sum^{4}_{i=1}x^{2}_{i}=\sum^{4}_{j=1}y^{2}_{j},\sum^{4}_{i=1}x_{i}y_{i}=0
\}.
\end{equation}
where $f_{1}=\sum^{4}_{i=1}(x^{2}_{i}-y^{2}_{i})$ and
$f_{2}=\sum^{4}_{i=1}x_{i}y_{i}$. This can be seen by defining
$z=x+iy$ and identifying imaginary and real part of equation
$\sum^{4}_{i=1}z^{2}_{i}=0$. As a real space, the conifold
is cone in $\mathbb{R}^{8}$ with top the origin and base space the
compact manifold $\mathbb{S}^{2}\times\mathbb{S}^{3}$.
One can reformulate this relation in
term of a theorem. The conifold $
\mathcal{V}_{\mathbb{C}}(\sum^{4}_{i=1}z^{2}_{i}) $ is the complex
cone over the Segre variety $\mathbb{CP}^{1}
\times\mathbb{CP}^{1}\longrightarrow\mathbb{CP}^{3}$. To see this let us make a complex linear
change of coordinate
\begin{equation}
\left(
\begin{array}{cc}
\alpha^{'}_{00} & \alpha^{'}_{01} \\
\alpha^{'}_{10} & \alpha^{'}_{11}\\
\end{array}
\right)\longrightarrow \left(
\begin{array}{cc}
z_{1}+iz_{2} & -z_{4}+iz_{3} \\
z_{4}+iz_{3} & z_{1}-iz_{2}\\
\end{array}
\right).
\end{equation}
Thus after this linear
coordinate transformation we have
\begin{equation}\label{Conifold}
\mathcal{V}_{\mathbb{C}}(\alpha^{'}_{00}\alpha^{'}_{11}-\alpha^{'}_{01}\alpha^{'}_{10})
=\mathcal{V}_{\mathbb{C}}(\sum^{4}_{i=1}z^{2}_{i})\subset\mathbb{C}^{4}.
\end{equation}
Thus we can think of conifold as a complex cone over $\mathbb{CP}^{1}
\times\mathbb{CP}^{1}$ see Figure 1.
\begin{figure}\label{fig1a}
\begin{center}
\includegraphics[scale=0.450]{Conifold1a.jpg}
\end{center}
\caption{Complex cone over $\mathbb{CP}^{1}
\times\mathbb{CP}^{1}$.}
\end{figure}
We will comeback to this result in section \ref{coifoldsec} where
we establish a relation between these varieties, two-qubit state, resolution of singulary, and deformation theory. We can also define a metric on conifold as
$dS^{2}_{6}=d r^{2}+r^{2}d S^{2}_{T^{1,1}}$, where
\begin{equation}
d S^{2}_{T^{1,1}}=\frac{1}{9}\left(d\psi +\sum^{2}_{i=1}\cos
\theta_{i}d\phi_{i}\right)^{2}+\frac{1}{6}\sum^{2}_{i=1}\left(d\phi^{2}_{i}
+\sin^{2} \theta_{i}d\phi^{2}_{i}\right)^{2},
\end{equation}
is the metric on the Einstein manifold $T^{1,1}=\frac{SU(2)\times
SU(2)}{U(1)}$, with $U(1)$ being a diagonal subgroup of the
maximal torus of $SU(2)\times SU(2)$. Moreover, $T^{1,1}$ is a
$U(1)$ bundle over $\mathbb{S}^{2}\times\mathbb{S}^{2}$, where
$0\leq \psi\leq 4$ is an angular coordinate and
$(\theta_{i},\phi_{i})$ for all $i=1,2$ parameterize the two
$\mathbb{S}^{2}$, see Ref. \cite{Kleb,Morr,Hosh6}.
\section{Toric varieties}
\label{sec:2}
The construction of toric varieties usually are based on two different branches of mathematics, namely, combinatorial geometry and algebraic geometry. Here, we will review the basic notations and structures of toric varieties \cite{Ewald,GKZ,Fulton}.
A general toric variety is an irreducible variety $\mathbb{X}$ that satisfies the following conditions. First of all $(\mathbb{C}^{*})^{n}$ is a Zariski open subset of $\mathbb{X}$ and the action of
$(\mathbb{C}^{*})^{n}$ on itself can extend to an action of $(\mathbb{C}^{*})^{n}$ on the variety
$\mathbb{X}$. As an example we will show that the complex projective space $\mathbb{P}^{n}$ is a toric variety. If $z_{0},z_{1},
\ldots,z_{n}$ are homogeneous coordinate of $\mathbb{P}^{n}$. Then, the map
$(\mathbb{C}^{*})^{n}\longrightarrow\mathbb{P}^{n}$ is defined by $(t_{1},t_{2},
\ldots,t_{n})\mapsto(1,t_{1},
\ldots,t_{n})$ and we have
\begin{equation}
(t_{1},t_{2},
\ldots,t_{n})\cdot (a_{0},a_{1},
\ldots,a_{n})=(a_{0},t_{1}a_{1},
\ldots,t_{n}a_{n})
\end{equation} which proof our claim that $\mathbb{P}^{n}$ is a toric variety.
We can also define toric varieties with combinatorial information such as polytope and fan.
But first we will give a short introduction to the basic of combinatorial geometry which is important in definition of toric varieties. Let $S\subset \mathbb{R}^{n}$ be finite subset, then a convex polyhedral cone is defined by
\begin{equation}
\sigma=\mathrm{Cone}(S)=\left\{\sum_{v\in S}\lambda_{v}v|\lambda_{v}\geq0\right\}.
\end{equation}
In this case $\sigma$ is generated by $S$. In a similar way we define a polytope by
\begin{equation}
P=\mathrm{Conv}(S)=\left\{\sum_{v\in S}\lambda_{v}v|\lambda_{v}\geq0, \sum_{v\in S}\lambda_{v}=1\right\}.
\end{equation}
We also could say that $P$ is convex hull of $S$. A convex polyhedral cone is called simplicial if it is generated by linearly independent set. Now, let $\sigma\subset \mathbb{R}^{n}$ be a convex polyhedarl cone and $\langle u,v\rangle$ be a natural pairing between $u\in \mathbb{R}^{n}$ and $v\in\mathbb{R}^{n}$. Then, the dual cone of the $\sigma$ is define by
\begin{equation}
\sigma^{\wedge}=\left\{u\in \mathbb{R}^{n*}|\langle u,v\rangle\geq0~\forall~v\in\sigma\right\},
,
\end{equation}
where $\mathbb{R}^{n*}$ is dual of $\mathbb{R}^{n}$. We also define the polar of $\sigma$ as
\begin{equation}
\sigma^{\circ}=\left\{u\in \mathbb{R}^{n*}|\langle u,v\rangle\geq-1~\forall~v\in\sigma\right\}.
\end{equation}
We call a convex polyhedral cone strongly convex if $\sigma\cap(-\sigma)=\{0\}$.
Next we will define rational polyhedral cones. A free Abelian group of finite rank is called a lattice, e.g., $N\simeq\mathbb{Z}^{n}$. The dual of a lattice $N$ is defined by
$M=\mathrm{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ which has rank $n$. We also define a vector space and its dual by $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\simeq \mathbb{R}^{n}$ and $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}\simeq \mathbb{R}^{n*}$ respectively.
Moreover, if $\sigma=\mathrm{Cone}(S)$ for some finite set $S\subset N$, then $\sigma\subset N_{\mathbb{R}}$ is a rational polydehral cone. Furthermore, if $\sigma\subset N_{\mathbb{R}}$ is a rational polyhedral cone, then $S_{\sigma}=\sigma^{\wedge}\cap M$ is a semigroup under addition with $0\in S_{\sigma}$ as additive identity which is finitely generated by Gordan's lemma \cite{Ewald}.
Here we will define a fan which is important in the construction of toric varieties. Let $\Sigma\subset N_{\mathbb{R}}$ be a finite non-empty set of strongly convex rational polyhedral cones. Then $\Sigma$ is called a fan if each face of a cone in $\Sigma$ belongs to $\Sigma$ and the intersection of any two cones in $\Sigma$ is a face of each.
Now, we can obtain the coordinate ring of a variety by associating to the semigroup $S$ a finitely generated commutative $\mathbb{C}$-algebra without nilpotent as follows. We associate to an arbitrary additive semigroup its semigroup algebra $\mathbb{C}[S]$ which as a vector space has the set $S$ as basis. The elements of $\mathbb{C}[S]$ are linear combinations
$\sum_{u\in S}a_{u}\chi^{u}$ and the product in $\mathbb{C}[S]$ is determined by the addition in $S$ using $\chi^{u}\chi^{u^{'}}=\chi^{u+u^{'}}$ which is called the exponential rule. Moreover, a set of semigroup generators $\{u_{i}: i\in I\}$ for $S$ gives algebra generators $\{\chi^{u_{i}}: i\in I\}$ for $\mathbb{C}[S]$.
Now, let $\sigma\subset N_{\mathbb{R}}$ be a strongly convex rational polyhedral cone and $A_{\sigma}=\mathbb{C}[S_{\sigma}]$ be an algebra which is a normal domain. Then,
\begin{equation}
\mathbb{X}_{\sigma}=\mathrm{Spec}(\mathbb{C}[S_{\sigma}])=\mathrm{Spec}(A_{\sigma})
\end{equation}
is called a affine toric variety. Next we need to define Laurent polynomials and monomial algebras. But first we observe that the dual cone $\sigma^{\vee}$ of the zero cone $\{0\}\subset N_{\mathbb{R}}$ is all of $ M_{\mathbb{R}}$ and the associated semigroup $S_{\sigma}$ is the group $M\simeq \mathbb{Z}^{n}$. Moreover, let $(e_{1},e_{2},\ldots,e_{n})$ be a basis of $N$ and
$(e^{*}_{1},e^{*}_{2},\ldots,e^{*}_{n})$ be its dual basis for $M$. Then, the elements $\pm e^{*}_{1},\pm e^{*}_{2},\ldots,\pm e^{*}_{n}$ generate $M$ as semigroup. The algebra of Laurent polynomials is defined by
\begin{equation}
\mathbb{C}[z,z^{-1}]=\mathbb{C}[z_{1},z^{-1}_{1},\ldots,z_{n},z^{-1}_{n}],
\end{equation}
where $z_{i}=\chi^{e^{*}_{i}}$. The terms of the form $\lambda \cdot z^{\beta}=\lambda z^{\beta_{1}}_{1}z^{\beta_{2}}_{2}\cdots z^{\beta_{n}}_{n}$ for $\beta=(\beta_{1},\beta_{2},\ldots,\beta_{n})\in \mathbb{Z}$ and $\lambda\in \mathbb{C}^{*}$ are called Laurent monomials. A ring $R$ of Laurent polynomials is called a monomial algebra if it is a $\mathbb{C}$-algebra generated by Laurent monomials. Moreover, for a lattice cone $\sigma$, the ring
$R_{\sigma}=\{f\in \mathbb{C}[z,z^{-1}]:\mathrm{supp}(f)\subset \sigma\}
$
is a finitely generated monomial algebra, where the support of a Laurent polynomial $f=\sum\lambda_{i}z^{i}$ is defined by
$\mathrm{supp}(f)=\{i\in \mathbb{Z}^{n}:\lambda_{i}\neq0\}.
$ Now, for a lattice cone $\sigma$ we can define an affine toric variety to be the maximal spectrum $\mathbb{X}_{\sigma}=\mathrm{Spec}R_{\sigma}$. A toric variety
$\mathbb{X}_{\Sigma}$ associated to a fan $\Sigma$ is the result of gluing affine varieties
$\mathbb{X}_{\sigma}=\mathrm{Spec}R_{\sigma}$ for all $\sigma\in \Sigma$ by identifying $\mathbb{X}_{\sigma}$ with the corresponding Zariski open subset in $\mathbb{X}_{\sigma^{'}}$ if
$\sigma$ is a face of $\sigma^{'}$. That is,
first we take the disjoint union of all affine toric varieties $\mathbb{X}_{\sigma}$ corresponding to the cones of $\Sigma$.
Then by gluing all these affine toric varieties together we get $\mathbb{X}_{\Sigma}$.
A affine toric variety $\mathbb{X}_{\sigma}$ is non-singular if and only if the normal polytope has a
unit volume.
\section{Conifold and resolution of toric singulrity for two-qubits}\label{coifoldsec}
In this section we study the simplicial decomposition of affine toric variety. For two qubits this simplicial decomposition coincides with desingularizing a conifold \cite{Closset}. We also show that resolved conifold is space of an entangles two-qubit state.
For a pairs of qubits $\ket{\Psi}=\sum^{1}_{x_{2}=0}\sum^{1}_{x_{1}=0}
\alpha_{x_{2}x_{1}} \ket{x_{2}x_{1}}$ we can also construct following simplex. For this two qubit state the separable state is given by the Segre embedding of $\mathbb{CP}^{1}\times\mathbb{CP}^{1}=
\{((\alpha^{1}_{0},\alpha^{1}_{1}),(\alpha^{2}_{0},\alpha^{2}_{1})): (\alpha^{1}_{0},\alpha^{1}_{1})\neq0,~(\alpha^{2}_{0},\alpha^{2}_{1})\neq0\}$. Let $z_{1}=\alpha^{1}_{1}(\alpha^{1}_{0})^{-1}$ and $z_{2}=\alpha^{2}_{1}(\alpha^{2}_{0})^{-1}$. Then we can cover $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ by four charts
\begin{equation}
\mathbb{X}_{\check{\Delta}_{1}}=\{(z_{1},z_{2})\},
~\mathbb{X}_{\check{\Delta}_{2}}=\{(z^{-1}_{1},z_{2})\},~
\mathbb{X}_{\check{\Delta}_{3}}=\{(z_{1},z^{-1}_{2})\},~
\mathbb{X}_{\check{\Delta}_{4}}=\{(z^{-1}_{1},z^{-1}_{2})\},
\end{equation}
The fan $\Sigma$ for $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ has edges spanned by $(1,0),(0,1),(-1,0),(0,-1)$. Next we observe that the space $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ and the conifold have the same toric variety.
If we split the conifold into a fan which has two cones as shown in Figure 2. Then this process converts the conifold into a resolved conifold. The cones are three dimensional and the dual cones are two copies of $\mathbb{C}^{3}$. The procedure of replacing an isolated singularity by a holomorphic cycle is called a resolution of the singularity. The resolved conifold has a Ricci-flat K\"{a}hler metric which was derived by Candelas and de la Ossa \cite{Candelas}
\begin{eqnarray}
\nonumber
ds^{2}_{res}&=& \widetilde{\varrho}^{'}d\widetilde{r}^{2}
+\frac{\widetilde{\varrho}^{'}}{4}\widetilde{r}^{2}(d\widetilde{\psi}
+\cos\widetilde{\theta}_{1}d\widetilde{\phi}_{1}
+\cos\widetilde{\theta}_{2}d\widetilde{\phi}_{2})^{2}\\\nonumber&+&
\frac{\widetilde{\varrho}}{4}(d\widetilde{\theta}^{2}_{1}
+\sin^{2}\widetilde{\theta}_{1}d\widetilde{\phi}^{2}_{1})+
\frac{\widetilde{\varrho}+4a^{2}}{4}(d\widetilde{\theta}^{2}_{2}
+\sin^{2}\widetilde{\theta}_{2}d\widetilde{\phi}^{2}_{2}),
\end{eqnarray}
where $\widetilde{\psi}=0\cdots4\pi$ is a $U(1)$ fiber over $S^{2}$, $(\widetilde{\phi}_{i},\widetilde{\theta}_{i}),~i=1,2$ are Euler angles on two sphere $S^{2}$, and $\widetilde{\varrho}=\widetilde{\varrho}(\widetilde{r})$ goes to zero as $\widetilde{r}\rightarrow0$.
Thus we can propose that the resolution of the singularity of a toric variety is equivalent to the space of entangled states.
\begin{figure}\label{fig1}
\begin{center}
\includegraphics[scale=0.450]{Conifold1b.jpg}
\end{center}
\caption{Two-qubit system. a) toric polytope of a two-qubit systems. b) and c) two ways of removing the singularity of conifold.}
\end{figure}
We can also remove the singularity by deformation. The process of deformation modifies the complex structure manifolds or
algebraic varieties. Based on our discussion of conifold we know that this space is defined by
$\alpha_{00}\alpha_{11}-\alpha_{01}\alpha_{10}$. Now, if we rewrite this equation in the following form
\begin{equation}
\alpha_{00}\alpha_{11}-\Gamma\alpha_{01}\alpha_{10}+\Lambda\alpha_{10}=0,
\end{equation}
then the constant $\Gamma$ and $\Lambda$ can be absorbed in new definition of $\alpha_{10}$ such as
$\alpha^{'}_{10}=\Gamma\alpha_{10}-\Lambda$. Next let $T_{n}$ be the group of translations. Then an affine variety over complex field of dimension $n$ can be transformed using the following action
$GL(n,\mathbb{C})\ltimes T_{n}$. For a generic polynomial of degree two we have 15 possible parameters, but most of them can be removed with the action of $GL(4,\mathbb{C})\ltimes T_{4}$. However, we cannot remove the constant term with such transformation and we end up with the following variety
\begin{equation}
\alpha_{00}\alpha_{11}-\alpha_{01}\alpha_{10}=\Omega.
\end{equation}
which is called deformed conifold. This space is now non-singular, but it is not a toric variety since the deformation break one action of torus. Thus we also could proposed that the deformed conifold is the space of an entangled pure two-qubit state. Moreover, if we take the absolute value of this equation that is $|\Omega|$, then this value is proportional to concurrence which is a measure of entanglement for a pure two-qubit state, that is
\begin{equation}
|\alpha_{00}\alpha_{11}-\alpha_{01}\alpha_{10}|=|\Omega|=C(\Psi)/2.
\end{equation}
In general let $X$ be an algebraic variety, then the space of all complex deformations of $X$ is called the complex moduli space of $X$.
\section{Three-qubit states}
Next, we will discuss a three-qubit state $\ket{\Psi}=\sum^{1}_{x_{3},x_{2},x_{1}=0}
\alpha_{x_{3}x_{2}x_{1}} \ket{x_{3}x_{2}x_{1}}$.
For this state the separable state is given by the Segre embedding of $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}=
\{((\alpha^{1}_{0},\alpha^{1}_{1}),(\alpha^{2}_{0},\alpha^{2}_{1}),(\alpha^{3}_{0},\alpha^{3}_{1}))): (\alpha^{1}_{0},\alpha^{1}_{1})\neq0,~(\alpha^{2}_{0},\alpha^{2}_{1})\neq0
,~(\alpha^{3}_{0},\alpha^{3}_{1})\neq0\}$.
Now, for example, let $z_{1}=\alpha^{1}_{1}/\alpha^{1}_{0}$,
$z_{2}=\alpha^{2}_{1}/\alpha^{2}_{0}$, and $z_{3}=\alpha^{3}_{1}/\alpha^{3}_{0}$.
Then we can cover $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ by eight charts
\begin{eqnarray}
\nonumber &&
\mathbb{X}_{\check{\Delta}_{1}}=\{(z_{1},z_{2},z_{3})\},
~\mathbb{X}_{\check{\Delta}_{2}}=\{(z^{-1}_{1},z_{2},z_{3})\},~
\mathbb{X}_{\check{\Delta}_{3}}=\{(z_{1},z^{-1}_{2},z_{3})\},~\\\nonumber&&
\mathbb{X}_{\check{\Delta}_{4}}=\{(z_{1},z_{2},z^{-1}_{3})\},
\mathbb{X}_{\check{\Delta}_{5}}=\{(z^{-1}_{1},z^{-1}_{2},z_{3})\},
~\mathbb{X}_{\check{\Delta}_{6}}=\{(z^{-1}_{1},z_{2},z^{-1}_{3})\},~\\\nonumber&&
\mathbb{X}_{\check{\Delta}_{7}}=\{(z_{1},z^{-1}_{2},z^{-1}_{3})\},~
\mathbb{X}_{\check{\Delta}_{8}}=\{(z^{-1}_{1},z^{-1}_{2},z^{-1}_{3})\},
\end{eqnarray}
The fan $\Sigma$ for $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ has edges spanned by $(\pm1,\pm1,\pm1)$.
Now, let $S=\mathbb{Z}^{3}$ and consider the polytope $\Delta$ centered at the origin with vertices $(\pm1,\pm1,\pm1)$. This gives the toric variety $\mathbb{X}_{\Delta}=\mathrm{Spec}\mathbb{C}[S_{\Delta}]$. To describe the fan of $\mathbb{X}_{\Delta}$, we observe that the polar $\Delta^{\circ}$ is the octahedron with vertices $\pm e_1,\pm e_2, \pm e_3$. Thus the normal fan is formed from the faces of the octahedron which gives a fan $\Sigma$ whose 3-dimensional cones are octants of $\mathbb{R}^{3}$. Thus this shows that the toric variety $\mathbb{X}_{\Sigma}=\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$.
\begin{figure}\label{fig2}
\begin{center}
\includegraphics[scale=0.450]{cube5a.jpg}
\end{center}
\caption{Three-qubit systems. a) toric polytope of a separable three-qubit systems. b) resolved space of entangled state, where each diagonal line is equivalent to the resolution of singularity of a conifold. }
\end{figure}
In this case
we split the faces of $3$-cube $E_{2,3}=2^{3-2}\frac{3(3-1)}{2}=6$ into two cones see Figure 3. Then this process converts the $3$-cube into a nonsingular space which is not anymore toric variety.
Following the same procedure, we can also remove all singularities of toric variety of three-qubits by deformation. Based on our discussion of conifold we can write six equations describing the faces of 3-cube. Here we will analyze one face of this 3-cube, namely
\begin{eqnarray}
\nonumber
&&\alpha_{000}\alpha_{011}-\alpha_{001}\alpha_{010}=\alpha_{0}\otimes(\alpha_{00}
\alpha_{11}-\alpha_{01}\alpha_{10})
\end{eqnarray}
Now, if we rewrite these equations e.g., in the following form
\begin{equation}
\alpha_{0}(\alpha_{00}\alpha_{11}-\Gamma\alpha_{01}\alpha_{10}+\Lambda\alpha_{10})=0,
\end{equation}
then the constant $\Gamma$ and $\Lambda$ can be absorbed in new definition of $\alpha_{10}$ such as
$\alpha^{'}_{10}=\Gamma\alpha_{10}-\Lambda$. At the end e.g., we have the following variety
\begin{equation}
\alpha_{000}\alpha_{011}-\alpha_{001}\alpha_{010}=\Omega
\end{equation}
which is equivalent to the deformed conifold. If we do this procedure for all faces of the 3-cube, then the whole space becomes non-singular, but it is not a toric variety anymore. Thus we also could proposed that the deformed conifold is the space of an entangled pure three-qubit state. There are other relations between toric variety and measures of quantum entanglement that can be seen from the toric structures of multipartite systems. For example three-tangle or 3-hyperdeterminant can be constructed from the toric variety.
\section{Multi-qubit states}
Next, we will discuss a multi-qubit state $\ket{\Psi}$ defined by equation (\ref{qubit}).
For this state the separable state is given by the Segre embedding of $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\cdots\times\mathbb{CP}^{1}=
\{((\alpha^{1}_{0},\alpha^{1}_{1}),(\alpha^{2}_{0},\alpha^{2}_{1}),
\ldots,(\alpha^{m}_{0},\alpha^{m}_{1}))): (\alpha^{1}_{0},\alpha^{1}_{1})\neq0,~(\alpha^{2}_{0},\alpha^{2}_{1})\neq0,\ldots,
,~(\alpha^{m}_{0},\alpha^{m}_{1})\neq0\}$.
Now, for example, let $z_{1}=\alpha^{1}_{1}/\alpha^{1}_{0}, z_{2}=\alpha^{2}_{1}/\alpha^{2}_{0},\ldots, z_{m}=\alpha^{m}_{1}/\alpha^{m}_{0}$.
Then we can cover $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\cdots\times\mathbb{CP}^{1}$ by $2^{m}$ charts
\begin{eqnarray}
\nonumber &&
\mathbb{X}_{\check{\Delta}_{1}}=\{(z_{1},z_{2},\ldots,z_{m})\},
~\mathbb{X}_{\check{\Delta}_{2}}=\{(z^{-1}_{1},z_{2},\ldots,z_{m})\}\\\nonumber&,\cdots,&
\mathbb{X}_{\check{\Delta}_{2^{m}-1}}=\{(z_{1},z^{-1}_{2},\ldots,z^{-1}_{m})\},~
\mathbb{X}_{\check{\Delta}_{2^{m}}}=\{(z^{-1}_{1},z^{-1}_{2},\ldots,z^{-1}_{m})\}
\end{eqnarray}
The fan $\Sigma$ for $\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\cdots\times\mathbb{CP}^{1}$ has edges spanned by $(\overbrace{\pm1,\pm1,\ldots,\pm1}^{m})$.
Now, let $S=\mathbb{Z}^{m}$ and consider the polytope $\Delta$ centered at the origin with vertices $(\pm1,\pm1,\ldots,\pm1)$. This gives the toric variety $\mathbb{X}_{\Delta}=\mathrm{Spec}\mathbb{C}[S_{\Delta}]$. To describe the fan of $\mathbb{X}_{\Delta}$, we observe that the polar $\Delta^{\circ}$ is the octahedron with vertices $\pm e_1,\pm e_2, \ldots,\pm e_m$. Thus this shows that the toric variety $\mathbb{X}_{\Sigma}=\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\cdots\times\mathbb{CP}^{1}$.
In this case
we split the faces of $m$-cube
\begin{equation}
E_{2,m}=2^{m-2}\frac{m(m-1)}{2}
\end{equation}
into two cones. Then this process converts the $m$-cube into a nonsingular space which is not anymore toric variety.
Following the same procedure, we can also remove all singularities of toric variety of multi-qubits by deformation. Based on our discussion of conifold we can write six equations describing the faces of $m$-cube. For example for one face (2-cube) of this $m$-cube, we ahve
\begin{equation}
\alpha_{00\cdots0}\alpha_{0\cdots011}-\alpha_{0\cdots01}\alpha_{0\cdots010}=\Omega
\end{equation}
which is equivalent to the deformed conifold, since e.g., we could have $\ket{\Psi}=\frac{1}{\sqrt{2}} (\ket{00\cdots000}+\ket{00\cdots011})=\frac{1}{\sqrt{2}} \ket{00\cdots0}\otimes(\ket{00}+\ket{11})$. If we do this procedure for all faces of the m-cube, then the whole space becomes non-singular, but it is not a toric variety anymore. Thus we also could proposed that this space is the space of an entangled pure multi-qubit state.
In this paper we have investigated the geometrical and combinatorial structures of entangled multipartite systems. We have shown that by removing singularity of conifold or by deforming the conifold we obtain the space of a pure entangled two-qubit state. We have also generalized the construction into multipartite entangled systems. The space of multipartite systems are difficult to visualize but the transformation from complex spaces to the combinatorial one makes this task much easier to realize. Hence our results give new insight about multipartite systems and also a new way of representing quantum entangled bipartite and multipartite systems.
\begin{flushleft}
\textbf{Acknowledgments:} This work was supported by the Swedish Research Council (VR).
\end{flushleft}
| 126,181
|
\begin{document}
\title[Semidirect products of C\textsuperscript{*}-quantum groups]{Semidirect products of C\textsuperscript{*}-quantum groups: multiplicative unitaries approach}
\author{Ralf Meyer}
\email{rmeyer2@uni-goettingen.de}
\address{Mathematisches Institut\\
Georg-August Universität Göttingen\\
Bunsenstraße 3--5\\
37073 Göttingen\\
Germany}
\author{Sutanu Roy}
\email{sutanu.roy@carleton.ca}
\address{School of Mathematics and Statistics\\
Carleton University\\
1125 Colonel By Drive\\
K1S 5B6 Ottawa\\
Canada.}
\author{Stanisław Lech Woronowicz}
\email{Stanislaw.Woronowicz@fuw.edu.pl}
\address{Instytut Matematyczny Polskiej Akademii Nauk\\ul.\@ Śniadeckich 8\\00-656 Warszawa\\Poland, and\\Katedra Metod Matematycznych Fizyki, Wydział Fizyki\\Uniwersytet Warszawski\\ul.\@ Pasteura 5\\02-093 Warszawa\\Poland}
\begin{abstract}
\(\Cst\)\nb-quantum groups with projection are the noncommutative
analogues of semidirect products of groups. Radford's Theorem about
Hopf algebras with projection suggests that any \(\Cst\)\nb-quantum
group with projection decomposes uniquely into an ordinary
\(\Cst\)\nb-quantum group and a ``braided'' \(\Cst\)\nb-quantum
group. We establish this on the level of manageable multiplicative
unitaries.
\end{abstract}
\subjclass[2000]{46L89 (81R50 18D10 )}
\keywords{quantum group, braided quantum group, semidirect product,
bosonisation, multiplicative unitary, braided multiplicative
unitary, quantum E(2) group}
\thanks{Supported by the German Research Foundation (Deutsche
Forschungsgemeinschaft (DFG)) through the Research Training Group
1493. The second author was also supported by a Fields--Ontario
postdoctoral fellowship. The third author was partially supported
by the Alexander von Humboldt-Stiftung and the National Science
Center (NCN), grant 2015/17/B/ST1/00085.}
\maketitle
\section{Introduction}
\label{sec:introduction}
Many important Lie groups like the Poincar\'e group or the group of
motions of Euclidean space are defined as semidirect products of
smaller building blocks. What is the quantum group analogue of a
semidirect product? Such a notion should be useful to understand
quantum deformations of semidirect products.
For a semidirect product of groups, we need two groups \(G\)
and~\(H\) and an action of~\(G\) on~\(H\) by group automorphisms. Since
non-commutative quantum groups cannot act on other quantum groups by
automorphisms, we need a different point of view: semidirect product
groups are the same as groups with a projection. A semidirect
product of groups~\(G\ltimes H\) comes with a canonical group
homomorphism
\[
p\colon G\ltimes H\to G\ltimes H,\qquad
(g,h)\mapsto (g,1_H),
\]
which is idempotent, that is, \(p^2=p\).
Its kernel and image are \(H\subseteq G\ltimes H\)
and \(G\subseteq G\ltimes H\),
respectively. The conjugation action of~\(G\)
on~\(H\)
needed for a semidirect product is the restriction of the conjugation
action of~\(G\ltimes H\)
on itself. Therefore, an idempotent group homomorphism
\(p\colon K\to K\)
on a group~\(K\)
is equivalent to a semidirect product decomposition of~\(K\).
Now consider a quantum group with a projection, that is, with an
idempotent quantum group endomorphism. What corresponds to the
building blocks \(G\) and~\(H\) in a semidirect product of groups? If
``quantum group'' means ``Hopf algebra,'' then a theorem by
Radford~\cite{Radford:Hopf_projection} answers this question. Here we
consider \(\Cst\)\nb-quantum groups, meaning
\(\Cst\)\nb-bialgebras coming from manageable multiplicative unitaries
(see~\cites{Woronowicz:Mult_unit_to_Qgrp,
Soltan-Woronowicz:Multiplicative_unitaries}). More precisely, we
work on the level of the multiplicative unitaries themselves to avoid
analytical difficulties.
Let us first recall Radford's Theorem. It splits a Hopf algebra~\(C\)
with a projection \(p\colon C\to C\) into two pieces \(A\) and~\(B\).
The ``image'' of the projection~\(A\) is a Hopf algebra as well. The
``kernel'' of the projection~\(B\) is only a Hopf algebra in a certain
braided monoidal category, namely, the category of Yetter--Drinfeld
modules over~\(A\). The tensor product of two Yetter--Drinfeld
algebras is again a Yetter--Drinfeld algebra, for the diagonal
Yetter--Drinfeld module structure and a certain deformed
multiplication. The comultiplication on~\(B\) is a homomorphism to
the deformed tensor product \(B\boxtimes B\).
Radford's Theorem contains two constructions. One puts together \(A\)
and~\(B\) into their ``semidirect product''~\(C\) and describes the
projection~\(p\) on~\(C\). The other splits~\(C\) into the two
factors \(A\) and~\(B\), with the Hopf algebra structure on~\(A\) and
the \(A\)-Yetter--Drinfeld algebra and braided Hopf algebra structure
on~\(B\). The first construction is called ``bosonisation'' by
Majid~\cite{Majid:Hopfalg_in_BrdCat}. The analogue of this
construction for \(\Cst\)\nb-quantum groups is described
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}, except for the
projection that we expect on this semidirect product. In particular,
the appropriate analogues of Yetter--Drinfeld algebras and their
deformed tensor product~\(\boxtimes\) are described
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} for arbitrary
\(\Cst\)\nb-quantum groups. For regular \(\Cst\)\nb-quantum groups
with Haar weights, this is already done by Nest and
Voigt~\cite{Nest-Voigt:Poincare}.
The ``projections'' on \(\Cst\)\nb-quantum groups that we use are
morphisms as introduced in \cites{Ng:Morph_of_Mult_unit,
Meyer-Roy-Woronowicz:Homomorphisms}. That is, a quantum group
morphism from \((C,\Comult[C])\) to \((A,\Comult[A])\) is a
bicharacter in \(\U\Mult(\hat{C}\otimes A)\). Several equivalent
descriptions of such morphisms are given
in~\cite{Meyer-Roy-Woronowicz:Homomorphisms}, including functors
between the categories of \(\Cst\)\nb-algebra coactions that preserve
the underlying \(\Cst\)\nb-algebra, and Hopf \Star{}homomorphisms
between the associated universal quantum groups. These are more
general than Hopf \Star{}homomorphisms between the reduced quantum
group \(\Cst\)\nb-algebras.
Thus a \(\Cst\)\nb-quantum group with projection consists of a
\(\Cst\)\nb-quantum group~\((C,\Comult[C])\) with a unitary multiplier
\(\projbichar\in\U\Mult(\hat{C}\otimes C)\) with certain properties.
To express these, we use a manageable multiplicative unitary
\(\Multunit\in\U(\Hils\otimes\Hils)\) that generates~\(C\); in
particular, \(\Multunit\) satisfies the pentagon equation
\begin{equation}
\label{eq:Multunit_pentagon}
\Multunit_{23}\Multunit_{12} =
\Multunit_{12}\Multunit_{13}\Multunit_{23}
\qquad\text{in }\U(\Hils\otimes\Hils\otimes\Hils).
\end{equation}
Then \(C\) and~\(\hat{C}\) act faithfully on~\(\Hils\).
Write~\(\ProjBichar\) for~\(\projbichar\) viewed as an operator on
\(\Hils\otimes\Hils\). The condition that~\(\projbichar\) is a
bicharacter is equivalent to
\begin{equation}
\label{eq:intro_pentagonal}
\ProjBichar_{23}\Multunit_{12}
= \Multunit_{12}\ProjBichar_{13}\ProjBichar_{23}
\quad\text{and}\quad
\Multunit_{23} \ProjBichar_{12}
= \ProjBichar_{12}\ProjBichar_{13}\Multunit_{23}
\qquad\text{in }\U(\Hils\otimes\Hils\otimes\Hils).
\end{equation}
The condition that~\(\projbichar\) is idempotent for the composition
of quantum group homomorphisms is equivalent to the pentagon equation
for~\(\ProjBichar\):
\begin{equation}
\label{eq:PR_pentagon}
\ProjBichar_{23}\ProjBichar_{12} =
\ProjBichar_{12}\ProjBichar_{13}\ProjBichar_{23}
\qquad\text{in }\U(\Hils\otimes\Hils\otimes\Hils).
\end{equation}
Thus a \(\Cst\)\nb-quantum group with projection is determined by two
unitaries \(\Multunit,\ProjBichar\in
\U(\Hils\otimes\Hils)\) that satisfy
\eqref{eq:Multunit_pentagon}--\eqref{eq:PR_pentagon}; in addition,
\(\Multunit\) must be manageable. Equation~\eqref{eq:PR_pentagon}
means that~\(\ProjBichar\) is a multiplicative unitary in its own
right. It is manageable if~\(\Multunit\) is. The \(\Cst\)\nb-quantum
group~\((A,\Comult[A])\) it generates is the image of the projection.
It is much more difficult to describe the other
factor~\(B\). As a \(\Cst\)\nb-algebra, it should be the generalised
fixed-point algebra for a canonical coaction of~\((A,\Comult[A])\)
on~\((C,\Comult[C])\). In the group case, this says that
\(\Cont_0(H)\) is the generalised fixed-point algebra for the left or
right translation action of~\(G\) on~\(\Cont_0(G\ltimes H)\).
Unless~\(G\) is compact, this requires Rieffel's generalisation of
fixed-point algebras to group actions that are ``proper'' in a
suitable sense (see \cites{Rieffel:Integrable_proper,
Meyer:Generalized_Fixed}).
Buss~\cites{Buss:thesis,Buss:GFPAQuantum} has generalised this theory
to locally compact quantum groups. We only need the special case of
quantum homogeneous spaces, which is also treated by
Vaes~\cite{Vaes:Induction_Imprimitivity}. All these approaches need
some regularity assumptions on~\((A,\Comult[A])\) and are technically
difficult.
We may avoid these difficulties by staying on the level of
multiplicative unitaries. We already described a \(\Cst\)\nb-quantum
group with projection through two multiplicative unitaries
\(\Multunit,\ProjBichar\in \U(\Hils\otimes\Hils)\) on the
same Hilbert space that are linked by the
conditions~\eqref{eq:intro_pentagonal}. We find that any such pair
comes from a ``braided multiplicative unitary'' over the
\(\Cst\)\nb-quantum group \((A,\Comult[A])\) generated
by~\(\ProjBichar\).
A braided multiplicative unitary is a unitary
\(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\) for a Hilbert
space~\(\Hils[L]\) with a Yetter--Drinfeld module structure over
\((A,\Comult[A])\). That is, \(\Hils[L]\) carries corepresentations
\(\corep{U}\in\U(\Comp(\Hils[L])\otimes A)\) and
\(\corep{V}\in\U(\Comp(\Hils[L])\otimes \hat{A})\) that are linked by
a Yetter--Drinfeld commutation relation. In addition, \(\BrMultunit\)
is equivariant for the tensor product corepresentations
\(\corep{U}\tenscorep\corep{U}\) and \(\corep{V}\tenscorep\corep{V}\)
on~\(\Hils[L]\otimes\Hils[L]\) and satisfies the \emph{braided
pentagon equation}:
\begin{equation}
\label{eq:intro_braided_pentagon}
\BrMultunit_{23}\BrMultunit_{12}
= \BrMultunit_{12}(\Braiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{12}(\Braiding{\Hils[L]}{\Hils[L]})^*_{23}
\BrMultunit_{23}
\qquad\text{in }\U(\Hils[L]\otimes\Hils[L]\otimes\Hils[L]).
\end{equation}
Here~\(\Braiding{\Hils[L]}{\Hils[L]}\) denotes the braiding operator
on the tensor product of the Yetter--Drinfeld Hilbert
space~\(\Hils[L]\) with itself,
see~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}.
Since \(A\) and~\(\hat{A}\) are represented faithfully on~\(\Hils\),
the unitaries \(\corep{U}\) and~\(\corep{V}\) are determined by
their images \(\Corep{U}\) and~\(\Corep{V}\)
in~\(\U(\Hils[L]\otimes\Hils)\). It is convenient to
replace~\(\Corep{V}\) by \(\DuCorep{V} \defeq \Sigma \Corep{V}^*
\Sigma\in\U(\Hils\otimes\Hils[L])\). We also write~\(\Multunit\)
instead of~\(\ProjBichar\); the multiplicative unitary for the
semidirect product quantum group will be denoted by~\(\Multunit[C]\).
Thus a braided multiplicative unitary is a family of four unitaries
\(\Multunit\in\U(\Hils\otimes\Hils)\),
\(\Corep{U}\in\U(\Hils[L]\otimes\Hils)\),
\(\DuCorep{V}\in\U(\Hils\otimes\Hils[L])\), and
\(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\) for two Hilbert spaces
\(\Hils\) and~\(\Hils[L]\); these unitaries satisfy seven conditions:
the pentagon condition for~\(\Multunit\); one corepresentation
condition each for \(\Corep{U}\) and~\(\DuCorep{V}\), which link them
to~\(\Multunit\); the Yetter--Drinfeld condition linking
\(\Corep{U}\) and~\(\DuCorep{V}\); the equivariance of~\(\BrMultunit\)
with respect to \(\Corep{U}\tenscorep\Corep{U}\)
and~\(\DuCorep{V}\tenscorep\DuCorep{V}\); and the braided pentagon
equation for~\(\BrMultunit\). We show that given these four unitaries
subject to these seven conditions, the unitary
\begin{equation}
\label{eq:crossed_product_multunit}
\Multunit[C]_{1234}\defeq
\Multunit_{13}\Corep{U}_{23}\DuCorep{V}^*_{34}
\BrMultunit_{24}\DuCorep{V}_{34}
\qquad\text{in } \U(\Hils\otimes\Hils[L]\otimes\Hils\otimes\Hils[L])
\end{equation}
is multiplicative. Furthermore, the unitaries \(\Multunit[C]\)
and \(\ProjBichar \defeq \Multunit_{13} \Corep{U}_{23}\)
on \(\Hils\otimes\Hils[L] \otimes\Hils\otimes\Hils[L]\)
satisfy the conditions
\eqref{eq:Multunit_pentagon}--\eqref{eq:PR_pentagon} that characterise
\(\Cst\)\nb-quantum
groups with projection. The only analytic issue is to prove
that~\(\Multunit[C]\)
is manageable if the braided multiplicative unitary is manageable in a
suitable sense. Otherwise, the claim is proved by a direct
computation. This has to be lengthy, however, because all seven
conditions on our four unitaries must play their role.
Conversely, let \(\Multunit[C],\ProjBichar\in\U(\Hils\otimes\Hils)\)
be unitaries satisfying the conditions
\eqref{eq:Multunit_pentagon}--\eqref{eq:PR_pentagon},
with~\(\Multunit[C]\)
manageable. Then we construct a braided multiplicative unitary based
on the unitary \(\Multunit=\ProjBichar\in\U(\Hils\otimes\Hils)\),
that is, we construct a Hilbert space~\(\Hils[L]\)
and unitaries \(\Corep{U}\in\U(\Hils[L]\otimes\Hils)\),
\(\DuCorep{V}\in\U(\Hils\otimes\Hils[L])\),
and \(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\)
satisfying the conditions for a braided multiplicative unitary, and we
check that this braided multiplicative unitary is manageable. When we
construct a pair \((\Multunit[C],\ProjBichar)\)
out of this data as in~\eqref{eq:crossed_product_multunit}, then we do
not get back the same data we started with because the underlying
Hilbert spaces have changed. We show, however, that the resulting
\(\Cst\)\nb-quantum
groups with projection are the same. This isomorphism is also
implemented by a quantum group isomorphism in the category constructed
in~\cite{Meyer-Roy-Woronowicz:Homomorphisms}.
When we start with a manageable braided multiplicative unitary, form
the crossed product as in~\eqref{eq:crossed_product_multunit} and go
back, we also get a different braided multiplicative unitary, which
should be ``equivalent'' to the one we started with. Since we do not
discuss how a braided multiplicative unitary generates a braided
\(\Cst\)\nb-quantum bialgebra, we cannot yet express this equivalence.
We treat one example of a braided multiplicative unitary in detail,
namely, the one that defines the simplified quantum
\(\textup{E}(2)\) group, a variant of the quantum \(\textup{E}(2)\)
group introduced by Woronowicz
in~\cite{Woronowicz:Qnt_E2_and_Pontr_dual}. We write down the
braided multiplicative unitary and check that it is manageable.
Similar computations appear in
\cites{Baaj:Regular-Representation-E-2,
Woronowicz:Qnt_E2_and_Pontr_dual}.
\section{Projections on Quantum Groups}
\label{sec:proj}
A \emph{\(\Cst\)\nb-quantum group} is, by definition, a
\(\Cst\)\nb-bialgebra that is generated by a manageable multiplicative
unitary,
see~\cites{Woronowicz:Mult_unit_to_Qgrp,
Soltan-Woronowicz:Multiplicative_unitaries}. We do not assume a
\(\Cst\)\nb-quantum group to have Haar weights. We fix a
\(\Cst\)\nb-quantum group \(\G[H] = (C,\Comult[C])\) and let
\(\Multunit\in\U(\Hils\otimes\Hils)\) be a manageable multiplicative
unitary on a Hilbert space~\(\Hils\) that generates it. Let
\(\hat{\G[H]} = (\hat{C},\DuComult[C])\) be the dual quantum group.
A \emph{bialgebra morphism} \((A,\Comult[A])\to (C,\Comult[C])\)
between two \(\Cst\)\nb-bialgebras is a \(\Cst\)\nb-algebra morphism
\(f\colon A\to C\) (that is, a nondegenerate \Star{}homomorphism
\(A\to\Mult(C)\)) making the following diagram commute:
\[
\begin{tikzpicture}
\matrix(m)[cd]{
A&A\otimes A\\
C&C\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[A]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(f\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Comult[C]\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(f\otimes f\)} (m-2-2);
\end{tikzpicture}
\]
This notion of morphism is too restrictive, however, because a group
homomorphism \(G\to H\)
need not induce a morphism \(\Cred(G)\to\Cred(H)\).
When we speak of morphisms of \(\Cst\)\nb-quantum
groups, we will mean those introduced by
Ng~\cite{Ng:Morph_of_Mult_unit}, and we shall use the equivalent
characterisations of these morphisms
in~\cite{Meyer-Roy-Woronowicz:Homomorphisms}.
\begin{definition}
\label{def:quantum_group_proj}
A \emph{\(\Cst\)\nb-quantum group with projection} is a
\(\Cst\)\nb-quantum group with an idempotent quantum group
endomorphism.
\end{definition}
Before we make this definition explicit, we consider the commutative
case. It allows us to view \(\Cst\)\nb-quantum groups with projection
as \(\Cst\)\nb-quantum group analogues of semidirect products of
groups.
\begin{proposition}
\label{pro:commutative_qg_projection}
Let~\((C,\Comult[C])\)
be a commutative \(\Cst\)\nb-quantum
group with projection. Then \(C\cong \Cont_0(G\ltimes H)\)
for a semidirect product group, with the corresponding
comultiplication, and the projection on~\(C\)
comes from the group homomorphism \(G\ltimes H\to G\ltimes H\),
\((g,h)\mapsto (g,1_H)\);
here \(G\)
and~\(H\)
are locally compact groups and~\(G\)
acts continuously on~\(H\)
by automorphisms. Conversely, any semidirect product group gives a
commutative \(\Cst\)\nb-quantum group with projection in this way.
\end{proposition}
\begin{proof}
Since~\(C\) is commutative, \(C\cong\Cont_0(K)\) for a locally
compact group~\(K\). A quantum group homomorphism from~\(C\) to
itself is equivalent to a group homomorphism \(K\to K\), and the
composition of quantum group homomorphisms also corresponds to the
composition of group homomorphisms. Thus a projection on~\(C\)
corresponds to a group homomorphism \(p\colon K\to K\) with
\(p\circ p=p\). Let \(G\subseteq K\) and \(H\subseteq K\) be the
image and kernel of~\(p\), respectively; these are locally compact
groups as well. Since~\(H\) is a normal subgroup, conjugation
in~\(K\) lets \(G\subseteq K\) act continuously on~\(H\) by
automorphisms. The continuous maps \(m\colon G\times H\to K\),
\((g,h)\mapsto g\cdot h\), and \(n\colon K\to G\times H\),
\(k\mapsto (p(k),p(k^{-1})k)\), are inverse to each other and
hence homeomorphisms. The multiplication is given by
\[
m(g_1,h_1)\cdot m(g_2,h_2)
= g_1h_1g_2h_2
= g_1g_2 (g_2^{-1}h_1g_2)h_2
= m(g_1g_2, (g_2^{-1}h_1g_2)h_2).
\]
Thus the homeomorphism~\(m\) is also a group isomorphism \(K\cong
G\ltimes H\). The converse assertion is routine to check.
\end{proof}
Now we make Definition~\ref{def:quantum_group_proj} explicit in
several different ways, corresponding to some of the equivalent
characterisations of quantum group morphisms
in~\cite{Meyer-Roy-Woronowicz:Homomorphisms}. First we use unitaries
satisfying pentagon equations.
\begin{proposition}
\label{pro:qg_projection_mu}
A \(\Cst\)\nb-quantum group with projection is given by a Hilbert
space~\(\Hils\) and two unitaries
\(\ProjBichar,\Multunit\in\U(\Hils\otimes\Hils)\) that satisfy
\begin{align*}
\Multunit_{23}\Multunit_{12}
&= \Multunit_{12}\Multunit_{13}\Multunit_{23},\\
\ProjBichar_{23}\Multunit_{12}
&= \Multunit_{12}\ProjBichar_{13}\ProjBichar_{23},\\
\Multunit_{23} \ProjBichar_{12}
&= \ProjBichar_{12}\ProjBichar_{13}\Multunit_{23},\\
\ProjBichar_{23}\ProjBichar_{12}
&= \ProjBichar_{12}\ProjBichar_{13}\ProjBichar_{23}
\qquad\quad\text{in }\U(\Hils\otimes\Hils\otimes\Hils).
\end{align*}
In addition, \(\Multunit\) is manageable as a multiplicative
unitary.
\end{proposition}
All four equations in Proposition~\ref{pro:qg_projection_mu} are variants
of the pentagon equation.
\begin{proof}
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Lemma 3.2} describes a
quantum group morphism from~\(\G[H]\) to itself by a unitary
\(\ProjBichar\in\U(\Hils\otimes\Hils)\) on the same Hilbert
space~\(\Hils\) on which the manageable multiplicative
unitary~\(\Multunit\) lives, subject to the two
conditions~\eqref{eq:intro_pentagonal}, which are the second and
third equation in our statement. The first equation is the pentagon
equation for~\(\Multunit\). The fourth equation says that the
quantum group endomorphism associated to~\(\ProjBichar\) is
idempotent by \cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Definition
3.5}.
\end{proof}
Our first goal is to prove the following structural result:
\begin{proposition}
\label{pro:idempotents_split}
Any idempotent endomorphism \(p\colon \G[H]\to\G[H]\)
of a \(\Cst\)\nb-quantum
group~\(\G[H]\)
splits. That is, there are a \(\Cst\)\nb-quantum
group~\(\G\)
and quantum group morphisms \(a\colon \G\to\G[H]\),
\(b\colon \G[H]\to\G\)
with \(a\circ b=p\) and \(b\circ a=\Id_{\G}\).
\end{proposition}
The \(\Cst\)\nb-quantum
group~\(\G\)
is called the \emph{image} of the idempotent endomorphism~\(p\).
We first construct this image, then we describe \(a\)
and~\(b\)
and then we prove \(a\circ b=p\)
and \(b\circ a=\Id_{\G}\).
The proof of Proposition~\ref{pro:idempotents_split} will be finished
by Lemma~\ref{lem:projection_from_ij}.
The fourth equation in Proposition~\ref{pro:qg_projection_mu} says
that~\(\ProjBichar\) is a multiplicative unitary.
\begin{proposition}
\label{pro:ProjBichar_manageable}
The multiplicative unitary \(\ProjBichar\in\U(\Hils\otimes\Hils)\)
is manageable.
\end{proposition}
\begin{proof}
The multiplicative unitary~\(\Multunit\) is manageable by
assumption. This requires the existence of certain auxiliary
operators \(Q\) and~\(\widetilde{\Multunit}\). We use the same
operator~\(Q\) for~\(\ProjBichar\).
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Theorem
1.6} gives a unitary
\(\widetilde{\ProjBichar}\in\U(\conj{\Hils}\otimes\Hils)\) with
\[
\bigl(x\otimes u\mid \ProjBichar\mid z\otimes y\bigr)
= \bigl(\conj{z}\otimes Q u\mid \widetilde{\ProjBichar}\mid
\conj{x}\otimes Q^{-1}y\bigr)
\]
for all \(x,z\in\Hils\), \(u\in\dom(Q)\) and \(y\in\dom(Q^{-1})\).
Lemma~\ref{lem:V_tilde_commute_Q} shows that~\(\ProjBichar\)
commutes with \(Q\otimes Q\). So \(\widetilde{\ProjBichar}\)
and~\(Q\) witness the manageability of the multiplicative
unitary~\(\ProjBichar\) (see
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Definition 1.2}).
\end{proof}
Proposition~\ref{pro:ProjBichar_manageable} shows that~\(\ProjBichar\)
generates a \(\Cst\)\nb-quantum
group \(\G=(A,\Comult[A])\),
which is called the \emph{image of~\(\ProjBichar\)}.
Let \(\hat{\G}=(\hat{A},\DuComult[A])\) be its dual.
The unitary~\(\ProjBichar\) is the image of a unitary multiplier
\(\projbichar\in\U(\hat{C}\otimes C)\) by
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Lemma 3.2}. Hence slices
of~\(\ProjBichar\) are multipliers of
\(\hat{C}\subseteq\Bound(\Hils)\) and \(C\subseteq\Bound(\Hils)\),
respectively. These slices generate \(\hat{A}\) and~\(A\),
respectively, so \(A\subseteq\Mult(C)\) and
\(\hat{A}\subseteq\Mult(\hat{C})\).
\begin{lemma}
\label{lem:A_into_C_Hopf}
The embeddings \(i\colon A\to\Mult(C)\) and \(j\colon
\hat{A}\to\Mult(\hat{C})\) are bialgebra morphisms \(\G\to\G[H]\)
and \(\hat{\G}\to\hat{\G[H]}\).
\end{lemma}
\begin{proof}
First we claim that \(i\) and~\(j\) are \(\Cst\)\nb-algebra morphisms,
that is, \(i(A)\cdot C = C\) and \(j(\hat{A})\cdot \hat{C} = \hat{C}\).
The third condition in Proposition~\ref{pro:qg_projection_mu} is
equivalent to
\[
\ProjBichar_{12}^*\Multunit_{23}\ProjBichar_{12}
=\ProjBichar_{13}\Multunit_{23}
\qquad\text{in }\U(\Hils\otimes\Hils\otimes\Hils).
\]
When we slice the first two legs on both sides
by~\(\omega_1\otimes\omega_2\) for
\(\omega_1, \omega_2\in\Bound(\Hils)_*\)
and close in norm, we get \(C=A\cdot C\).
The same argument works for~\(j\).
The conditions in Proposition~\ref{pro:qg_projection_mu} also imply
\begin{gather*}
\Multunit_{23} \ProjBichar_{12}\Multunit[*]_{23}
= \ProjBichar_{12}\ProjBichar_{13}
= \ProjBichar_{23}\ProjBichar_{12} \ProjBichar_{23}^*,\\
\Multunit[*]_{12} \ProjBichar_{23}\Multunit_{12}
= \ProjBichar_{13}\ProjBichar_{23}
= \ProjBichar_{12}^* \ProjBichar_{23}\ProjBichar_{12}.
\end{gather*}
Since \((\Id_D\otimes \Comult[C])(x) = \Multunit_{23}
x_{12}\Multunit[*]_{23}\) for all \(x\in D\otimes C\) and
\((\Id_D\otimes \Comult[A])(x) = \ProjBichar_{23}
x_{12}\ProjBichar^*_{23}\) for all \(x\in D\otimes A\), the first
equation says that \(\Id_{\hat{A}}\otimes\Comult[C]\) and
\(\Id_{\hat{A}}\otimes\Comult[A]\) agree on~\(\ProjBichar\). Since
slices of~\(\ProjBichar\) generate~\(A\), this implies
\(\Comult[C]|_A=\Comult[A]\), that is, \(i\) is a bialgebra
morphism. So is~\(j\) by a similar argument.
\end{proof}
The bialgebra morphisms \(i\) and~\(j\) give quantum group morphisms
\begin{alignat*}{2}
V_i&=(\Id\otimes i)(\multunit[A]) \in\U(\hat{A} \otimes C)&\qquad
&\text{from }A\text{ to }C,\\
\hat{V}_j &=(j\otimes \Id)(\multunit[A]) \in\U(\hat{C} \otimes A)&\qquad
&\text{from }C\text{ to }A.
\end{alignat*}
The quantum groups \(\G\)
and~\(\G[H]\)
may be generated by the multiplicative unitaries \(\ProjBichar\)
and~\(\Multunit\)
on the same Hilbert space~\(\Hils\).
Then the unitaries \(V_i\)
and~\(\hat{V}_j\)
are both represented by the same unitary~\(\ProjBichar\)
on~\(\Hils\otimes\Hils\);
the conditions in Proposition~\ref{pro:qg_projection_mu} allow us to
view~\(\ProjBichar\)
as a quantum group homomorphism \(\G\to\G[H]\),
\(\G[H]\to \G\),
\(\G[H]\to\G[H]\),
or as the identity quantum group homomorphism on~\(\G\).
\begin{lemma}
\label{lem:projection_from_ij}
The composite quantum group homomorphism \(V_i\circ \hat{V}_j\colon
\G[H]\to\G\to\G[H]\) is the given projection
\(\projbichar\in\U(\hat{C}\otimes C)\) on~\(\G[H]\). The other composite
\(\G\to\G[H]\to\G\) is the identity on~\(\G\).
\end{lemma}
\begin{proof}
The composition of quantum group homomorphisms is described
in~\cite{Meyer-Roy-Woronowicz:Homomorphisms} by a
pentagon-like equation. The two claims in the lemma are both
equivalent to the pentagon equation for~\(\ProjBichar\).
\end{proof}
The description of a projection on a \(\Cst\)\nb-quantum group by a pair of bialgebra morphisms \((i,j)\) is unwieldy because it mixes quantum groups and their duals and because the composition \(\G\to\G[H]\to\G\) is computed only indirectly.
The quantum group morphism \(\G[H]\to\G\) is usually not representable by a bialgebra morphism \(C\to A\). We may, however, also represent the quantum group morphism~\(j\) by a
bialgebra morphism \(\hat{j}^\univ\colon C^\univ\to A^\univ\)
between the universal quantum groups, see
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Theorem 4.8}. Similarly,
\(i\) lifts to a bialgebra morphism \(i^\univ\colon A^\univ\to
C^\univ\). A \(\Cst\)\nb-quantum group with projection is
equivalent to a \(\Cst\)\nb-quantum group~\(\G[H]\) with a bialgebra
morphism \(p\colon C^\univ\to C^\univ\) satisfying \(p\circ p=p\) by
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Theorem 4.8}. Our
analysis above shows that for any such~\(p\) there are a \(\Cst\)\nb-quantum group~\((A,\Comult[A])\) and bialgebra morphisms
\(\hat{j}^\univ\colon C^\univ\to A^\univ\) and \(i^\univ\colon A^\univ\to
C^\univ\) with \(p=i^\univ\circ \hat{j}^\univ\) and \(\hat{j}^\univ \circ i^\univ = \Id_A\). Thus a quantum group with projection is equivalent to two \(\Cst\)\nb-quantum groups with bialgebra morphisms
\(\hat{j}^\univ\colon C^\univ\to A^\univ\) and \(i^\univ\colon A^\univ\to
C^\univ\) with \(\hat{j}^\univ \circ i^\univ = \Id_A\).
Next we replace~\(\hat{j}\) by right and left quantum group morphisms:
\begin{proposition}
\label{pro:projection_via_action}
A \(\Cst\)\nb-quantum group with projection is equivalent to two
\(\Cst\)\nb-\alb{}quantum groups \(\G[H]=(C,\Comult[C])\)
and \(\G=(A,\Comult[A])\) with morphisms \(i\colon A\to C\) and
\(\Delta_R\colon C\to C\otimes A\) such that the following
diagrams commute:
\[
\begin{tikzpicture}
\matrix(m)[cd]{
A&A\otimes A\\
C&C\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[A]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(i\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Comult[C]\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(i\otimes i\)} (m-2-2);
\end{tikzpicture}\qquad
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&C\otimes A\\
C\otimes C&C\otimes C\otimes A\\
};
\draw[cdar] (m-1-1) -- node {\(\Delta_R\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Comult[C]\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Id_C\otimes\Delta_R\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Comult[C]\otimes\Id_A\)} (m-2-2);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&C\otimes A\\
C\otimes A&C\otimes A\otimes A\\
};
\draw[cdar] (m-1-1) -- node {\(\Delta_R\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Delta_R\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_R\otimes\Id_A\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Id_C\otimes\Comult[A]\)} (m-2-2);
\end{tikzpicture}\qquad
\begin{tikzpicture}
\matrix(m)[cd]{
A&A\otimes A\\
C&C\otimes A\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[A]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(i\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_R\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(i\otimes\Id_A\)} (m-2-2);
\end{tikzpicture}
\]
Another equivalent set of data is a pair of morphisms \(i\colon
A\to C\) and \(\Delta_L\colon C\to A\otimes C\) with commutative diagrams
\[
\begin{tikzpicture}
\matrix(m)[cd]{
A&A\otimes A\\
C&C\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[A]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(i\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Comult[C]\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(i\otimes i\)} (m-2-2);
\end{tikzpicture}\qquad
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&A\otimes C\\
C\otimes C&A\otimes C\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Delta_L\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Comult[C]\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_L\otimes\Id_C\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Id_A\otimes\Comult[C]\)} (m-2-2);
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&A\otimes C\\
A\otimes C&A\otimes A\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Delta_L\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Delta_L\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Id_A\otimes\Delta_L\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Comult[A]\otimes\Id_C\)} (m-2-2);
\end{tikzpicture}\qquad
\begin{tikzpicture}
\matrix(m)[cd]{
A&A\otimes A\\
C&A\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[A]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(i\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_L\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Id_A\otimes i\)} (m-2-2);
\end{tikzpicture}
\]
Finally, the quantum group with projection is equivalent to a
triple of morphisms \(i\colon A\to C\), \(\Delta_R\colon C\to
C\otimes A\) and \(\Delta_L\colon C\to A\otimes C\) satisfying all
the above conditions and, in addition,
\[
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&C\otimes C\\
C\otimes C&C\otimes A\otimes C\\
};
\draw[cdar] (m-1-1) -- node {\(\Comult[C]\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Comult[C]\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_R\otimes \Id_C\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Id_C\otimes \Delta_L\)} (m-2-2);
\end{tikzpicture}
\]
Then the following diagram also commutes:
\[
\begin{tikzpicture}
\matrix(m)[cd,column sep=4.5em]{
C&A\otimes C\\
C\otimes A&A\otimes C\otimes A\\
};
\draw[cdar] (m-1-1) -- node {\(\Delta_L\)} (m-1-2);
\draw[cdar] (m-1-1) -- node[swap] {\(\Delta_R\)} (m-2-1);
\draw[cdar] (m-2-1) -- node {\(\Delta_L\otimes \Id_A\)} (m-2-2);
\draw[cdar] (m-1-2) -- node {\(\Id_A\otimes \Delta_R\)} (m-2-2);
\end{tikzpicture}
\]
\end{proposition}
\begin{proof}
We have already seen that any projection on a \(\Cst\)\nb-quantum
group~\(\G[H]\) has an image~\(\G\) and that there are a bialgebra
morphism \(i\colon A\to C\) and a quantum group morphism
\(\hat{j}\colon \G[H]\to\G\) with \(\hat{j}\circ i=\Id_{\G}\) and
\(i\circ\hat{j} = p\), where~\(p\) denotes the given projection
on~\(\G[H]\). Now we describe~\(\hat{j}\) by a right quantum
group morphism~\(\Delta_R\) as in
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Definition 5.1}.
The first diagram above says that~\(i\)
is a bialgebra morphism. The second and third diagram together say
that~\(\Delta_R\)
is a right quantum group homomorphism from~\(C\)
to~\(A\).
The fourth diagram says that the composite \(A\to C\to A\)
of these quantum group morphisms is the identity map. Therefore,
the other composite \(C\to A\to C\)
is idempotent, hence a projection. Thus \(i\)
and~\(\Delta_R\)
give a projection on~\(\G[H]\)
with image~\(\G\).
Conversely, any projection on a \(\Cst\)\nb-quantum
group~\(\G[H]\)
has an image by Proposition~\ref{pro:idempotents_split}, which
gives~\(i\) and~\(\Delta_R\) as above.
Replacing right by left quantum group morphisms shows that
pairs~\((i,\Delta_L)\)
as above are also equivalent to \(\Cst\)\nb-quantum
groups with projection. Of the two diagrams that relate
\(\Delta_R\)
and~\(\Delta_L\),
the first one characterises when the right and left quantum group
homomorphisms \(\Delta_R\)
and~\(\Delta_L\)
describe the same quantum group morphism, and the second one
commutes automatically, see
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Lemma 5.7}.
\end{proof}
Let \(A\) and~\(B\) be~\(\Cst\)\nb-algebras and \(T\in\U(A\otimes
B)\). Then~\(B\) is \emph{generated by~\(T\)} in the sense of
\cite{Woronowicz:Cstar_generated}*{Definition 4.1} if, for any
representation \(\xi\colon B\to\Bound(\Hils)\) and any
\(\Cst\)\nb-algebra \(C\subset\Bound(\Hils)\), the condition
\((\Id_A\otimes\xi)T\in\Mult(A\otimes C)\) implies that
\(\xi\in\Mor(B,C)\).
\begin{definition}[\cite{Daws-Kasprzak-Skalski-Soltan:Closed_qnt_subgrps}*{Definition
3.2}]
\label{def:closed_qnt_sb_grp}
Let \(\Qgrp{I}{C}\) and \(\Qgrp{G}{A}\) be quantum groups. We
call~\(\G\) a \emph{closed quantum subgroup of\/~\(\G[I]\) in the
sense of Woronowicz} if there is a bicharacter
\(\bichar\in\U(\hat{C}\otimes A)\) that generates~\(\G\).
\end{definition}
In the situation of Proposition~\ref{pro:projection_via_action},
\((A,\Comult[A])\)
is indeed a closed quantum subgroup of~\((C,\Comult[C])\)
because the bicharacter
\((j\otimes\Id_A)(\multunit[A])\in \U(\hat{C}\otimes A)\)
generates~\(A\).
This is to be expected because \((A,\Comult[A])\)
is even a retract of~\((C,\Comult[C])\)
in the category of quantum group morphisms.
\subsection{Semidirect products}
\label{sec:semidirect}
In this section, we are going to show that the semidirect product
construction in \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Section
6} gives examples of \(\Cst\)\nb-quantum
groups with projection. Since we do not use this construction in the
rest of the article, we do not recall the notation and setup
from~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}. Readers unfamiliar
with the semidirect product construction
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} may skip this
section.
Let \(\G=(A,\Comult[A])\)
be a \(\Cst\)\nb-quantum
group. Let~\((B,\beta,\hat\beta)\)
be an \(A\)\nb-Yetter--Drinfeld
algebra, that is, \(\beta\colon B\to B\otimes A\)
and \(\hat\beta\colon B\to B\otimes \hat{A}\)
are continuous coactions of \(A\)
and~\(\hat{A}\)
that satisfy the compatibility condition in
\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Definition 5.11}. The
twisted tensor product \(B\boxtimes B = B\boxtimes_{\multunit} B\)
is defined in \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}. We also
require a coassociative comultiplication
\(\Comult[B]\colon B\to B\boxtimes B\).
Then \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Theorem 6.8}
describes a coassociative comultiplication~\(\Comult[C]\)
on \(C\defeq A\boxtimes B\)
and shows that the \(\Cst\)\nb-bialgebra
\(\G[H]= (C,\Comult[C])\)
is bisimplifiable if~\((B,\Comult[B])\)
is bisimplifiable. Furthermore, \(\Comult[C]\)
is injective if and only if~\(\Comult[B]\)
is injective. It is not studied
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} when
\((C,\Comult[C])\)
is a \(\Cst\)\nb-quantum
group: by our definition, this would require a multiplicative unitary
that generates it. If~\(C\)
is unital, then this automatically exists and we are dealing with a
compact quantum group. In the non-compact case, we need some sort of
multiplicative unitary for~\(B\) to get one for~\(C\).
For now, we disregard this issue. We want to describe a projection
on~\(\G[H]\)
with image~\(\G\),
and the description of projections in
Proposition~\ref{pro:projection_via_action} makes sense in our
situation. Thus we are going to define morphisms
\[
i\colon A\to C,\qquad
\Delta_R\colon C\to C\otimes A,\qquad
\Delta_L\colon C\to A\otimes C
\]
with the properties listed in
Proposition~\ref{pro:projection_via_action}. If we know for some
reason that~\(\G[H]\)
is a \(\Cst\)\nb-quantum
group, that is, comes from a manageable multiplicative unitary, then
\((i,\Delta_L,\Delta_R)\)
as in Proposition~\ref{pro:projection_via_action} give a projection
on~\(\G[H]\)
with image~\(\G\).
Actually, we only need either \(\Delta_L\)
or~\(\Delta_R\)
for this purpose. We provide both, however, and check all conditions in
Proposition~\ref{pro:projection_via_action}.
The morphism \(i\colon A\to A\boxtimes B = C\) is the canonical
embedding from the twisted tensor product, which is denoted~\(j_1\)
or~\(\iota_A\) in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}. The
right coaction \(\Delta_R\colon C\to C\otimes A\) is the one
constructed in \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Lemma
6.5}. It is the unique one for which the embeddings
\(i=\iota_A\colon A\to C\) and \(\iota_B\colon B\to C\) are
equivariant; that is,
\[
\Delta_R(\iota_A(a)\cdot \iota_B(b))
= (\iota_A\otimes\Id_A)(\Comult[A](a))
\cdot (\iota_B\otimes\Id_A)(\beta(b)).
\]
To construct~\(\Delta_L\), we equip \(A\otimes A\) with the right
\(A\)\nb-coaction \(\Id_A\otimes\Comult[A]\) on the second tensor
factor; this is a continuous \(A\)\nb-coaction, and
\(\Comult[A]\colon A\to A\otimes A\) is an \(A\)\nb-equivariant
morphism. Therefore, there is an \(A\)\nb-equivariant morphism
\(\Comult[A]\boxtimes \Id_B\colon A\boxtimes B \to (A\otimes
A)\boxtimes B\). We let~\(\Delta_L\) be the composite of
\(\Comult[A]\boxtimes \Id_B\) with the isomorphism \((A\otimes
A)\boxtimes B \cong A\otimes (A\boxtimes B) = A\otimes C\) from
\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Lemma 3.14}.
We may also rewrite
\[
A\boxtimes_{\multunit} B \cong B\boxtimes_{\Dumultunit} A
\]
by \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Proposition 5.1}.
This is exactly the reduced crossed product for the
\(\hat{A}\)\nb-coaction
on~\(B\)
by \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Section 6.3}. After
this identification, \(\Delta_L\)
becomes the dual coaction on the reduced crossed product as described
in \cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Section 6.3}.
\begin{proposition}
\label{pro:projection_on_semidirect_product}
The morphisms \(i\), \(\Delta_R\) and~\(\Delta_L\) constructed
above make all the diagrams in
Proposition~\textup{\ref{pro:projection_via_action}} commute.
\end{proposition}
\begin{proof}
Of the ten diagrams in
Proposition~\ref{pro:projection_via_action}, the last one commutes
automatically if the others do, and the first and fifth one
are the same. So we have to check eight commuting diagrams. The
maps \(\Comult[C]\), \(\Delta_R\) and~\(\Delta_L\) are defined to
have certain composites with \(\iota_A\) and~\(\iota_B\):
\begin{alignat*}{2}
\Comult[C]\circ\iota_A &= (\iota_A\otimes\iota_A)\Comult[A],&\qquad
\Comult[C]\circ\iota_B &= \Psi_{23}\circ\Comult[B],\\
\Delta_R \circ\iota_A &= (\iota_A\otimes\Id_A) \Comult[A],&\qquad
\Delta_R \circ\iota_B &= (\iota_B\otimes\Id_A) \beta,\\
\Delta_L \circ\iota_A &= (\Id_A \otimes\iota_A)\Comult[A],&\quad
\Delta_L \circ\iota_B &= 1_A \otimes\iota_B,
\end{alignat*}
where \(\Psi_{23}\colon B\boxtimes B\to C\otimes C\) is the
restriction of the map~\(\Psi\) in
\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}*{Proposition 6.6} to
the second two legs; that is, \(\Psi_{23}j_1(b) =
(\iota_B\otimes\iota_A)\beta(b)\) and \(\Psi_{23}j_2(b) =
(1\otimes\iota_B)(b)\) for all \(b\in B\).
In particular, \((\iota_A\otimes\iota_A)\circ \Comult[A] =
\Comult[C]\circ\iota_A\) says that the first and fifth diagram
commute, \(\Delta_R \circ\iota_A = (\iota_A\otimes\Id_A)
\Comult[A]\) says that the fourth diagram commutes, and \(\Delta_L
\circ\iota_A = (\Id_A \otimes\iota_A)\Comult[A]\) says that the
eighth diagram commutes.
The remaining diagrams in
Proposition~\ref{pro:projection_via_action} involve equalities of
two maps defined on~\(C\). Two maps \(f,f'\) defined on~\(C\) are
equal if and only if \(f\circ\iota_A = f'\circ\iota_A\) and
\(f\circ\iota_B = f'\circ\iota_B\). For all remaining diagrams, it is
trivial to check that they commute after composing
with~\(\iota_A\) because of the explicit formulas above. The
third and seventh diagram do not involve~\(\Comult[C]\), so the
composites with~\(\iota_B\) are also given explicitly, which makes
them trivial to check; in fact, they say simply that \(\Delta_R\)
and~\(\Delta_L\) are a right and a left coaction, respectively, which
is already checked in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}.
The condition on~\(B\) for the sixth diagram is also trivial
because~\(\Delta_L\) only does something complicated
on~\(\iota_A(A)\) and \(\Comult[C]\) maps~\(\iota_B(B)\) into
\(\iota_B(B)\otimes C\).
For the second diagram, we must check
\((\Comult[C]\otimes\Id_A)\Delta_R\iota_B =
(\Id_C\otimes\Delta_R)\Comult[C]\iota_B\). Since~\(\Comult[B]\)
is \(A\)\nb-equivariant, \((\Comult[B]\otimes \Id_A)\circ \beta =
(\beta\bowtie\beta)\circ\Comult[B]\). Using the definition
of~\(\Comult[C]\), we may rewrite our goal as
\((\Psi_{23}\otimes\Id_A)(\beta\bowtie\beta)\Comult[B] =
(\Id_C\otimes\Delta_R)\Psi_{23}\Comult[B]\). From this, we may
cancel the factor~\(\Comult[B]\), so it suffices to check that
\[
(\Psi_{23}\otimes\Id_A)(\beta\bowtie\beta)
= (\Id_C\otimes\Delta_R)\Psi_{23}.
\]
This is an equality of maps \(B\boxtimes B\to C\otimes C\otimes A\),
which we may check on both legs separately. On the first leg, this
reduces to the condition
\((\Id_B\otimes\Comult[A])\beta= (\beta\otimes\Id_A)\beta\)
that says that~\(\beta\)
is a coaction, and on the second leg this is trivial. This finishes
the proof that the second diagram commutes
In the condition from the ninth diagram on~\(B\), we may cancel
the factor~\(\Comult[B]\) from~\(\Comult[C]\), so it suffices to
check that \((\Id_C\otimes\Delta_L)\Psi_{23} =
(\Delta_R\otimes\Id_C)\Psi_{23}\) as maps \(B\boxtimes B\to
C\otimes A\otimes C\). This is once again checked separately on
the two factors~\(B\). So we must check that the maps
\(\Id_C\otimes\Delta_L\) and \(\Delta_R\otimes\Id_C\) take the
same values both on \((\iota_B\otimes\iota_A)\beta(b)\) and on
\(1\otimes\iota_B(b)\) for all \(b\in B\). This reduces to the
coaction condition for~\(\beta\) on
\((\iota_B\otimes\iota_A)\beta(b)\) and is trivial on
\(1\otimes\iota_B(b)\).
\end{proof}
\section{Braided Multiplicative Unitaries}
\label{sec:braided_mu}
The definition of a braided multiplicative unitary is as complicated
as the definition of a braided \(\Cst\)\nb-quantum
group. Recall that the latter is relative to a \(\Cst\)\nb-quantum
group \(\G=(A,\Comult[A])\)
which generates the braiding. The underlying
\(\Cst\)\nb-algebra~\(B\)
of a braided \(\Cst\)\nb-quantum
group carries continuous coactions \(\beta\)
and~\(\hat{\beta}\)
of \(\G\)
and~\(\DuG\),
respectively, which satisfy the Yetter--Drinfeld compatibility
condition which characterises coactions of the quantum codouble
of~\(\G\).
Finally, there is the comultiplication
\(\Comult[B]\colon B\to B\boxtimes B\),
which is equivariant with respect to \(\beta\)
and~\(\hat{\beta}\)
and coassociative. Thus a braided \(\Cst\)\nb-quantum
group contains four coactions or comultiplications \(\Comult[A]\),
\(\beta\),
\(\hat{\beta}\),
\(\Comult[B]\), which must satisfy seven algebraic conditions:
\begin{enumerate}
\item \(\Comult[A]\) is coassociative;
\item \(\beta\) is a coaction of \((A,\Comult[A])\);
\item \(\hat{\beta}\) is a coaction of \((\hat{A},\DuComult[A])\);
\item \(\beta\) and~\(\hat{\beta}\) satisfy the Drinfeld commutation
relation, so that they give a coaction of the quantum codouble;
\item \(\Comult[B]\) is equivariant with respect to the
coaction~\(\beta\);
\item \(\Comult[B]\) is equivariant with respect to the
coaction~\(\hat{\beta}\);
\item \(\Comult[B]\) is coassociative.
\end{enumerate}
The tensor product~\(\boxtimes\) is not symmetric unless~\(\G\) is
trivial. Thus \(X \boxtimes' Y \defeq Y\boxtimes X\) gives another
equally reasonable tensor product. We may also consider braided
quantum groups where the comultiplication takes values in
\(B\boxtimes' B\) instead of \(B\boxtimes B\). Actually, these
\(\Cst\)\nb-algebras are canonically isomorphic through the flip map,
which interchanges the two factors~\(B\). Thus there are two kinds of
braided \(\Cst\)-quantum group, and taking the ``coopposite,'' that
is, composing~\(\Comult[B]\) with the flip map~\(\Sigma\) and leaving
everything else the same, gives a bijection between the two types.
\begin{remark}
\label{rem:quasitriangular_simplify}
The definition above simplifies somewhat if~\(\G\) is
quasitriangular. Then a corepresentation~\(\beta\) determines a
corepresentation~\(\hat\beta\) so as to form a coaction of the
quantum codouble. Since~\(\hat\beta\) is a coaction constructed
naturally from~\(\beta\), the conditions (3), (4) and~(6) above are
redundant. A similar simplification occurs for braided
multiplicative unitaries. Since we are concerned with the general
theory here, we do not explore this situation any further.
\end{remark}
When we turn to multiplicative unitaries, we replace
\(\Cst\)\nb-algebras
by Hilbert spaces on which they act faithfully; comultiplications and
coactions are replaced by unitaries on appropriate tensor product
Hilbert spaces that implement the coactions through conjugation. So
to specify a braided multiplicative unitary, we need two Hilbert
spaces and four unitaries that satisfy seven conditions, which
correspond to the seven conditions for the comultiplications and
coactions listed above. Moreover, there are two slightly different
kinds of braided multiplicative unitaries, depending on whether we use
the ``standard'' braiding or its opposite; which braiding
is standard and which is opposite is, of course, a mere convention.
The following
definition contains the details:
\begin{definition}
\label{def:braided_multiplicative_unitary}
Let \(\Hils\) and~\(\Hils[L]\) be Hilbert spaces and let
\(\Multunit \in \U(\Hils\otimes\Hils)\) be a manageable
\emph{multiplicative unitary}; in particular, \(\Multunit\)
satisfies the \emph{pentagon equation}
\begin{equation}
\label{eq:pentagon}
\Multunit_{23}\Multunit_{12}
= \Multunit_{12}\Multunit_{13}\Multunit_{23}.
\end{equation}
A \emph{top-braided multiplicative unitary on~\(\Hils[L]\) relative
to~\(\Multunit\)} is given by unitaries
\[
\Corep{U}\in\U(\Hils[L]\otimes\Hils),\qquad
\DuCorep{V}\in\U(\Hils\otimes\Hils[L]),\qquad
\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])
\]
which satisfy the following conditions:
\begin{itemize}
\item \(\Corep{U}\) is a \emph{right corepresentation}
of~\(\Multunit\):
\begin{equation}
\label{eq:U_corep}
\Multunit_{23} \Corep{U}_{12}
= \Corep{U}_{12} \Corep{U}_{13} \Multunit_{23}
\quad\text{in }\U(\Hils[L]\otimes\Hils\otimes\Hils);
\end{equation}
\item \(\DuCorep{V}\) is a \emph{left corepresentation}
of~\(\Multunit\):
\begin{equation}
\label{eq:V_corep}
\DuCorep{V}_{23} \Multunit_{12}
= \Multunit_{12} \DuCorep{V}_{13} \DuCorep{V}_{23}
\quad\text{in }\U(\Hils\otimes\Hils\otimes\Hils[L]);
\end{equation}
\item the corepresentations \(\Corep{U}\) and~\(\DuCorep{V}\) are
\emph{Drinfeld compatible}:
\begin{equation}
\label{eq:U_V_compatible}
\Corep{U}_{23} \Multunit_{13} \DuCorep{V}_{12}
= \DuCorep{V}_{12} \Multunit_{13} \Corep{U}_{23}
\quad\text{in }\U(\Hils\otimes\Hils[L]\otimes\Hils);
\end{equation}
\item \(\BrMultunit\) is \emph{invariant} with respect to the
right corepresentation \(\Corep{U} \tenscorep \Corep{U} \defeq
\Corep{U}_{13}\Corep{U}_{23}\) of~\(\Multunit\)
on~\(\Hils[L]\otimes\Hils[L]\):
\begin{equation}
\label{eq:F_U-invariant}
\Corep{U}_{13} \Corep{U}_{23} \BrMultunit_{12}
= \BrMultunit_{12} \Corep{U}_{13} \Corep{U}_{23}
\quad\text{in }\U(\Hils[L]\otimes\Hils[L]\otimes \Hils);
\end{equation}
\item \(\BrMultunit\) is \emph{invariant} with respect to the left
corepresentation \(\DuCorep{V} \tenscorep \DuCorep{V} \defeq
\DuCorep{V}_{13}\DuCorep{V}_{12}\) of~\(\Multunit\)
on~\(\Hils[L]\otimes\Hils[L]\):
\begin{equation}
\label{eq:F_V-invariant}
\DuCorep{V}_{13} \DuCorep{V}_{12} \BrMultunit_{23}
= \BrMultunit_{23} \DuCorep{V}_{13} \DuCorep{V}_{12}
\quad\text{in }\U(\Hils\otimes\Hils[L]\otimes\Hils[L]);
\end{equation}
\item \(\BrMultunit\) satisfies the \emph{top-braided pentagon
equation}
\begin{equation}
\label{eq:top-braided_pentagon}
\BrMultunit_{23} \BrMultunit_{12}
= \BrMultunit_{12} (\Braiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{12} (\Dualbraiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{23}
\quad\text{in }\U(\Hils[L]\otimes\Hils[L]\otimes\Hils[L]);
\end{equation}
here the braiding \(\Braiding{\Hils[L]}{\Hils[L]} \in
\U(\Hils[L]\otimes \Hils[L])\) and
\(\Dualbraiding{\Hils[L]}{\Hils[L]} =
(\Braiding{\Hils[L]}{\Hils[L]})^*\) are defined as
\(\Braiding{\Hils[L]}{\Hils[L]} = Z \Flip\) for the
flip~\(\Flip\), \(x\otimes y\mapsto y\otimes x\), and the unique
unitary \(Z \in \U(\Hils[L]\otimes \Hils[L])\) that satisfies
\begin{equation}
\label{eq:braiding}
Z_{13} = \DuCorep{V}_{23} \Corep{U}_{12}^*
\DuCorep{V}_{23}^* \Corep{U}_{12}
\quad\text{in }\U(\Hils[L]\otimes\Hils\otimes\Hils[L]).
\end{equation}
\end{itemize}
A \emph{bottom-braided multiplicative unitary on~\(\Hils[L]\)
relative to~\(\Multunit\)} is given by the same unitaries
\(\Corep{U}\), \(\DuCorep{V}\), \(\BrMultunit\) satisfying
\eqref{eq:U_corep}--\eqref{eq:F_V-invariant} and the
\emph{bottom-braided pentagon equation}
\begin{equation}
\label{eq:bottom-braided_pentagon}
\BrMultunit_{23} \BrMultunit_{12}
= \BrMultunit_{12} (\Dualbraiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{12} (\Braiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{23}
\quad\text{in }\U(\Hils[L]\otimes\Hils[L]\otimes\Hils[L]).
\end{equation}
\end{definition}
Two corepresentations \(\Corep{U}\) and~\(\DuCorep{V}\) on a Hilbert
space~\(\Hils[L]\) satisfying~\eqref{eq:U_V_compatible} are equivalent
to a corepresentation of the quantum codouble of the quantum group
associated to~\(\Multunit\). It is shown
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} that these
corepresentations form a braided monoidal category. Our conventions
differ from those in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2}
because we use a left corepresentation~\(\DuCorep{V}\) instead of the
corresponding right corepresentation \(\Corep{V} \defeq \Sigma
\DuCorep{V}^* \Sigma\). The compatibility
condition~\eqref{eq:U_V_compatible} and the definition of the braiding
operator above are equivalent to those
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} up to this change of
notation. The operator~\(Z\) in~\eqref{eq:braiding} exists
because~\(\Multunit\) is manageable. It is shown
in~\cite{Meyer-Roy-Woronowicz:Twisted_tensor_2} that the
operators~\(\Braiding{\Hils[L]_1}{\Hils[L]_2}\) defined as above form
a braiding on the tensor category of triples
\((\Hils[L],\Corep{U},\DuCorep{V})\); the
operators~\(\Dualbraiding{\Hils[L]_1}{\Hils[L]_2}\) give the opposite
braiding.
In a braided monoidal category, the leg numbering notation should use
the braiding operators. This explains why we
replace~\(\BrMultunit_{13}\) by \((\Braiding{\Hils[L]}{\Hils[L]})_{23}
\BrMultunit_{12} (\Dualbraiding{\Hils[L]}{\Hils[L]})_{23}\) or
\((\Dualbraiding{\Hils[L]}{\Hils[L]})_{23} \BrMultunit_{12}
(\Braiding{\Hils[L]}{\Hils[L]})_{23}\) in the two braided pentagon
equations \eqref{eq:top-braided_pentagon}
and~\eqref{eq:bottom-braided_pentagon}. We should also have
replaced~\(\BrMultunit_{23}\) by
\(\Braiding{\Hils[L]}{\Hils[L]\otimes \Hils[L]} \BrMultunit_{12}
\Dualbraiding{\Hils[L]}{\Hils[L] \otimes \Hils[L]}\); the braiding
operator~\(\Braiding{\Hils[L]}{\Hils[L]\otimes \Hils[L]}\) is defined
as \(Z' \Sigma^{\Hils[L],\Hils[L]\otimes\Hils[L]}\), where~\(Z'\) is
the unique operator on \((\Hils[L]\otimes\Hils[L])\otimes\Hils[L]\)
with
\[
Z'_{134} = (\DuCorep{V} \tenscorep \DuCorep{V})_{234} \Corep{U}_{12}^*
(\DuCorep{V} \tenscorep \DuCorep{V})_{234} ^* \Corep{U}_{12}
\quad\text{in }
\U(\Hils[L]\otimes\Hils\otimes\Hils[L]\otimes\Hils[L]).
\]
Since we are dealing with a braided monoidal category, we also have
\[
\Braiding{\Hils[L]}{\Hils[L]\otimes \Hils[L]}
= \Braiding{\Hils[L]}{\Hils[L]}_{23}
\Braiding{\Hils[L]}{\Hils[L]}_{12},\qquad
\Braiding{\Hils[L]\otimes \Hils[L]}{\Hils[L]}
= \Braiding{\Hils[L]}{\Hils[L]}_{12}
\Braiding{\Hils[L]}{\Hils[L]}_{23}.
\]
Since~\(\BrMultunit\) is invariant with respect to both
corepresentations, it commutes with any operator that is constructed
in a natural way out of them, such as~\(Z'\). This implies
\[
\BrMultunit_{23}
= \Braiding{\Hils[L]}{\Hils[L]\otimes \Hils[L]}
\BrMultunit_{12} \Dualbraiding{\Hils[L]}{\Hils[L] \otimes \Hils[L]}
= \Dualbraiding{\Hils[L]}{\Hils[L]\otimes \Hils[L]}
\BrMultunit_{12} \Braiding{\Hils[L]}{\Hils[L] \otimes \Hils[L]},
\]
so here the braiding has no effect. This also implies
\[
\Braiding{\Hils[L]}{\Hils[L]}_{23} \BrMultunit_{12}
\Dualbraiding{\Hils[L]}{\Hils[L]}_{23} =
\Dualbraiding{\Hils[L]}{\Hils[L]}_{12} \BrMultunit_{23}
\Braiding{\Hils[L]}{\Hils[L]}_{12},\qquad
\Dualbraiding{\Hils[L]}{\Hils[L]}_{23} \BrMultunit_{12}
\Braiding{\Hils[L]}{\Hils[L]}_{23} =
\Braiding{\Hils[L]}{\Hils[L]}_{12} \BrMultunit_{23}
\Dualbraiding{\Hils[L]}{\Hils[L]}_{12}.
\]
Such equations are easier to digest as pictures:
\[
\begin{tikzpicture}[baseline=(current bounding box.west),scale=.4]
\draw (0,4)--(0,3.1);
\draw (1,4) to[out=315, in=90] (2,2.5) to[out=270,, in=45] (1,1);
\draw[overar] (2,4)--(1,3.1);
\draw[overar] (1,1.9)--(2,1);
\draw (0,1.9)--(0,1);
\draw (-.1,3.1)--(1.1,3.1)--(1.1,1.9)--(-.1,1.9)--(-.1,3.1);
\draw (0.5,2.4) node {$\BrMultunit$};
\draw (1.5,0) node {$\Braiding{\Hils[L]}{\Hils[L]}_{23} \BrMultunit_{12} \Dualbraiding{\Hils[L]}{\Hils[L]}_{23}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=(current bounding box.west),scale=.4]
\draw (1,4) to[out=225, in=90] (0,2.5) to[out=270,, in=135] (1,1);
\draw (2,4)--(2,3.1);
\draw[overar] (1,1.9)--(0,1);
\draw[overar] (0,4)--(1,3.1);
\draw (2,1.9)--(2,1);
\draw (.9,3.1)--(2.1,3.1)--(2.1,1.9)--(.9,1.9)--(.9,3.1);
\draw (1.5,2.4) node {$\BrMultunit$};
\draw (1.5,0) node {$\Dualbraiding{\Hils[L]}{\Hils[L]}_{12} \BrMultunit_{23} \Braiding{\Hils[L]}{\Hils[L]}_{12}$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=(current bounding box.west),scale=.4]
\draw (0,4)--(0,3.1);
\draw (2,4)--(1,3.1);
\draw (1,1.9)--(2,1);
\draw (0,1.9)--(0,1);
\draw[overar] (1,4) to[out=315, in=90] (2,2.5) to[out=270,, in=45] (1,1);
\draw (-.1,3.1)--(1.1,3.1)--(1.1,1.9)--(-.1,1.9)--(-.1,3.1);
\draw (0.5,2.4) node {$\BrMultunit$};
\draw (1.5,0) node {$\Dualbraiding{\Hils[L]}{\Hils[L]}_{23} \BrMultunit_{12} \Braiding{\Hils[L]}{\Hils[L]}_{23}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=(current bounding box.west),scale=.4]
\draw (2,4)--(2,3.1);
\draw (1,1.9)--(0,1);
\draw (0,4)--(1,3.1);
\draw (2,1.9)--(2,1);
\draw[overar] (1,4) to[out=225, in=90] (0,2.5) to[out=270,, in=135] (1,1);
\draw (.9,3.1)--(2.1,3.1)--(2.1,1.9)--(.9,1.9)--(.9,3.1);
\draw (1.5,2.4) node {$\BrMultunit$};
\draw (1.5,0) node {$\Braiding{\Hils[L]}{\Hils[L]}_{12} \BrMultunit_{23} \Dualbraiding{\Hils[L]}{\Hils[L]}_{12}$};
\end{tikzpicture}
\]
The top-braided pentagon equation~\eqref{eq:top-braided_pentagon}
uses the version of~\(\BrMultunit_{13}\) where~\(\BrMultunit\) acts
on the two top strands, whereas the bottom-braided pentagon
equation~\eqref{eq:bottom-braided_pentagon} uses the version
of~\(\BrMultunit_{13}\) where~\(\BrMultunit\) acts on the two bottom
strands; this explains our notation.
The braided pentagon equation is the usual pentagon equation if and
only if~\(\BrMultunit\) commutes with \(\Sigma Z\Sigma\). Sufficient
conditions for this are \(Z=1\), \(\Corep{U}=1\) or \(\DuCorep{V}=1\).
From now on, we restrict attention to top-braided multiplicative
unitaries, so \emph{braided multiplicative unitary} means
\emph{top-braided multiplicative unitary}.
\begin{definition}
\label{def:dual_brmult}
The \emph{dual} of a braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\) over~\(\Multunit\) is
\((\Corep{V},\DuCorep{U},\DuBrMultunit)\) over~\(\DuMultunit\),
where \(\DuMultunit\defeq \Sigma \Multunit[*] \Sigma\), \(\Corep{V}
\defeq \Sigma \DuCorep{V}^* \Sigma\), \(\DuCorep{U} \defeq \Sigma
\Corep{U}^* \Sigma\), and
\[
\DuBrMultunit\defeq \Dualbraiding{\Hils[L]}{\Hils[L]}
\BrMultunit^*\Braiding{\Hils[L]}{\Hils[L]} \in
\U(\Hils[L]\otimes\Hils[L]).
\]
\end{definition}
The braiding operator~\(\Braiding{\Hils[L]}{\Hils[L]}\) for
\((\DuMultunit,\Corep{V},\DuCorep{U})\) is the opposite
braiding~\(\Dualbraiding{\Hils[L]}{\Hils[L]}\) for
\((\Multunit,\Corep{U},\DuCorep{V})\). Therefore, the dual of the
dual is the braided multiplicative unitary that we started with,
even if the braiding is not symmetric.
\begin{proposition}
\label{pro:dual_brmult}
Let \((\Corep{U},\DuCorep{V},\BrMultunit)\) be a top-braided
multiplicative unitary over~\(\Multunit\). Its dual
\((\Corep{V},\DuCorep{U},\DuBrMultunit)\) is a top-braided
multiplicative unitary over \(\DuMultunit \defeq \Sigma
\Multunit[*] \Sigma\).
\end{proposition}
\begin{proof}
It is well-known that the dual~\(\DuMultunit\) is again a
multiplicative unitary, that~\(\Corep{U}\) is a right
corepresentation of~\(\Multunit\) if and only if~\(\DuCorep{U}\)
is a left corepresentation of~\(\DuMultunit\), and
that~\(\DuCorep{V}\) is a left corepresentation of~\(\Multunit\)
if and only if~\(\Corep{V}\) is a right corepresentation
of~\(\DuMultunit\). Routine computations show that the Drinfeld
compatibility condition and the invariance conditions are also
preserved by the duality. The top-braided (or bottom-braided)
pentagon equation for the dual is equivalent to the top-braided
(or bottom-braided) pentagon equation for the original braided
multiplicative unitary because the duality replaces the braiding
by the opposite braiding.
\end{proof}
Now we define when a braided multiplicative unitary
\((\Multunit,\Corep{U},\DuCorep{V},\BrMultunit)\)
is manageable. This requires~\(\Multunit\)
to be manageable, that is, there are a strictly positive
operator~\(Q\)
on~\(\Hils\)
and a unitary
\(\widetilde{\Multunit}\in \U(\conj{\Hils}\otimes\Hils)\)
with \(\Multunit[*](Q\otimes Q)\Multunit = Q\otimes Q\) and
\begin{equation}
\label{eq:Multunit_manageable}
\bigl(x\otimes u\mid \Multunit\mid z\otimes y\bigr)
= \bigl(\conj{z}\otimes Q u\mid \widetilde{\Multunit} \mid
\conj{x}\otimes Q^{-1}y\bigr)
\end{equation}
for all \(x,z\in\Hils\),
\(u\in\dom(Q)\)
and \(y\in\dom(Q^{-1})\)
(see \cite{Woronowicz:Mult_unit_to_Qgrp}*{Definition 1.2}).
Here~\(\conj{\Hils}\)
is the conjugate Hilbert space, and an operator is \emph{strictly
positive} if it is positive and self-adjoint with trivial kernel.
The condition \(\Multunit[*](Q\otimes Q)\Multunit = Q\otimes Q\)
means that the unitary~\(\Multunit\)
commutes with the unbounded operator~\(Q\otimes Q\).
\begin{definition}
\label{def:braided_manageable}
Let \(\Multunit\in\U(\Hils\otimes\Hils)\) be a manageable
multiplicative unitary and let \(Z\) and~\(Q\) be as above. A
braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\) over~\(\Multunit\) is
\emph{manageable} if there are a strictly positive
operator~\(Q_{\Hils[L]}\) on~\(\Hils[L]\) and a unitary
\(\widetilde{\BrMultunit} \widetilde{Z}{}^*\in
\U(\conj{\Hils[L]}\otimes\Hils[L])\) such that
\begin{align}
\label{eq:br_manag_commute_U}
\Corep{U}(Q_{\Hils[L]}\otimes Q)\Corep{U}^* &= Q_{\Hils[L]}\otimes Q,\\
\label{eq:br_manag_commute_V}
\DuCorep{V}(Q\otimes Q_{\Hils[L]})\DuCorep{V}^* &= Q\otimes Q_{\Hils[L]},\\
\label{eq:br_manag_commute_F}
\BrMultunit(Q_{\Hils[L]}\otimes Q_{\Hils[L]})\BrMultunit^* &= Q_{\Hils[L]}\otimes Q_{\Hils[L]},\\
\label{eq:br_manag}
(x\otimes u\mid Z^* \BrMultunit \mid y\otimes v) &=
(\conj{y}\otimes Q_{\Hils[L]}(u) \mid \widetilde{\BrMultunit} \widetilde{Z}{}^*
\mid \conj{x}\otimes Q_{\Hils[L]}^{-1}(v))
\end{align}
for all \(x,y\in\Hils[L]\), \(u\in\dom(Q_{\Hils[L]})\) and
\(v\in\dom(Q_{\Hils[L]}^{-1})\).
\end{definition}
We have written~\(\widetilde{\BrMultunit} \widetilde{Z}{}^*\) and
not~\(\widetilde{\BrMultunit}\) in~\eqref{eq:br_manag} to
make the formula more symmetric and to clarify the manageability of
the dual of a braided multiplicative unitary.
We now describe the operator~\(\widetilde{Z}\) that we want to use. The
corepresentation~\(\Corep{U}\) of~\(\Multunit\) on~\(\Hils[L]\)
induces a contragradient corepresentation on~\(\conj{\Hils[L]}\).
This is of the form~\(\widetilde{\Corep{U}}{}^*\), where
\(\widetilde{\Corep{U}}\in \U(\conj{\Hils[L]}\otimes\Hils)\)
satisfies a variant of~\eqref{eq:Multunit_manageable}, see
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Theorem 1.6} and
\cite{Soltan-Woronowicz:Multiplicative_unitaries}*{Proposition 10}.
Since~\(\widetilde{\Corep{U}}{}^*\) is a right corepresentation
of~\(\Multunit\) on~\(\conj{\Hils[L]}\), there is a unique unitary
\(\widetilde{Z}\in\U(\conj{\Hils[L]}\otimes \Hils[L])\) that
satisfies
\begin{equation}
\label{eq:braiding-manag}
\widetilde{Z}_{13} = \DuCorep{V}_{23}\widetilde{\Corep{U}}_{12}
\DuCorep{V}{}_{23}^*\widetilde{\Corep{U}}{}_{12}^*
\qquad
\text{in }\U(\conj{\Hils[L]}\otimes\Hils\otimes\Hils[L]).
\end{equation}
We use this unitary in~\eqref{eq:br_manag}. Of course, it does not
matter which unitary~\(\widetilde{Z}\) we use because we may absorb it
in~\(\widetilde{\BrMultunit}\).
\begin{proposition}
\label{prop:br_dual_manag}
The dual of a manageable braided multiplicative unitary is again
manageable.
\end{proposition}
\begin{proof}
Let \((\Corep{U},\DuCorep{V},\BrMultunit)\) be a manageable
top-braided multiplicative unitary over~\(\Multunit\), let \(Z\)
and~\(\widetilde{Z}\) be as in \eqref{eq:braiding}
and~\eqref{eq:braiding-manag}. Let \(\widetilde{\Multunit},Q\)
witness the manageability of~\(\Multunit\) and let
\(\widetilde{\BrMultunit}\) and~\(Q_{\Hils[L]}\) witness the manageability
of \((\Corep{U},\DuCorep{V},\BrMultunit)\).
On \(\Hils[L]\otimes\Hils\otimes\Hils[L]\), both
\(\Corep{U}_{12}\) and~\(\DuCorep{V}_{23}\) commute with
\(Q_{\Hils[L]}\otimes Q\otimes Q_{\Hils[L]}\) by \eqref{eq:br_manag_commute_U}
and~\eqref{eq:br_manag_commute_V}. Hence so does~\(Z\)
by~\eqref{eq:braiding}. Thus
\begin{equation}
\label{eq:Z_comm_Q}
Z(Q_{\Hils[L]}\otimes Q_{\Hils[L]})Z^* = Q_{\Hils[L]}\otimes Q_{\Hils[L]}.
\end{equation}
Together with~\eqref{eq:br_manag_commute_F}, this implies
that~\(Z^* \BrMultunit\) commutes with~\(Q_{\Hils[L]}\otimes
Q_{\Hils[L]}\). This together with~\eqref{eq:br_manag} implies
that~\(\widetilde{\BrMultunit} \widetilde{Z}{}^*\) commutes
with~\(Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1}\), compare
the proof of Lemma~\ref{lem:V_tilde_commute_Q} or
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Proposition 1.4.(1)}.
The unitary~\(\widetilde{\Corep{U}}\) commutes
with~\(Q_{\Hils[L]}^\transpose\otimes Q^{-1}\), compare
Lemma~\ref{lem:V_tilde_commute_Q} or
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Proposition 1.4.(1)}. This
together with \eqref{eq:br_manag_commute_V}
and~\eqref{eq:braiding-manag} implies
\begin{equation}
\label{eq:tilde-Z-W-comm_Q1}
\widetilde{Z}(Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1})\widetilde{Z}{}^*
= Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1},
\end{equation}
compare the proof of~\eqref{eq:Z_comm_Q}. Hence
\begin{equation}
\label{eq:tilde-Z-W-comm_Q2}
\widetilde{\BrMultunit}(Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1})
\widetilde{\BrMultunit}{}^*
= Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1}
\end{equation}
because~\(\widetilde{\BrMultunit} \widetilde{Z}{}^*\) commutes
with~\(Q_{\Hils[L]}^\transpose\otimes Q_{\Hils[L]}^{-1}\) as well.
If \(y\in\dom(Q_{\Hils[L]})\), \(x\in\dom(Q_{\Hils[L]}^{-1})\),
and \(u,v\in\Hils[L]\), then
\begin{equation}
\label{eq:equiv_br_manag_mod}
(x\otimes u\mid Z^* \BrMultunit \mid y\otimes v) =
(\conj{Q_{\Hils[L]}(y)}\otimes u \mid \widetilde{\BrMultunit}\widetilde{Z}{}^*
\mid \conj{Q_{\Hils[L]}^{-1}(x)}\otimes v);
\end{equation}
this is proved like
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Proposition 1.4 (2)}. We
rewrite this using the unitaries \(\widetilde{\hat{Z}},
\widetilde{\DuBrMultunit}\in \U(\conj{\Hils[L]}\otimes\Hils[L])\)
defined by
\[
\widetilde{\hat{Z}}\defeq
\bigl(\Flip\widetilde{Z}{}^*\Flip\bigr)^{\transpose\otimes\transpose},
\qquad
\widetilde{\DuBrMultunit}\defeq
\bigl(\Flip\widetilde{\BrMultunit}{}^*\Flip\bigr)^{\transpose\otimes\transpose}.
\]
By definition, \(\hat{Z}{}^* \DuBrMultunit = \Flip\BrMultunit{}^*
Z\Flip\) and \(\widetilde{\DuBrMultunit}\widetilde{\hat{Z}}{}^* =
(\Flip\widetilde{Z}\widetilde{\BrMultunit}{}^*
\Flip)^{\transpose\otimes\transpose}\).
Thus~\eqref{eq:equiv_br_manag_mod} gives
\begin{align*}
(x\otimes u\mid \hat{Z}{}^* \DuBrMultunit \mid y\otimes v) &=
(x\otimes u\mid \Flip\BrMultunit{}^*Z\Flip \mid y\otimes v)\\
&=\overline{ (y\otimes v\mid \Flip Z^*\BrMultunit\Flip \mid x\otimes u)}\\
&=\overline{(v\otimes y\mid Z^*\BrMultunit \mid u\otimes x)}\\
&=\overline{(\conj{Q_{\Hils[L]}(u)}\otimes y\mid
\widetilde{\BrMultunit} \widetilde{Z}{}^*\mid
\conj{Q_{\Hils[L]}^{-1}(v)} \otimes x)}\\
&=(\conj{Q_{\Hils[L]}^{-1}(v)}\otimes x\mid
\widetilde{Z}\widetilde{\BrMultunit}{}^*\mid
\conj{Q_{\Hils[L]}(u)}\otimes y)\\
&= (x\otimes\conj{Q_{\Hils[L]}^{-1}(v)}\mid
\Flip\widetilde{Z}\widetilde{\BrMultunit}{}^*\Flip\mid
y\otimes\conj{Q_{\Hils[L]}(u)})\\
&= (\conj{y}\otimes Q_{\Hils[L]}(u)\mid
(\Flip\widetilde{Z}\widetilde{\BrMultunit}{}^*\Flip)^{\transpose\otimes\transpose}
\mid \conj{x}\otimes Q_{\Hils[L]}^{-1}(v))\\
&= (\conj{y}\otimes Q_{\Hils[L]}(u) \mid
\widetilde{\DuBrMultunit}\widetilde{\hat{Z}}{}^*
\mid \conj{x}\otimes Q_{\Hils[L]}^{-1}(v)).
\end{align*}
Since the unitary~\(Z\) for the dual braided multiplicative
unitary becomes~\(\hat{Z}\), the operators \(Q_{\Hils[L]}\)
and~\(\widetilde{\DuBrMultunit}\) witness the manageability
of~\(\DuBrMultunit\).
\end{proof}
\subsection{Semidirect product multiplicative unitaries}
\label{sec:semidirect_product_mu}
In this section, we construct a semidirect product multiplicative
unitary~\(\Multunit[C]\)
and a projection~\(\ProjBichar\)
out of a braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\)
over a multiplicative unitary~\(\Multunit\).
We show that the semidirect product multiplicative
unitary~\(\Multunit[C]\)
is manageable if the braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\) is manageable.
The formulas and proofs below are explicit but lengthy because all
four unitaries \(\Multunit\),
\(\Corep{U}\),
\(\DuCorep{V}\),
\(\BrMultunit\)
must enter in the definitions of \(\Multunit[C]\)
and~\(\ProjBichar\)
and all seven conditions on them must be used in the proofs.
\begin{theorem}
\label{the:standard_mult_from_braided_and_standard}
Let \((\Corep{U},\DuCorep{V},\BrMultunit)\) be a braided
multiplicative unitary over a multiplicative unitary~\(\Multunit\).
Define \(\Multunit[C]_{1234},\ProjBichar \in
\U(\Hils\otimes\Hils[L]\otimes\Hils\otimes\Hils[L])\) by
\begin{align}
\label{eq:Stand_multunit_frm_brd}
\Multunit[C]_{1234} &\defeq
\Multunit_{13} \Corep{U}_{23} \DuCorep{V}^*_{34}
\BrMultunit_{24} \DuCorep{V}_{34},\\
\ProjBichar_{1234} &\defeq \Multunit_{13} \Corep{U}_{23}.
\end{align}
Then \(\Multunit[C]\) and~\(\ProjBichar\) satisfy the four
pentagon-like equations in
Proposition~\textup{\ref{pro:qg_projection_mu}}. Thus they give a
\(\Cst\)\nb-quantum group with projection when~\(\Multunit[C]\) is
manageable.
\end{theorem}
\begin{proof}
We first verify the pentagon equation~\eqref{eq:pentagon}
for~\(\Multunit[C]_{1234}\). Let
\[
XXX = \Multunit[C]_{3456} \Multunit[C]_{1234}
(\Multunit[C])^*_{3456}.
\]
We will rewrite this in several steps using the conditions in
Definition~\ref{def:braided_multiplicative_unitary}. We use
\(\{\ldots\}\) to highlight which part of the formula we are
modifying in the following step.
Definition~\eqref{eq:Stand_multunit_frm_brd} gives
\[
XXX
= \Multunit_{35} \{\Corep{U}_{45} \DuCorep{V}^*_{56}
\BrMultunit_{46} \DuCorep{V}_{56}\} \{\Multunit_{13}
\Corep{U}_{23}\} \DuCorep{V}^*_{34} \BrMultunit_{24}
\DuCorep{V}_{34} \DuCorep{V}^*_{56} \BrMultunit_{46}^*
\DuCorep{V}_{56} \Corep{U}_{45}^* \Multunit[*]_{35}.
\]
Since \(\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56}\) and \(\Multunit_{13} \Corep{U}_{23}\) commute,
\[
XXX =
\{\Multunit_{35} \Multunit_{13}\} \Corep{U}_{23} \Corep{U}_{45}
\DuCorep{V}^*_{56} \BrMultunit_{46} \{\DuCorep{V}_{56}\}
\DuCorep{V}^*_{34} \BrMultunit_{24} \DuCorep{V}_{34}
\{\DuCorep{V}^*_{56}\} \BrMultunit_{46}^*
\DuCorep{V}_{56} \Corep{U}_{45}^* \Multunit[*]_{35}.
\]
Now we use the pentagon equation~\eqref{eq:pentagon}
for~\(\Multunit\)
and commute \(\DuCorep{V}_{56}\)
with \(\DuCorep{V}^*_{34} \BrMultunit_{24} \DuCorep{V}_{34}\):
\[
XXX =
\Multunit_{13} \Multunit_{15} \{\Multunit_{35} \Corep{U}_{23}\}
\Corep{U}_{45} \DuCorep{V}^*_{56}
\{\BrMultunit_{46} \DuCorep{V}^*_{34}\} \BrMultunit_{24}
\{\DuCorep{V}_{34} \BrMultunit_{46}^*\}
\DuCorep{V}_{56} \Corep{U}_{45}^* \Multunit[*]_{35}.
\]
Equations \eqref{eq:U_corep} and~\eqref{eq:F_V-invariant} turn
this into
\[
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
\Multunit_{35} \Corep{U}_{45} \{\DuCorep{V}^*_{56}
\DuCorep{V}^*_{34}\} \DuCorep{V}^*_{36} \BrMultunit_{46}
\{\DuCorep{V}_{36} \BrMultunit_{24} \DuCorep{V}^*_{36}\}
\BrMultunit_{46}^* \DuCorep{V}_{36} \{\DuCorep{V}_{34}
\DuCorep{V}_{56}\} \Corep{U}_{45}^* \Multunit[*]_{35}.
\]
Commuting \(\DuCorep{V}_{56}\) with~\(\DuCorep{V}_{34}\) and
\(\DuCorep{V}_{36}\) with~\(\BrMultunit_{24}\) gives
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
\{\Multunit_{35} \Corep{U}_{45}
\DuCorep{V}^*_{34}\} \DuCorep{V}^*_{56} \DuCorep{V}^*_{36}
\BrMultunit_{46} \BrMultunit_{24}
\BrMultunit_{46}^* \DuCorep{V}_{36} \DuCorep{V}_{56}
\{\DuCorep{V}_{34} \Corep{U}_{45}^* \Multunit[*]_{35}\}.
\]
Now~\eqref{eq:U_V_compatible} gives
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
\DuCorep{V}^*_{34} \Corep{U}_{45} \{\Multunit_{35}
\DuCorep{V}^*_{56} \DuCorep{V}^*_{36}\}
\BrMultunit_{46} \BrMultunit_{24}
\BrMultunit_{46}^* \{\DuCorep{V}_{36} \DuCorep{V}_{56}
\Multunit[*]_{35}\} \Corep{U}_{45}^* \DuCorep{V}_{34}.
\]
We transform this using~\eqref{eq:V_corep}:
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
\DuCorep{V}^*_{34} \Corep{U}_{45} \DuCorep{V}^*_{56}
\{\Multunit_{35}\} \BrMultunit_{46} \BrMultunit_{24}
\BrMultunit_{46}^* \{\Multunit[*]_{35}\} \DuCorep{V}_{56}
\Corep{U}_{45}^* \DuCorep{V}_{34}.
\]
We commute~\(\Multunit_{35}\) with~\(\BrMultunit_{46}
\BrMultunit_{24} \BrMultunit_{46}^*\):
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
\DuCorep{V}^*_{34} \Corep{U}_{45} \DuCorep{V}^*_{56}
\{\BrMultunit_{46} \BrMultunit_{24} \BrMultunit_{46}^*\}
\DuCorep{V}_{56} \Corep{U}_{45}^* \DuCorep{V}_{34}.
\]
Now we use the braided pentagon
equation~\eqref{eq:top-braided_pentagon} and the definition of the
braiding through~\(Z\):
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \{\Corep{U}_{25}
\DuCorep{V}^*_{34}\} \Corep{U}_{45}
\{\DuCorep{V}_{56}^* \BrMultunit_{24}\} Z_{46} \BrMultunit_{26} Z^*_{46}
\DuCorep{V}_{56} \Corep{U}_{45}^* \DuCorep{V}_{34}.
\]
Now we commute \(\Corep{U}_{25}\) with~\(\DuCorep{V}^*_{34}\),
\(\DuCorep{V}_{56}^*\) with~\(\BrMultunit_{24}\):
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \DuCorep{V}^*_{34}
\Corep{U}_{25} \Corep{U}_{45} \BrMultunit_{24}
\{\DuCorep{V}_{56}^* Z_{46}\} \BrMultunit_{26} \{Z^*_{46}
\DuCorep{V}_{56} \Corep{U}_{45}^*\} \DuCorep{V}_{34}.
\]
Equation~\eqref{eq:braiding} implies \(\Corep{U}_{45}
\DuCorep{V}_{56}^* Z_{46} = \DuCorep{V}_{56}^* \Corep{U}_{45}\),
so this becomes
\[
XXX =
\Multunit_{13} \Multunit_{15} \Corep{U}_{23} \DuCorep{V}^*_{34}
\{\Corep{U}_{25} \Corep{U}_{45}
\BrMultunit_{24} \Corep{U}^*_{45}\} \DuCorep{V}^*_{56}
\{\Corep{U}_{45} \BrMultunit_{26}\Corep{U}^*_{45}\}
\DuCorep{V}_{56} \DuCorep{V}_{34}.
\]
Now we use~\eqref{eq:F_U-invariant} and
commute~\(\BrMultunit_{26}\) with~\(\Corep{U}_{45}\):
\[
XXX =
\Multunit_{13} \{\Multunit_{15}\} \{\Corep{U}_{23} \DuCorep{V}^*_{34}
\BrMultunit_{24}\} \{\Corep{U}_{25} \DuCorep{V}^*_{56} \BrMultunit_{26}
\DuCorep{V}_{56}\} \{\DuCorep{V}_{34}\}.
\]
Finally, we commute \(\Multunit_{15}\) with~\(\Corep{U}_{23}
\DuCorep{V}^*_{34} \BrMultunit_{24}\) and \(\Multunit_{15}
\Corep{U}_{25} \DuCorep{V}^*_{56} \BrMultunit_{26}
\DuCorep{V}_{56}\) with~\(\DuCorep{V}_{34}\) to get
\[
XXX
= \{\Multunit_{13} \Corep{U}_{23} \DuCorep{V}^*_{34}
\BrMultunit_{24} \DuCorep{V}_{34}\} \{\Multunit_{15}
\Corep{U}_{25} \DuCorep{V}^*_{56} \BrMultunit_{26} \DuCorep{V}_{56}\}
= \Multunit[C]_{1234} \Multunit[C]_{1256}.
\]
This is the desired pentagon equation for~\(\Multunit[C]_{1234}\).
Next we show that~\(\ProjBichar\) satisfies the pentagon equation:
\begin{align*}
\ProjBichar_{3456} \ProjBichar_{1234} \ProjBichar_{3456}
&= \Multunit_{35} \Corep{U}_{45} \Multunit_{13} \Corep{U}_{23}
\Corep{U}_{45}^* \Multunit[*]_{35}
= \Multunit_{35} \Multunit_{13} \Corep{U}_{23} \Multunit[*]_{35}
\\&= \Multunit_{13} \Multunit_{15} \Multunit_{35} \Corep{U}_{23}
\Multunit[*]_{35}
= \Multunit_{13} \Multunit_{15} \Corep{U}_{23} \Corep{U}_{25}
= \Multunit_{13} \Corep{U}_{23} \Multunit_{15} \Corep{U}_{25}
\\&= \ProjBichar_{1234} \ProjBichar_{1256}.
\end{align*}
The first and last equalities are the definition
of~\(\ProjBichar\); the second step commutes \(\Corep{U}_{45}\)
with~\(\Multunit_{13} \Corep{U}_{23}\); the third step uses the
pentagon equation~\eqref{eq:pentagon} for~\(\Multunit\); the
fourth step uses~\eqref{eq:U_corep}; the fifth step commutes
\(\Multunit_{15}\) with~\(\Corep{U}_{23}\).
Next we prove
\(\ProjBichar_{3456} \Multunit[C]_{1234} = \Multunit[C]_{1234}
\ProjBichar_{1256} \ProjBichar_{3456}\)
or, equivalently,
\(\ProjBichar_{3456} \Multunit[C]_{1234} \ProjBichar_{3456}^* =
\Multunit[C]_{1234} \ProjBichar_{1256}\):
\begin{align*}
\ProjBichar_{3456} \Multunit[C]_{1234} \ProjBichar_{3456}^*
&= \Multunit_{35} \Corep{U}_{45} \Multunit_{13}
\Corep{U}_{23} \DuCorep{V}^*_{34} \BrMultunit_{24}
\DuCorep{V}_{34} \Corep{U}_{45}^* \Multunit[*]_{35}
\\&= \Multunit_{35} \Multunit_{13} \Corep{U}_{23}
\Corep{U}_{45} \DuCorep{V}^*_{34} \BrMultunit_{24}
\DuCorep{V}_{34} \Corep{U}_{45}^* \Multunit[*]_{35}
\\&= \Multunit_{13} \Multunit_{15} \Multunit_{35}
\Corep{U}_{23} \Corep{U}_{45} \DuCorep{V}^*_{34}
\BrMultunit_{24} \DuCorep{V}_{34}
\Corep{U}_{45}^* \Multunit[*]_{35}
\\&= \Multunit_{13} \Multunit_{15} \Corep{U}_{23}
\Corep{U}_{25} \Multunit_{35} \Corep{U}_{45}
\DuCorep{V}^*_{34} \BrMultunit_{24} \DuCorep{V}_{34}
\Corep{U}_{45}^* \Multunit[*]_{35}
\\&= \Multunit_{13} \Corep{U}_{23} \Multunit_{15}
\Corep{U}_{25} \DuCorep{V}^*_{34} \Corep{U}_{45}
\Multunit_{35} \BrMultunit_{24} \Multunit[*]_{35} \Corep{U}_{45}^*
\DuCorep{V}_{34}
\\&= \Multunit_{13} \Corep{U}_{23} \DuCorep{V}^*_{34}
\Multunit_{15} \Corep{U}_{25} \Corep{U}_{45} \BrMultunit_{24}
\Corep{U}_{45}^* \DuCorep{V}_{34}
\\&= \Multunit_{13} \Corep{U}_{23} \DuCorep{V}^*_{34}
\Multunit_{15} \BrMultunit_{24} \Corep{U}_{25}
\DuCorep{V}_{34}
\\&= \Multunit_{13} \Corep{U}_{23} \DuCorep{V}^*_{34}
\BrMultunit_{24} \DuCorep{V}_{34} \Multunit_{15}
\Corep{U}_{25}
= \Multunit[C]_{1234} \ProjBichar_{1256}.
\end{align*}
The first and last equalities are the definitions of
\(\Multunit[C]\) and~\(\ProjBichar\); the second, sixth, and
eighth steps commute unitaries in different legs; the third step
uses the pentagon equation~\eqref{eq:pentagon} for~\(\Multunit\);
the fourth step uses~\eqref{eq:U_corep}; the fifth step
uses~\eqref{eq:U_V_compatible} and commutes unitaries in different
legs; the seventh step uses~\eqref{eq:F_U-invariant}.
Finally, we prove \(\Multunit[C]_{3456} \ProjBichar_{1234} =
\ProjBichar_{1234} \ProjBichar_{1256} \Multunit[C]_{3456}\) by
computing \(\ProjBichar_{1234}^* \Multunit[C]_{3456}
\ProjBichar_{1234}\):
\begin{align*}
\ProjBichar_{1234}^* \Multunit[C]_{3456} \ProjBichar_{1234}
&= \Corep{U}_{23}^* \Multunit[*]_{13} \Multunit_{35}
\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56} \Multunit_{13} \Corep{U}_{23}
\\&=\Corep{U}_{23}^* \Multunit[*]_{13} \Multunit_{35}\Multunit_{13}
\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56}\Corep{U}_{23}
\\&= \Corep{U}_{23}^* \Multunit_{15}\Multunit_{35}
\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56}\Corep{U}_{23}
\\&=\Multunit_{15}\Corep{U}_{23}^* \Multunit_{35} \Corep{U}_{23}
\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56}
\\&=\Multunit_{15}\Corep{U}_{25}\Multunit_{35}
\Corep{U}_{45} \DuCorep{V}^*_{56} \BrMultunit_{46}
\DuCorep{V}_{56}=\ProjBichar_{1256}\Multunit[C]_{3456}.
\end{align*}
The first and last steps are the definitions of \(\Multunit[C]\)
and~\(\ProjBichar\); the second and fourth steps commute unitaries
in different legs; the third step uses the pentagon
equation~\eqref{eq:pentagon} for~\(\Multunit\); the fifth step
uses~\eqref{eq:U_corep}.
\end{proof}
\begin{theorem}
\label{the:mang_mult_from_braided_and_standard}
Let \(\Multunit\) be a manageable multiplicative unitary and let
\((\Corep{U},\DuCorep{V},\BrMultunit)\) be a manageable braided
multiplicative unitary over~\(\Multunit\). Then the multiplicative
unitaries \(\Multunit[C] \defeq \Multunit_{13} \Corep{U}_{23}
\DuCorep{V}^*_{34} \BrMultunit_{24} \DuCorep{V}_{34}\) and
\(\ProjBichar \defeq \Multunit_{13} \Corep{U}_{23}\) are manageable.
\end{theorem}
\begin{proof}
Let \(\widetilde{\Multunit}\)
and~\(Q\)
witness the manageability of~\(\Multunit\),
and let~\(\widetilde{\BrMultunit}\)
and~\(Q_{\Hils[L]}\)
witness the manageability of~\(\BrMultunit\).
The construction of the unitary~\(\widetilde{\Corep{U}}\)
in~\eqref{eq:second_leg_bichar_adapted} works for any right
corepresentation of~\(\Multunit\)
by the same argument; in particular, it works for~\(\Corep{U}\),
so we get
\(\widetilde{\Corep{U}} \in \U(\conj{\Hils[L]}\otimes\Hils)\) with
\begin{equation}
\label{eq:Corep_U_tilde}
\bigl<x\otimes u\big| \Corep{U}\big| z\otimes y\bigr>
= \bigl<\conj{z}\otimes Q u\big| \widetilde{\Corep{U}}\big|
\conj{x}\otimes Q^{-1}y\bigr>
\end{equation}
for all \(x,z\in\Hils[L]\), \(u\in\dom(Q)\) and
\(y\in\dom(Q^{-1})\).
Let \(Q^C \defeq Q\otimes Q_{\Hils[L]}\in\U(\Hils\otimes\Hils[L])\) and
\begin{equation}
\label{eq:manageable_W_1234}
\widetilde{\Multunit}{}^C_{1234} \defeq
\widetilde{\Multunit}_{13} \DuCorep{V}^*_{34}
\widetilde{\BrMultunit}_{24} \widetilde{\Corep{U}}_{23}
\DuCorep{V}_{34}
\in \U(\conj{\Hils}\otimes\conj{\Hils[L]}\otimes
\Hils\otimes\Hils[L]).
\end{equation}
We claim that these operators witness the manageability
of~\(\Multunit[C]\). It is clear that~\(Q^C\) is strictly positive.
The operators \(\Multunit_{13}\), \(\DuCorep{V}_{34}\),
\(\Corep{U}_{23}\) and~\(\BrMultunit_{24}\) all commute with
\(Q^C\otimes Q^C = Q\otimes Q_{\Hils[L]}\otimes Q\otimes
Q_{\Hils[L]}\) by the manageability assumptions.
Hence~\(\Multunit[C]\) commutes with~\(Q^C\otimes Q^C\). It
remains to check~\eqref{eq:Multunit_manageable} for
\(\Multunit[C]\), \(\widetilde{\Multunit}{}^C\) and~\(Q^C\). We
relegate this technical computation to
Lemma~\ref{lem:mang_mult_from_braided_and_standard} in the
appendix. This finishes the proof that~\(\Multunit[C]\) is
manageable. Now Proposition~\ref{pro:ProjBichar_manageable} shows
that~\(\ProjBichar\) is manageable as well.
\end{proof}
\subsection{Analysis of a quantum group with projection}
\label{sec:Qnt_grp_with_proj}
In this section, we construct a braided multiplicative unitary from a
quantum group with projection. Our starting point is a Hilbert
space~\(\Hils\) with two unitaries
\(\Multunit[C],\ProjBichar\in\U(\Hils\otimes\Hils)\) satisfying the
conditions in Proposition~\ref{pro:qg_projection_mu}. We must
construct another Hilbert space~\(\Hils[L]\) with operators
\(\Corep{U}\in\U(\Hils[L]\otimes\Hils)\),
\(\DuCorep{V}\in\U(\Hils\otimes\Hils[L])\) and
\(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\) as in
Definition~\ref{def:braided_multiplicative_unitary}.
In particular, the corepresentations \(\Corep{U}\)
and~\(\DuCorep{V}\) form a Drinfeld pair for the multiplicative
unitary~\(\ProjBichar\). The simplest
general construction of such a Drinfeld pair lives on the tensor
product Hilbert space \(\Hils[L] \defeq \conj{\Hils} \otimes
\Hils\), where~\(\conj{\Hils}\) denotes the conjugate Hilbert space
of~\(\Hils\). Therefore, we will use this rather large Hilbert
space.
Let \(\G = (A,\Comult[A])\) be the \(\Cst\)\nb-quantum group generated
by~\(\ProjBichar\), which is manageable by
Proposition~\ref{pro:ProjBichar_manageable}.
Let \(\G[H] = (C,\Comult[C])\)
be the \(\Cst\)\nb-quantum
group generated by the manageable multiplicative
unitary~\(\Multunit[C]\).
By construction, we have inclusion maps
\(\iota \colon C\to \Bound(\Hils)\)
and \(\hat{\iota} \colon C\to \Bound(\Hils)\),
which are non-degenerate \Star{}homomorphisms. The reduced
bicharacter is the unique unitary
\(\multunit[C]\in\U(\hat{C}\otimes C)\)
with \(\Multunit[C] = (\hat{\iota}\otimes \iota)(\multunit[C])\)
or, briefly, \(\Multunit[C]= \multunit[C]_{\hat{\iota}\iota}\).
By construction, \(A\subseteq \Mult(C)\)
and \(\hat{A}\subseteq \Mult(\hat{C})\)
as \(\Cst\)\nb-subalgebras of \(\Bound(\Hils)\).
The representations \((\iota,\hat{\iota})\)
form a Heisenberg pair for the quantum group~\((C,\Comult[C])\)
in the notation of~\cite{Meyer-Roy-Woronowicz:Twisted_tensor}. This
Heisenberg pair generates an anti-Heisenberg pair
\(\alpha\colon C\to\Bound(\conj{\Hils})\),
\(\hat\alpha\colon \hat{C} \to \Bound(\conj{\Hils})\)
by \cite{Meyer-Roy-Woronowicz:Twisted_tensor}*{Lemma 3.6}. Thus
\begin{equation}
\label{eq:anti-Heisenberg_MultunitC}
\multunit[C]_{1 \alpha} \multunit[C]_{\hat\alpha 3}
= \multunit[C]_{\hat\alpha 3} \multunit[C]_{1 3} \multunit[C]_{1 \alpha}
\qquad\text{in }\U(\hat{C} \otimes \Comp(\conj{\Hils}) \otimes \hat{C}).
\end{equation}
The restriction of a Heisenberg or anti-Heisenberg pair for~\(\G[H]\)
to~\(\G\)
remains a Heisenberg or anti-Heisenberg pair, respectively. Thus
\begin{equation}
\label{eq:anti-Heisenberg_MultunitA}
\projbichar_{1 \alpha}\projbichar_{\hat\alpha 3}
= \projbichar_{\hat\alpha 3} \projbichar_{1 3} \projbichar_{1 \alpha}
\qquad\text{in } \U(\hat{A} \otimes \Comp(\conj{\Hils}) \otimes \hat{A}).
\end{equation}
To make computations shorter, we shall use leg numbering notation such
as
\(\projbichar_{ij}, \multunit[C]_{ij}\in \U(\conj{\Hils} \otimes \Hils
\otimes \conj{\Hils} \otimes\Hils)\)
for \(1\le i<j\le 4\).
This means the unitary acting on the \(i\)th
and \(j\)th
tensor factor by applying the appropriate representations of \(C\)
or~\(\hat{C}\)
to the two legs of \(\projbichar\)
or~\(\multunit[C]\),
respectively. For instance,
\(\projbichar_{12} = (\hat\alpha\otimes \iota)(\projbichar) \otimes
1_{\conj{\Hils} \otimes \Hils}\).
This notation is not ambiguous if we also specify the Hilbert space on
which the operator acts because we have given one representation of
\(C\) and~\(\hat{C}\) on \(\Hils\) and~\(\conj{\Hils}\) each. We let
\begin{alignat}{2}
\label{eq:U_from_projection}
\Corep{U} &\defeq \projbichar_{23} \projbichar_{13}
\defeq (\hat{\iota}\otimes\iota)\projbichar_{23} \cdot
(\hat{\alpha}\otimes\iota)\projbichar_{13}
&\qquad &\text{in } \U(\conj{\Hils} \otimes \Hils \otimes \Hils),\\
\label{eq:V_from_projection}
\DuCorep{V} &\defeq \projbichar_{12} \projbichar_{13}
\defeq (\hat{\iota}\otimes\alpha)\projbichar_{12} \cdot
(\hat{\iota}\otimes\iota)\projbichar_{13}
&\qquad &\text{in } \U(\Hils \otimes \conj{\Hils} \otimes \Hils),\\
\label{eq:F_from_projection}
\BrMultunit &\defeq \projbichar_{14}^* \projbichar_{24}^*
\multunit[C]_{24} \multunit[C]_{14}
&\qquad &\text{in } \U(\conj{\Hils} \otimes \Hils \otimes
\conj{\Hils} \otimes \Hils).
\end{alignat}
\begin{theorem}
\label{the:analysis_qg_projection}
The unitaries \(\ProjBichar\in\U(\Hils\otimes\Hils)\),
\(\Corep{U}\in\U(\Hils[L]\otimes\Hils)\),
\(\DuCorep{V}\in\U(\Hils\otimes\Hils[L])\),
and \(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\)
form a braided multiplicative unitary.
\end{theorem}
The proof of Theorem~\ref{the:analysis_qg_projection} will take some
work. The precise formulas for \(\alpha\) and~\(\hat{\alpha}\) will
only matter in the end when we check the manageability of our braided
multiplicative unitary. Thus our construction really uses
that~\(\Multunit[C]\) is manageable (or at least modular). The
pentagon equation~\eqref{eq:pentagon} for~\(\ProjBichar\) holds by
assumption. Equations \eqref{eq:U_corep} and~\eqref{eq:V_corep},
which say that \(\Corep{U}\) and~\(\DuCorep{V}\) are
corepresentations, amount to
\begin{alignat*}{2}
\projbichar_{34} \projbichar_{23} \projbichar_{13}
&= \projbichar_{23} \projbichar_{13} \projbichar_{24} \projbichar_{14}
\projbichar_{34}
&\qquad&\text{in }\U(\conj{\Hils} \otimes \Hils \otimes \Hils\otimes
\Hils),\\
\projbichar_{23} \projbichar_{24} \projbichar_{12}
&= \projbichar_{12} \projbichar_{13} \projbichar_{14} \projbichar_{23}
\projbichar_{24}
&\qquad&\text{in }\U(\Hils \otimes \Hils\otimes \conj{\Hils} \otimes
\Hils).
\end{alignat*}
We get both equations using the pentagon equation for~\(\ProjBichar\)
twice, in legs where the representations of \(\G[H]\)
and hence of~\(\G\)
form a Heisenberg pair. Thus \(\Corep{U}\)
and~\(\DuCorep{V}\) are corepresentations of~\(\ProjBichar\). The
Drinfeld compatibility condition~\eqref{eq:U_V_compatible} becomes
\[
\projbichar_{34} \projbichar_{24} \projbichar_{14}
\projbichar_{12} \projbichar_{13}
= \projbichar_{12} \projbichar_{13} \projbichar_{14} \projbichar_{34}
\projbichar_{24}
\qquad\text{in }\U(\Hils\otimes \conj{\Hils} \otimes \Hils \otimes
\Hils\otimes \Hils).
\]
The anti-Heisenberg and Heisenberg properties of our representations
on the second and third leg give
\(\projbichar_{24} \projbichar_{14} \projbichar_{12} =
\projbichar_{12} \projbichar_{24}\)
and
\(\projbichar_{13} \projbichar_{14} \projbichar_{34} =
\projbichar_{34} \projbichar_{13}\). Hence
\[
\projbichar_{34} \projbichar_{24} \projbichar_{14}
\projbichar_{12} \projbichar_{13}
= \projbichar_{34} \projbichar_{12} \projbichar_{24} \projbichar_{13}
= \projbichar_{12} \projbichar_{34} \projbichar_{13} \projbichar_{24}
= \projbichar_{12} \projbichar_{13} \projbichar_{14} \projbichar_{34}
\projbichar_{24}
\]
as needed. Thus \(\Corep{U}\)
and~\(\DuCorep{V}\)
are Drinfeld compatible. The equivariance of~\(\BrMultunit\)
with respect to~\(\Corep{U}\) in~\eqref{eq:F_U-invariant} amounts to
\begin{equation}
\label{eq:F_U-invariant_analysis}
\projbichar_{25} \projbichar_{15} \projbichar_{45} \projbichar_{35}
\projbichar_{14}^* \projbichar_{24}^* \multunit[C]_{24} \multunit[C]_{14}
= \projbichar_{14}^* \projbichar_{24}^* \multunit[C]_{24} \multunit[C]_{14}
\projbichar_{25} \projbichar_{15} \projbichar_{45} \projbichar_{35}
\end{equation}
in \(\U(\conj{\Hils} \otimes \Hils \otimes \conj{\Hils} \otimes
\Hils\otimes \Hils)\).
Since we have a Heisenberg pair on the fourth leg, we may use the
conditions in Proposition~\ref{pro:qg_projection_mu} to simplify
\[
\multunit[C]_{24} \multunit[C]_{14}
\projbichar_{25} \projbichar_{15} \projbichar_{45}
= \multunit[C]_{24} \projbichar_{25} (\multunit[C]_{14} \projbichar_{15}
\projbichar_{45})
= \multunit[C]_{24} \projbichar_{25} \projbichar_{45} \multunit[C]_{14}
= \projbichar_{45} \multunit[C]_{24} \multunit[C]_{14}.
\]
Hence the right side in~\eqref{eq:F_U-invariant_analysis} becomes
\[
\projbichar_{14}^* \projbichar_{24}^* \multunit[C]_{24} \multunit[C]_{14}
\projbichar_{25} \projbichar_{15} \projbichar_{45} \projbichar_{35}
= \projbichar_{14}^* \projbichar_{24}^* \projbichar_{45}
\multunit[C]_{24} \multunit[C]_{14} \projbichar_{35}.
\]
Plugging this in and cancelling
\(\multunit[C]_{24} \multunit[C]_{14} \projbichar_{35}\),
we see that~\eqref{eq:F_U-invariant_analysis} is equivalent to
\[
\projbichar_{25} \projbichar_{15} \projbichar_{45} \projbichar_{14}^*
\projbichar_{24}^*
= \projbichar_{14}^* \projbichar_{24}^* \projbichar_{45}
\quad\text{or}\quad
\projbichar_{24} \projbichar_{25} \projbichar_{14} \projbichar_{15}
\projbichar_{45} = \projbichar_{45} \projbichar_{24} \projbichar_{14}.
\]
Since we have Heisenberg pairs on the second and fourth legs, this
follows by two applications of the pentagon equation
for~\(\projbichar\).
The condition~\eqref{eq:F_V-invariant} about~\(\BrMultunit\)
being equivariant with respect to~\(\DuCorep{V}\) becomes
\begin{equation}
\label{eq:F_V-invariant_analysis}
\projbichar_{14} \projbichar_{15} \projbichar_{12} \projbichar_{13}
\projbichar_{25}^* \projbichar_{35}^* \multunit[C]_{35} \multunit[C]_{25}
= \projbichar_{25}^* \projbichar_{35}^* \multunit[C]_{35} \multunit[C]_{25}
\projbichar_{14} \projbichar_{15} \projbichar_{12} \projbichar_{13}
\end{equation}
in
\(\U(\Hils\otimes \conj{\Hils} \otimes \Hils \otimes \conj{\Hils}
\otimes \Hils)\).
Since we have an anti-Heisenberg pair on the second and a Heisenberg
pair on the third leg, we get
\(\multunit[C]_{25} \projbichar_{15} \projbichar_{12} =
\projbichar_{12} \multunit[C]_{25}\)
and
\(\projbichar_{13} \projbichar_{15} \multunit[C]_{35} =
\multunit[C]_{35} \projbichar_{13}\).
Thus the right side of~\eqref{eq:F_V-invariant_analysis} becomes
\begin{align*}
&\phantom{{}={}}\projbichar_{25}^* \projbichar_{35}^* \multunit[C]_{35}
\multunit[C]_{25} \projbichar_{14} \projbichar_{15} \projbichar_{12}
\projbichar_{13}\\
&= \projbichar_{14} \projbichar_{25}^* \projbichar_{35}^*
\multunit[C]_{35} (\multunit[C]_{25} \projbichar_{15}
\projbichar_{12}) \projbichar_{13}
= \projbichar_{14} \projbichar_{25}^* \projbichar_{35}^*
\multunit[C]_{35} \projbichar_{12}
\multunit[C]_{25} \projbichar_{13}
\\&= \projbichar_{14} \projbichar_{25}^* \projbichar_{35}^* \projbichar_{12}
(\multunit[C]_{35} \projbichar_{13}) \multunit[C]_{25}
= \projbichar_{14} \projbichar_{25}^* \projbichar_{35}^* \projbichar_{12}
\projbichar_{13} \projbichar_{15} \multunit[C]_{35} \multunit[C]_{25}.
\end{align*}
Now we may cancel \(\projbichar_{14}\)
on the left and \(\multunit[C]_{35} \multunit[C]_{25}\)
on the right to transform our condition into
\(\projbichar_{15} \projbichar_{12} \projbichar_{13}
\projbichar_{25}^* \projbichar_{35}^* = \projbichar_{25}^*
\projbichar_{35}^* \projbichar_{12} \projbichar_{13}
\projbichar_{15}\) or, equivalently,
\[
\projbichar_{35} \projbichar_{25} \projbichar_{15} \projbichar_{12} \projbichar_{13}
= \projbichar_{12} \projbichar_{13} \projbichar_{15} \projbichar_{35} \projbichar_{25}
\]
in
\(\U(\Hils\otimes \conj{\Hils} \otimes \Hils \otimes \conj{\Hils}
\otimes \Hils)\).
The anti-Heisenberg pair condition on the second leg gives
\(\projbichar_{25} \projbichar_{15} \projbichar_{12} =
\projbichar_{12} \projbichar_{25}\),
the Heisenberg pair on the third leg gives
\(\projbichar_{13} \projbichar_{15} \projbichar_{35} =
\projbichar_{35} \projbichar_{13}\).
Plugging this in, our condition becomes
\[
\projbichar_{35} \projbichar_{12} \projbichar_{25} \projbichar_{13}
= \projbichar_{12} \projbichar_{35} \projbichar_{13}\projbichar_{25},
\]
which is manifestly true. Thus our operators
satisfy~\eqref{eq:F_V-invariant} as well.
Checking the braided pentagon equation~\eqref{eq:top-braided_pentagon}
is a long computation. We may omit it because of the following
trick. In the proof of
Theorem~\ref{the:standard_mult_from_braided_and_standard}, the braided
pentagon equation is used exactly once. Therefore, if all the other
conditions in Definition~\ref{def:braided_multiplicative_unitary}
hold, then the braided pentagon
equation~\eqref{eq:top-braided_pentagon} is both sufficient and
\emph{necessary} for the usual pentagon equation for the
unitary~\(\Multunit[D]\)
constructed in
Theorem~\ref{the:standard_mult_from_braided_and_standard}. Thus the
proof of Theorem~\ref{the:analysis_qg_projection} is finished up to
the braided pentagon equation, and this follows from the proof of the
following theorem.
\begin{theorem}
\label{the:there_and_back_again}
Let \(\Multunit[C],\ProjBichar\in\U(\Hils\otimes\Hils)\)
define a \(\Cst\)\nb-quantum
group with projection as in
Proposition~\textup{\ref{pro:qg_projection_mu}}. Construct a
braided multiplicative unitary
\((\ProjBichar,\Corep{U},\DuCorep{V},\BrMultunit)\)
on the Hilbert space \(\Hils[L]= \conj{\Hils}\otimes\Hils\)
with the unitaries defined in
\eqref{eq:U_from_projection}--\eqref{eq:F_from_projection}. From
this, construct a multiplicative unitary~\(\Multunit[D]\)
with a projection~\(\ProjBichar^D\) on the Hilbert space
\[
\Hils\otimes\Hils[L]\otimes\Hils\otimes\Hils[L] \cong
\Hils\otimes\conj{\Hils}\otimes\Hils
\otimes\Hils\otimes\conj{\Hils} \otimes\Hils
\]
by
Theorem~\textup{\ref{the:standard_mult_from_braided_and_standard}}.
The braided multiplicative unitary
\((\ProjBichar,\Corep{U},\DuCorep{V},\BrMultunit)\)
and the multiplicative unitary~\(\Multunit[D]\)
are manageable. And~\(\Multunit[D]\)
generates the same \(\Cst\)\nb-quantum
group as~\(\Multunit[C]\).
The isomorphism between these \(\Cst\)\nb-quantum
groups maps~\(\ProjBichar\) to~\(\ProjBichar^D\).
\end{theorem}
Roughly speaking, going from a \(\Cst\)\nb-quantum
group with projection to a braided \(\Cst\)\nb-quantum
group and back gives an isomorphic \(\Cst\)\nb-quantum
group with projection.
The definitions in
Theorem~\ref{the:standard_mult_from_braided_and_standard} amount to
\begin{align*}
\Multunit[D] {}\defeq{} &
\projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{45}^* \projbichar_{26}^*
\projbichar_{36}^* \multunit[C]_{36} \multunit[C]_{26}
\projbichar_{45} \projbichar_{46}\\
{}={}& \projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{26}^*
\projbichar_{36}^* \multunit[C]_{36} \multunit[C]_{26}
\projbichar_{46},\\
\ProjBichar^D {}\defeq{}&
\projbichar_{14} \projbichar_{34} \projbichar_{24}.
\end{align*}
Our first task is to construct representations \(\pi\)
and~\(\hat{\pi}\) of \(C\) and~\(\hat{C}\) that form a Heisenberg pair
and that satisfy \((\hat{\pi}\otimes\pi)\multunit[C]= \Multunit[D]\).
This implies that~\(\Multunit[D]\) satisfies the pentagon equation.
As we remarked above, this implies the braided pentagon
equation~\eqref{eq:top-braided_pentagon} for~\(\BrMultunit\), which
still remained to be proven.
\begin{lemma}
\label{lemm:aux-C-heis}
There is a representation
\(\pi'\colon C\to\Bound(\Hils\otimes\conj{\Hils}\otimes\Hils)\)
such that
\begin{alignat}{2}
\label{eq:pi_on_WC}
(\Id_{\hat{C}}\otimes\pi')\multunit[C]
&=\projbichar_{12}\multunit[C]_{14}
&\qquad\text{in \(\U(\hat{C}\otimes\Comp(\Hils\otimes\conj{\Hils}\otimes\Hils))\),}\\
\label{eq:pi_on_P}
(\Id_{\hat{C}}\otimes\pi')\projbichar
&=\projbichar_{12}\projbichar_{14}
&\qquad\text{in \(\U(\hat{C}\otimes\Comp(\Hils\otimes\conj{\Hils}\otimes\Hils))\).}
\end{alignat}
\end{lemma}
\begin{proof}
Let
\(\Duprojbichar = \sigma(\projbichar)^* \in \U(C\otimes \hat{C})\)
for the flip~\(\sigma\).
Define \(\varphi\colon C\to\Mult(C\otimes\Comp(\conj{\Hils}))\)
by
\(\varphi(c)\defeq
\Duprojbichar_{1\hat{\alpha}} (1\otimes\alpha(c))
\Duprojbichar{}_{1\hat{\alpha}}^*\).
Then
\[
(\Id_{\hat{C}}\otimes\varphi) \multunit[C]
=\Duprojbichar_{2\hat{\alpha}}\multunit[C]_{1\alpha}
\Duprojbichar{}_{2\hat{\alpha}}^*
=\flip_{23}(\projbichar_{\hat{\alpha}3}^* \multunit[C]_{1\alpha}
\projbichar_{\hat{\alpha}3})
\qquad\text{in }\Mult(\hat{C}\otimes C\otimes\Comp(\conj{\Hils})).
\]
The second condition in Proposition~\ref{pro:qg_projection_mu} is
equivalent to
\(\multunit[C]_{1\alpha}\projbichar_{\hat{\alpha}3} =
\projbichar_{\hat{\alpha}3} \projbichar_{13}\multunit[C]_{1
\alpha}\)
in \(\U(\hat{C}\otimes\Comp(\conj{\Hils})\otimes C)\)
because we have an anti\nb-Heisenberg pair on the second leg. Thus
\[
(\Id_{\hat{C}}\otimes\varphi)\multunit[C]
= \flip_{23}(\projbichar_{13}\multunit[C]_{1\alpha})
=\projbichar_{12}\multunit[C]_{1\alpha}
\qquad\text{in }\Mult(\hat{C}\otimes C\otimes\Comp(\conj{\Hils})).
\]
We may define \(\pi'(c)\defeq
\bigl((\hat{\iota}\otimes\iota\circ\alpha^{-1})\varphi(c)\bigr)_{13}\)
because~\(\alpha\) is automatically injective (see
\cite{Roy:Codoubles}*{Proposition 3.7}). This is the unique
representation that satisfies~\eqref{eq:pi_on_WC}.
Replacing~\(\multunit[C]\) by~\(\projbichar\) in the above
computations gives~\eqref{eq:pi_on_P}.
\end{proof}
\begin{lemma}
\label{lem:aux-C-Heis}
Let~\(\pi'\)
be as in the previous lemma. The pair of representations
\((\pi,\hat{\pi})\)
of \(C\)
and~\(\hat{C}\)
on \(\Hils\otimes\conj{\Hils}\otimes\Hils\) defined by
\[
\pi(c)\defeq \projbichar_{13}^*\pi'(c)\projbichar_{13},
\qquad
\hat{\pi}(\hat{c})\defeq \projbichar_{13}^*
((\hat{\alpha}\otimes\hat{\iota}) \DuComult[C](\hat{c}))_{23}
\projbichar_{13}
\]
is an \(\G[H]\)\nb-Heisenberg pair.
\end{lemma}
\begin{proof}
Let
\(\hat{\pi}'(\hat{c})\defeq ((\hat{\alpha}\otimes\hat{\iota})
\DuComult[C](\hat{c}))_{23}\).
The lemma is equivalent to \((\pi',\hat{\pi}')\)
being \(\G[H]\)\nb-Heisenberg.
Recall that
\((\DuComult[C]\otimes\Id_{C})\multunit[C] =
\multunit[C]_{23}\multunit[C]_{13}\).
Lemma~\ref{lemm:aux-C-heis} gives
\[
\multunit[C]_{\hat{\pi}'5}\multunit[C]_{1\pi'}
=\multunit[C]_{45}\multunit[C]_{35}\projbichar_{12}\multunit[C]_{14}
=\projbichar_{12}\multunit[C]_{45}\multunit[C]_{14}\multunit[C]_{35}
\]
in
\(\U(\hat{C}\otimes
\Comp(\Hils\otimes\conj{\Hils}\otimes\Hils)\otimes C)\).
Since we have a Heisenberg pair on the fourth leg, the pentagon
equation~\eqref{eq:pentagon} gives
\[
\projbichar_{12} \multunit[C]_{45} \multunit[C]_{14} \multunit[C]_{35}
= \projbichar_{12} \multunit[C]_{14} \multunit[C]_{15} \multunit[C]_{45}
\multunit[C]_{35},
\]
which is equivalent to
\(\multunit[C]_{\hat{\pi}'5}\multunit[C]_{1\pi'} =
\multunit[C]_{1\pi'}\multunit[C]_{15}\multunit[C]_{\hat{\pi}'5}\)
in
\(\U(\hat{C}\otimes\Comp(\Hils\otimes\conj{\Hils}\otimes\Hils)\otimes
C)\).
\end{proof}
\begin{lemma}
\label{lem:analysis-multunit}
\(\Multunit[D] = (\hat{\pi}\otimes\pi)\multunit[C]\)
and \(\ProjBichar^D = (\hat{\pi}\otimes\pi)\projbichar\)
in \(\U(\Hils\otimes\Hils[L]\otimes\Hils\otimes \Hils[L])\).
\end{lemma}
\begin{proof}
Cancelling~\(\projbichar_{45}\)
gives
\(\Multunit[D]= \projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{26}^* \projbichar_{36}^*
\multunit[C]_{36} \multunit[C]_{26} \projbichar_{46}\).
Computing as in the proof of Lemma~\ref{lem:aux-C-Heis}, we get
\[
(\hat{\pi}\otimes\pi)\multunit[C]
= \projbichar_{13}^* \projbichar_{46}^* \projbichar_{34}
\multunit[C]_{36} \projbichar_{24} \multunit[C]_{26}
\projbichar_{13}\projbichar_{46}
= \projbichar_{46}^* \projbichar_{13}^* \projbichar_{34}
\projbichar_{24} \multunit[C]_{36} \projbichar_{13}
\multunit[C]_{26} \projbichar_{46}.
\]
Since we have a Heisenberg pair on the third leg,
\(\multunit[C]_{36} \projbichar_{13} = \projbichar_{13}
\projbichar_{16} \multunit[C]_{36}\). Hence
\[
(\hat{\pi}\otimes\pi)\multunit[C]
= \projbichar_{46}^* \projbichar_{13}^* \projbichar_{34}
\projbichar_{24} \projbichar_{13} \multunit[C]_{36} \multunit[C]_{26}
\projbichar_{46}.
\]
Since we have Heisenberg pairs on the third and fourth legs,
\(\projbichar_{34}\projbichar_{13} =
\projbichar_{13}\projbichar_{14}\projbichar_{34}\)
and
\(\projbichar_{46}\projbichar_{i4} = \projbichar_{i 6}\projbichar_{i
6} \projbichar_{4 6}\)
for all \(i=1,2,3\).
Using these identities (for the expressions within brackets in the
computation below) we get
\begin{multline*}
\projbichar_{46}^* \projbichar_{13}^* \projbichar_{34}
\projbichar_{24} \projbichar_{13} \projbichar_{16}
= \projbichar_{46}^* (\projbichar_{13}^* \projbichar_{34}
\projbichar_{13}) \projbichar_{24} \projbichar_{16}
= \projbichar_{46}^* \projbichar_{14} \projbichar_{34}
\projbichar_{24} \projbichar_{16} \\
= (\projbichar_{46}^* \projbichar_{14} \projbichar_{16})
\projbichar_{34} \projbichar_{24}
= \projbichar_{14} (\projbichar_{46}^* \projbichar_{34})
\projbichar_{24}
= \projbichar_{14} \projbichar_{34} \projbichar_{46}^*
\projbichar_{36}^* \projbichar_{24}\\
= \projbichar_{14} \projbichar_{34} (\projbichar_{46}^*
\projbichar_{24}) \projbichar_{36}^*
= \projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{26}^* \projbichar_{36}^* .
\end{multline*}
The last two computations together give
\[
(\hat{\pi}\otimes\pi)\multunit[C]
=\projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{26}^* \projbichar_{36}^*
\multunit[C]_{36}\multunit[C]_{26}\projbichar_{46}
=\Multunit[D].
\]
Equation~\eqref{eq:pi_on_P} allows a similar computation
with~\(\projbichar\) instead of~\(\multunit[C]\). This gives
\[
(\hat{\pi}\otimes\pi)\projbichar
=\projbichar_{14} \projbichar_{34} \projbichar_{24}
\projbichar_{46}^* \projbichar_{26}^* \projbichar_{36}^*
\projbichar_{36}\projbichar_{26}\projbichar_{46}
=\ProjBichar^D.\qedhere
\]
\end{proof}
The following remarks apply to any Heisenberg pair \((\pi,\hat{\pi})\)
for a \(\Cst\)\nb-quantum
group \(\G[H]= (C,\Comult[C])\)
on a Hilbert space~\(\Hils'\).
Being a Heisenberg pair means that
\((\hat{\pi}\otimes\pi)\multunit[C]\)
is a multiplicative unitary. It is unclear, in general, whether this
multiplicative unitary is manageable. If it is manageable, then we
claim that the \(\Cst\)\nb-quantum
group that it generates is isomorphic to the one we started with. The
representations in a Heisenberg pair are automatically faithful by
\cite{Roy:Codoubles}*{Proposition 3.7}. Hence we may view \(C\)
and~\(\hat{C}\)
as subalgebras of \(\Bound(\Hils')\),
and \((\hat{\pi}\otimes\pi)\multunit[C]\)
is a unitary multiplier of
\(\hat{C}\otimes C \subseteq \Bound(\Hils'\otimes\Hils')\).
It makes no difference whether we take slices on the first leg with
elements of \(\Bound(\Hils')_*\)
or~\(\hat{C}^*\):
both generate the same \(\Cst\)\nb-subalgebra
of \(\Bound(\Hils')\),
namely, \(\pi(C)\).
The comultiplication on the quantum group generated
by~\((\hat{\pi}\otimes\pi)\multunit[C]\)
is defined so that the isomorphism~\(\pi\)
is a Hopf \Star{}homomorphism.
Thus the \(\Cst\)\nb-quantum group generated
by~\((\hat{\pi}\otimes\pi)\multunit[C]\) is isomorphic to~\(\G[H]\)
for any Heisenberg pair for
which~\((\hat{\pi}\otimes\pi)\multunit[C]\) is manageable.
Furthermore, Lemma~\ref{lem:analysis-multunit}
shows that this Hopf \Star{}isomorphism maps~\(\projbichar\)
to~\(\ProjBichar^D\),
so we also get the same projection on our \(\Cst\)\nb-quantum group.
Thus the proof of Theorem~\ref{the:there_and_back_again} will be
finished once we show that~\(\Multunit[D]\)
and the braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\)
are manageable. By
Theorem~\ref{the:mang_mult_from_braided_and_standard},
\(\Multunit[D]\)
is manageable once \((\Corep{U},\DuCorep{V},\BrMultunit)\)
is manageable. So it remains to prove this.
The braiding on \(\Hils[L]\otimes\Hils[L]\)
comes from the unique unitary~\(Z\)
that verifies~\eqref{eq:braiding}. A simple computation shows that
\(Z = \projbichar_{14}^* \projbichar_{24}^* \projbichar_{13}^*
\projbichar_{23}^*\)
in \(\U(\conj{\Hils}\otimes\Hils\otimes\conj{\Hils}\otimes\Hils)\)
does the job. This gives
\[
Z^*\BrMultunit=\projbichar_{23}\projbichar_{13}\projbichar_{24}\projbichar_{14}
\projbichar_{14}^* \projbichar_{24}^* \multunit[C]_{24}\multunit[C]_{14}
= \projbichar_{23}\projbichar_{13} \multunit[C]_{24}\multunit[C]_{14}.
\]
Now we use that \((\iota,\hat{\iota})\)
is the standard Heisenberg pair, generated by~\(\Multunit[C]\),
and that the anti-Heisenberg pair \((\alpha,\hat{\alpha})\)
is constructed as in \cite{Meyer-Roy-Woronowicz:Twisted_tensor}*{Lemma
3.6}; that is,
\(\hat{\alpha}(\hat{a}) \defeq
\hat{a}^{\transpose\circ\hat{\iota}\circ\hat{R}_C}\)
and \(\alpha(a)\defeq a^{\transpose\circ\iota\circ R_C}\). Thus
\[
Z^*\BrMultunit =
\projbichar_{23}^{\hat{\iota}\otimes\transpose\circ\iota\circ R_{C}}
\projbichar_{13}^{\transpose\circ\hat{\iota}\circ\hat{R}_{C}\otimes\transpose\circ\iota\circ R_{C}}
(\multunit[C]_{24})^{\hat{\iota}\otimes\iota}
(\multunit[C]_{14})^{\transpose\circ\hat{\iota}\circ\hat{R}_{C}\otimes\iota}.
\]
Let \(Q_C\)
and \(\widetilde{\Multunit[C]} \in \U(\conj{\Hils}\otimes\Hils)\)
witness the manageability of
\(\Multunit[C] = (\hat{\iota}\otimes\iota)
\multunit[C]\in\U(\Hils\otimes\Hils)\),
see Appendix~\ref{app_sec:mang_bichar}. Since~\(\ProjBichar\)
is manageable by Proposition~\ref{pro:ProjBichar_manageable}, so is
the dual \(\DuProjBichar = \Sigma \ProjBichar^* \Sigma\).
This is witnessed by a certain unitary
\(\widetilde{\DuProjBichar} \in \U(\conj{\Hils}\otimes\Hils)\).
We have
\((\multunit[C])^{\transpose \hat\iota R_{\hat{C}} \otimes \iota} =
(\multunit[C])^{\transpose \hat\iota \otimes \iota R_C} =
(\widetilde{\Multunit}{}^C)^*\)
and
\(\projbichar^{\transpose \hat\iota R_{\hat{C}} \otimes \transpose
\iota R_C} = \projbichar^{\transpose \otimes \transpose}\)
by \cite{Woronowicz:Mult_unit_to_Qgrp}*{Theorem 1.6 (5)}
and~\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{(19)}. Similarly,
\((\Sigma \projbichar^{\hat{\iota}\otimes\transpose\circ\iota\circ
R_{C}} \Sigma)^* = \Duprojbichar{}^{\transpose\circ\iota\circ
R_C\otimes \hat{\iota}} = \widetilde{\DuProjBichar}\). Thus
\[
Z^*\BrMultunit = \Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\ProjBichar^{\transpose\otimes\transpose}_{13} \Multunit[C]_{24}
(\widetilde{\Multunit}{}^{C}_{14})^*
\qquad\text{in }\U(\conj{\Hils}\otimes\Hils\otimes\conj{\Hils}\otimes\Hils).
\]
Let \(Q\defeq Q_C^\transpose\otimes Q_C\).
Then \(Q\otimes Q\)
commutes with~\(\BrMultunit\),
\(Q\otimes Q_C\)
commutes with~\(\Corep{U}\),
and \(Q_C\otimes Q\)
commutes with~\(\DuCorep{V}\).
Define
\(\widetilde{\BrMultunit} \in \U(\conj{\Hils[L]}\otimes\Hils[L])\) by
\begin{equation}
\label{eq:mang-br-analysis}
\widetilde{\BrMultunit} \defeq
\widetilde{\Multunit}{}^C_{24}(\Multunit[C]_{14})^*
(\ProjBichar^*)^{\transpose\otimes\transpose}_{23}
\widetilde{\ProjBichar}^{\transpose\otimes\transpose}_{13}
\qquad\text{in }\U(\Hils\otimes\conj{\Hils}\otimes\conj{\Hils}\otimes\Hils).
\end{equation}
This unitary and~\(Q\) witness the
manageability of the braided multiplicative unitary
\((\Corep{U},\DuCorep{V},\BrMultunit)\). The rather technical proof
of this fact is relegated to the appendix, see
Lemma~\ref{lemm:brmanag_analysis}.
\section{Examples of Quantum Groups with Projections}
\label{sec:Ex_Qunt_grp_proj}
The simplest examples of semidirect products of connected Lie groups
are \(\textup{E}(2) = \R^2\rtimes \T\) and the real and complex
\(ax+b\)-groups \(\R\rtimes \R_{>0}^\times\) and \(\C\rtimes
\C^\times\), where the second, multiplicative factor acts by multiplication
on the first, additive factor. The group~\(\textup{E}(2)\) is the group of
isometries of the plane. Another very important example is the
Poincar\'e group, the semidirect product of the Lorentz group
with~\(\R^4\).
When quantising such groups, one may try to preserve the semidirect
product structure, that is, construct \(\Cst\)\nb-quantum groups
with projection. For instance, the quantum \(\textup{E}(2)\) groups by
Woronowicz~\cite{Woronowicz:Qnt_E2_and_Pontr_dual} have obvious
morphisms to the circle group~\(\T\) and back that compose to the
identity on~\(\T\). The quantum \(az+b\) groups introduced by
Woronowicz~\cite{Woronowicz:Quantum_azb} and
Sołtan~\cite{Soltan:New_az_plus_b} -- which deform
\(\C\rtimes\C^\times\) -- have obvious morphisms to the group
\(\C_q^\times = q^{\Z+\ima\R}\subseteq \C^\times\) (with
multiplication as group structure) and back, which compose to the
identity on~\(\C_q^\times\); see also
\cite{Kasprzak-Soltan:Quantum_projection}*{Example 3.7}. The
quantum \(ax+b\) group by Woronowicz and
Zakrzewski~\cite{Woronowicz-Zakrzewski:Quantum_axb} has an obvious
projection onto the group \(\R_{>0}^\times \cong \R\).
There are also quantum versions of semidirect product groups that
appear to have no such projection. This includes the \(az+b\)
groups by Baaj and Skandalis, see
\cite{Vaes-Vainerman:Extension_of_lcqg}*{Section 5.3}, the \(ax+b\)
groups by Stachura~\cite{Stachura:ax_plus_b} and the
\(\kappa\)\nb-Poincaré groups by
Stachura~\cite{Stachura:Kappa-Poincare}. These examples are all
constructed using the formalism of quantum group extensions
of~\cite{Vaes-Vainerman:Extension_of_lcqg}. Quantum group
extensions are compared with quantum groups with projection
in~\cite{Kasprzak-Soltan:Extension_vs_Qgrp_proj}.
As an example of our theory, we are going to construct a braided
multiplicative unitary that generates ``simplified quantum
\(\textup{E}(2)\),'' a variant of quantum \(\textup{E}(2)\) also due
to Woronowicz (unpublished); whereas the quantum \(\textup{E}(2)\)
groups in~\cite{Woronowicz:Qnt_E2_and_Pontr_dual} deform a double
cover of~\(\textup{E}(2)\), the simplified variants deform
\(\textup{E}(2)\) itself. A common feature of simplified quantum
\(\textup{E}(2)\) and the quantum groups with projection mentioned
above is that the image of the projection is a classical, Abelian
group. This is to be expected when deforming semidirect products by
Abelian groups because these cannot be deformed to quantum groups in
interesting ways. We begin by observing some common features of
braided multiplicative unitaries in case~\(\Multunit\) generates an
Abelian group~\(G\).
Let~\(\hat{G}\)
be the dual group. The corepresentations \(\Corep{U}\)
and~\(\DuCorep{V}\)
in a braided multiplicative unitary are equivalent to representations
of \(G\)
and~\(\hat{G}\)
on the Hilbert space~\(\Hils[L]\),
respectively. The compatibility
condition~\eqref{eq:U_V_compatible} for \(\Corep{U}\)
and~\(\DuCorep{V}\)
says here that the representations of \(G\)
and~\(\hat{G}\)
commute. Thus we may combine them to one representation of
\(\hat{G}\times G\)
on~\(\Hils[L]\).
We can further normalise this representation because the left
regular representation of any quantum group absorbs every other
representation. The operator \(\BrMultunit\otimes 1\) on
\(\Hils[L]\otimes L^2(\hat{G} \times G)\) is a braided
multiplicative unitary if and only if~\(\BrMultunit\) is, and it
generates an equivalent semidirect product quantum group. Thus we
may assume without loss of generality that our representation is a
multiple of the left regular representation:
\[
\Hils[L]= L^2(\hat{G}\times G) \otimes \Hils[L]_0
\]
for some separable Hilbert space~\(\Hils[L]_0\), with \(\Cst(\hat{G}
\times G)\) acting only on the first tensor factor, by the regular
representation. We may identify \(\Cont_0(G\times\hat{G}) \cong
\Cst(\hat{G}\times G)\) and \(L^2(\hat{G}\times G) \cong L^2(G\times
\hat{G})\) by the Fourier transform, and the regular representation
of \(\Cst(\hat{G}\times G)\) on \(L^2(\hat{G}\times G) \cong
L^2(G\times \hat{G})\) becomes the standard representation of
\(\Cont_0(G\times\hat{G})\) on \(L^2(G\times\hat{G})\) by pointwise
multiplication.
For some examples, a variant of the above is useful: if the
representation of~\(\hat{G}\times G\) on~\(\Hils[L]\) factors
through an Abelian locally compact group~\(H\), then we may
use~\(H\) instead of \(\hat{G}\times G\) in the above
simplification. That is, we seek a braided multiplicative unitary
on the Hilbert space \(L^2(H)\otimes\Hils[L]_0\) with
\(\hat{G}\times G\) acting only on the first tensor factor, through
the regular representation of~\(H\) and the given homomorphism
\(\hat{G}\times G\to H\).
For instance, the compact quantum group \(\textup{U}_q(2)\) is a
semidirect product of the braided quantum group \(\textup{SU}_q(2)\)
by the circle~\(\T\)
(see~\cite{Kasprzak-Meyer-Roy-Woronowicz:Braided_SU2}), and the
relevant representation of \(\Z\times\T\) factors through a
homomorphism \(\Z\times\T\to\T\), \((n,z)\mapsto \lambda^n\cdot z\)
for some \(\lambda\in\T\). For the quantum \(az+b\) groups,
\(G=\C_q\) is self-dual, and the relevant representation of
\(\hat{G}\times G\) factors through the map \(\hat{G}\times G \cong
G\times G \to G\), where the second map is the multiplication map
\((x,y)\mapsto x\cdot y\).
When we have simplified~\(\Hils[L]\) to \(L^2(G\times\hat{G})
\otimes \Hils[L]_0\) with \(\Cont_0(G\times\hat{G})\) acting by
pointwise multiplication, the braiding
operator~\(\Braiding{\Hils[L]}{\Hils[L]}\) is the operator of
pointwise multiplication with the circle-valued function
\begin{equation}
\label{eq:braiding_Abelian}
(g_1,\chi_1,g_2,\chi_2)\mapsto \chi_1(g_2).
\end{equation}
The conditions \eqref{eq:F_U-invariant} and~\eqref{eq:F_V-invariant}
for~\(\BrMultunit\) mean that
\(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\) is a \(\hat{G}\times
G\)-equivariant operator with respect to the tensor product
representation of \(\hat{G}\times G\) on
\(\Hils[L]\otimes\Hils[L]\). In terms of the above spectral
analysis, \(f\in \Cont_0(G\times\hat{G})\) acts on
\(\Hils[L]\otimes\Hils[L]\) by pointwise multiplication on each
fibre with the function
\[
\Delta^* f(g_1,\chi_1,g_2,\chi_2) = f(g_1g_2,\chi_1\chi_2)
\]
for all \((g_1,\chi_1,g_2,\chi_2)\in G\times\hat{G}\times
G\times\hat{G}\). An operator on \(\Hils[L]\otimes\Hils[L]\) is
\(\hat{G}\times G\)-equivariant if and only if it commutes with the
operators of pointwise multiplication by~\(\Delta^* f\) for \(f\in
\Cont_0(G\times\hat{G})\).
Summing up, it suffices to look for braided multiplicative unitaries
over an Abelian group~\(G\) on the Hilbert space \(\Hils[L]=
L^2(G\times\hat{G}, \ell^2(\N))\). Such a braided multiplicative
unitary \(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\) must commute
with the operators of pointwise multiplication by functions in
\(\Delta^*\Contb(G\times\hat{G}) \subseteq
\Contb(G\times\hat{G}\times G\times\hat{G})\), and it must satisfy
the braided pentagon equation~\eqref{eq:top-braided_pentagon}, where
the braiding is given by pointwise multiplication with the function
in~\eqref{eq:braiding_Abelian}.
\subsection{Simplified quantum E(2) groups}
\label{sec:qnt_E_2}
Now we specialise to simplified quantum E(2) groups.
They were already treated in~\cite{Roy:Qgrp_with_proj} except
for the manageability of the braided multiplicative unitary in
question. A \(\Cst\)\nb-algebraic version of this construction
appears in~\cite{Roy:Braided_Cstar}.
Here the image of the projection is the circle group \(G=\T\), so
\(\hat{G}=\Z\). The analysis above suggests to construct a braided
multiplicative unitary for quantum E(2) on a Hilbert space of the form
\(L^2(\T\times\Z) \otimes \Hils[L]_0\). Actually, we shall not
need~\(\Hils[L]_0\) and work on \(L^2(\T\times\Z)\) itself. First
we describe the standard multiplicative unitary~\(\Multunit\)
generating~\(\T\).
Let \(\Hils\defeq \ell^2(\Z)\) and let~\(\{e_p\}_{p\in\Z}\) be an
orthonormal basis of~\(\Hils\). Define
\[
u e_p\defeq e_{p+1} \quad\text{ and }\quad
\hat{N}e_p\defeq p e_p.
\]
The shift~\(u\) is unitary and generates the regular representation
of~\(\Z\) on~\(\Hils\). The operator~\(\hat{N}\) is self-adjoint
with spectrum~\(\Z\), and the resulting representation of
\(\Cont_0(\Z) \cong \Cst(\T)\) is the Fourier transform of the
regular representation of~\(\T\) on~\(L^2(\T)\). These operators
generate a representation of the crossed product
\(\Cont_0(\Z)\rtimes\Z \cong \Comp(\ell^2\Z)\). That is, they
satisfy the commutation relation
\begin{equation}
\label{eq:comm_u_hat_N}
u^*\hat{N}u=\hat{N}+1.
\end{equation}
The multiplicative unitary generating~\(\T\) is
\[
\Multunit\defeq(1\otimes u)^{\hat{N}\otimes 1}
= \int_{\Z\times\T} z^s \, \diff E_{\hat{N}}(s)\otimes\diff E_{u}(z),
\qquad e_k\otimes e_l \mapsto e_k \otimes e_{l+k},
\]
where \(\diff E_{\hat{N}}\) and~\(\diff E_{u}\) are the spectral
measures of \(\hat{N}\) and~\(u\), respectively. The commutation
relation~\eqref{eq:comm_u_hat_N} implies the pentagon equation
\[
(1\otimes 1\otimes u)^{1\otimes\hat{N}\otimes 1}
(1\otimes u\otimes 1)^{\hat{N}\otimes 1\otimes 1}
(1\otimes 1\otimes u^*)^{1\otimes\hat{N}\otimes 1}
= (1\otimes u\otimes u)^{\hat{N}\otimes 1\otimes 1}.
\]
for~\(\Multunit\), that is, \(\Multunit\) is a multiplicative
unitary. Simple computations show that \(Q=1\) and
\(\widetilde{\Multunit}\defeq (1\otimes u^*)^{\hat{N}\otimes 1}\)
witness the manageability of~\(\Multunit\). Slices of~\(\Multunit\)
in the first and second leg clearly generate the
\(\Cst\)\nb-algebras \(\Cst(\hat{N})\cong \Cont_0(\Z)\) and
\(\Cst(u)\cong \Cont_0(\T)\). The comultiplications defined
by~\(\Multunit\) satisfy \(\Comult[\Cont(\T)](u)\defeq u\otimes u\)
and \(\Comult[\Contvin(\Z)](\hat{N})\defeq \hat{N}\otimes 1\dotplus
1\otimes\hat{N}\), where \(\hat{N}\otimes 1\dotplus
1\otimes\hat{N}\) means the unbounded affiliated element of
\(\Contvin(\Z\times\Z)\) given by the closure of the essentially
self-adjoint operator \(\hat{N}\otimes 1+ 1\otimes\hat{N}\).
Let \(0<q<1\). We identify \(\Z\times\T \cong \C_q^\times =
q^{\Z+\ima \R}\subseteq \C^\times\) by mapping \((n,z)\mapsto
q^n\cdot z\). As suggested above, we are going to construct a
multiplicative unitary on the Hilbert space \(L^2(\Z\times\T) =
L^2(\C_q^\times)\) with the regular representation of
\(\Z\times\T\). We choose the orthonormal basis \(e_{k,l} \defeq
\delta_k\otimes z^l\) for \(k,l\in\Z\) in~\(L^2(\C_q^\times)\) and
thus identify \(\Hils[L] \cong \Hils\otimes \Hils\). In our chosen
basis, \(\Z\) acts by \(\hat\alpha_n(e_{k,l}) = e_{k+n,l}\) for
\(n,k,l\in\Z\) and~\(\T\) acts by \(\alpha_\zeta(e_{k,l}) = \zeta^l
\cdot e_{k,l}\) for \(k,l\in\Z\), \(\zeta\in\T\). Thus the right
and left corepresentations \(\Corep{U}\in\U(\Hils[L]\otimes\Hils)\)
and \(\DuCorep{V}\in\U(\Hils\otimes\Hils[L])\) and the resulting
braiding operator \(\Braiding{\Hils[L]}{\Hils[L]} \in
\U(\Hils[L]\otimes\Hils[L])\) are
\begin{alignat*}{2}
\Corep{U} &= \Multunit_{23},&\qquad
e_{k,l,m} &\mapsto e_{k,l,m+l}\\
\DuCorep{V} &= \Multunit_{12},&\qquad
e_{m,k,l} &\mapsto e_{m,k+m,l}\\
\Braiding{\Hils[L]}{\Hils[L]} = Z\Sigma
&= \Multunit[*]_{23} \Sigma,&\qquad
e_{k,l} \otimes e_{n,p} &\mapsto e_{n,p} \otimes e_{k-p,l}.
\end{alignat*}
We also describe the representations of \(\Cont(\T) \cong \Cst(\Z)\)
and \(\Cont_0(\Z) \cong \Cst(\T)\) on~\(\Hils[L]\) through a unitary
operator~\(\udrinf\) and a self-adjoint operator \(\Nhatdrinf\) with
spectrum~\(\Z\) and commuting with~\(\udrinf\):
\[
\udrinf(e_{k,l})\defeq e_{k+1,l}
\qquad
\Nhatdrinf(e_{k,l})\defeq l e_{k,l}.
\]
We define a closed operator \(\Upsilon = \Ph{\Upsilon}
\Mod{\Upsilon}\) on~\(\Hils[L]\) by
\[
\Ph{\Upsilon} e_{k,l}\defeq e_{k,l+1},\qquad
\Mod{\Upsilon} e_{k,l}\defeq q^{2k+l} e_{k,l},\qquad
\Upsilon e_{k,l}\defeq q^{2k+l} e_{k,l+1}.
\]
The operator~\(\Ph{\Upsilon}\) is unitary and~\(\Mod{\Upsilon}\) is
strictly positive with spectrum \(q^\Z\cup\{0\}\), and
\(\Ph{\Upsilon}\) and~\(\Mod{\Upsilon}\) satisfy the following
commutation relations:
\begin{equation}
\label{eq:commutation_Upsilon}
\left\{\begin{alignedat}{2}
\Ph{\Upsilon}\Mod{\Upsilon}\Ph{\Upsilon}^* &=
q^{-1}\Mod{\Upsilon},\\
\udrinf \Ph{\Upsilon} &= \Ph{\Upsilon} \udrinf,
&\qquad& \udrinf \Mod{\Upsilon} \udrinf^* = q^{-2} \Mod{\Upsilon},\\
\Ph{\Upsilon} \Nhatdrinf\Ph{\Upsilon}^* &= \Nhatdrinf-1,
&\qquad& \Mod{\Upsilon} \text{ and }
\Nhatdrinf \text{ strongly commute.}
\end{alignedat}\right.
\end{equation}
Thus \(\Upsilon^{-1}(e_{k,l}) = q^{-2k-l+1} e_{k,l-1}\). The closed
operator
\[
X\defeq \Upsilon q^{-2\Nhatdrinf}\otimes\Upsilon^{-1},\qquad
e_{k,l} \otimes e_{n,p} \mapsto
q^{2(k-n) - (l+p) + 1} e_{k,l+1} \otimes e_{n,p-1},
\]
on \(\Hils[L] \otimes \Hils[L]\) is normal because \(\Mod{X}\colon
e_{k,l} \otimes e_{n,p} \mapsto q^{2(k-n) - (l+p) + 1} e_{k,l}
\otimes e_{n,p}\) commutes with its phase \(\Ph{X}\colon e_{k,l}
\otimes e_{n,p} \mapsto e_{k,l+1} \otimes e_{n,p-1}\) in the polar
decomposition. The spectrum of~\(X\) is \(\cl{\C}_q \defeq
\C_q^\times \cup\{0\}\). Both \(\Mod{X}\) and~\(\Ph{X}\) strongly
commute with \(\udrinf\otimes \udrinf\) and \(\Nhatdrinf \otimes 1
\dotplus 1 \otimes \Nhatdrinf\). Thus~\(X\) is equivariant for the
tensor product representations \(\Corep{U} \tenscorep \Corep{U}\)
and \(\DuCorep{V} \tenscorep \DuCorep{V}\). Hence any circle-valued
function \(F\colon \cl{\C}_q \to \T\) gives a unitary~\(F(X)\) on
\(\Hils[L] \otimes \Hils[L]\) that is equivariant with respect to
\(\Corep{U} \tenscorep \Corep{U}\) and \(\DuCorep{V} \tenscorep
\DuCorep{V}\).
We want to choose~\(F\) so that~\(F(X)\) satisfies the braided
pentagon equation. Since the functional calculus is compatible with
conjugation by unitaries, the top-braided pentagon equation
for~\(F(X)\) says
\begin{multline}
\label{eq:braided_pentagon_Upsilon}
F(F(X_{23}) X_{12} F(X_{23})^*)
= F(X_{12}) \Braiding{\Hils[L]}{\Hils[L]}_{23} F(X_{12})
\Dualbraiding{\Hils[L]}{\Hils[L]}_{23}
\\= F(X_{12}) F(Z_{23} X_{13} Z_{23}^*).
\end{multline}
We compute
\[
Z_{23} X_{13} Z_{23}^* (e_{k,l} \otimes e_{n,p} \otimes e_{r,s})
= q^{2(k-p-r) - (l+s) + 1} (e_{k,l+1} \otimes e_{n,p} \otimes e_{p,s-1}).
\]
Thus
\[
Z_{23} X_{13} Z_{23}^*
= \Upsilon q^{-2\Nhatdrinf}\otimes q^{-2\Nhatdrinf}\otimes\Upsilon^{-1}
= X_{12} \cdot X_{23}.
\]
Hence~\eqref{eq:braided_pentagon_Upsilon} becomes
\begin{equation}
\label{eq:braided_pentagon_Upsilon2}
F(F(X_{23}) X_{12} F(X_{23})) = F(X_{12}) F(X_{12} X_{23}).
\end{equation}
The quantum exponential function is defined
in~\cite{Woronowicz:Operator_eq_E2} by
\begin{equation}
\label{eq:Qnt_exp}
\brmultunit_q(z)\defeq
\begin{cases}
\displaystyle \prod_{k=1}^{\infty}
\frac{1+q^{2k}\conj{z}}{1+q^{2k} z}
& z\in\cl{\C}_q \setminus\{-q^{2\Z}\},\\
-1 & \text{otherwise.}
\end{cases}
\end{equation}
This product converges absolutely outside \(-q^{2\Z}\), and
\(\bigl\lvert\frac{1+q^{2k}\conj{z}}{1+q^{2k} z}\bigr\rvert = 1\)
for all \(z\neq -q^{2k}\). Thus~\(\brmultunit_q\) is a unitary
multiplier of \(\Cont_0(\cl{\C_q})\), and
\[
\BrMultunit\defeq
\brmultunit_q(\Upsilon^{-1} q^{-2\Nhatdrinf}\otimes\Upsilon)
\]
is a unitary operator on~\(\Hils[L]\otimes\Hils[L]\).
\begin{theorem}
\label{the:Br_mult_Qnt_pl}
The triple \((\Corep{U},\DuCorep{V},\BrMultunit)\) is a manageable
braided multiplicative unitary on~\(\Hils[L]\) relative
to~\(\Multunit\).
\end{theorem}
The proof will occupy the rest of this section. For
\((\Corep{U},\DuCorep{V},\BrMultunit)\) to be a braided
multiplicative unitary, it only remains to verify the braided
pentagon equation, which is equivalent
to~\eqref{eq:braided_pentagon_Upsilon}. We shall use the properties of the
quantum exponential function established
in~\cite{Woronowicz:Operator_eq_E2}.
The operators
\[
R \defeq X_{12}
= \Upsilon q^{-2\Nhatdrinf}\otimes\Upsilon^{-1} \otimes 1,\qquad
S \defeq X_{12} X_{23}
= \Upsilon q^{-2\Nhatdrinf}\otimes q^{-2\Nhatdrinf}\otimes \Upsilon^{-1}
\]
are normal and satisfy the commutation relations
in~\cite{Woronowicz:Operator_eq_E2}*{(0.1)}, that is, their phases
and their absolute values strongly commute and
\[
\Ph{R}^* \abs{S} \Ph{R} = q \abs{S},\qquad
\Ph{S} \abs{R} \Ph{S}^* = q \abs{R}.
\]
Since \(R^{-1} S = X_{23}\) is also normal with
spectrum~\(\cl{\C}_q\), \cite{Woronowicz:Operator_eq_E2}*{Theorems
2.1--2} apply and show that \(R\dotplus S\) is normal with
spectrum~\(\cl{\C}_q\) and
\[
\brmultunit_q(X_{23}) \cdot X_{12}\cdot \brmultunit_q(X_{23})^*
= \brmultunit_q(R^{-1} S) \cdot R\cdot \brmultunit_q(R^{-1} S)^*
= R \dotplus S.
\]
Moreover, \cite{Woronowicz:Operator_eq_E2}*{Theorem 3.1} gives
\[
\brmultunit_q(R) \brmultunit_q(S) = \brmultunit_q(R \dotplus S).
\]
Both results of~\cite{Woronowicz:Operator_eq_E2} together
give~\eqref{eq:braided_pentagon_Upsilon2} for \(F=\brmultunit_q\);
this is equivalent to the braided pentagon equation
for~\(\BrMultunit\).
Now we turn to braided manageability. First we compute the
unitary~\(\widetilde{Z}\). It is the unique unitary on
\(\conj{\Hils[L]}\otimes\Hils[L]\) that
satisfies~\eqref{eq:braiding-manag}. The
contragradient~\(\widetilde{\corep{U}}{}^*\) of~\(\corep{U}\) is given
in the standard basis \(\conj{e_{k,l}} \otimes e_m\) of
\(\conj{\Hils[L]}\otimes\Hils\) by
\(\widetilde{\corep{U}}^*(\conj{e_{k,l}} \otimes e_m) =
\conj{e_{k,l}} \otimes e_{m-l}\). Hence \(\widetilde{Z} \in
\U(\conj{\Hils[L]}\otimes\Hils[L])\) acts on the standard basis by
\[
\widetilde{Z}(\conj{e_{k,l}} \otimes e_{n,p}) =
\conj{e_{k,l}} \otimes e_{n-l,p}.
\]
Equivalently, \(\widetilde{Z} = (1\otimes
\udrinf)^{\Nhatdrinf^{\transpose}\otimes 1}\).
Next we define the operator~\(Q_{\Hils[L]}\) required by
Definition~\ref{def:braided_manageable}:
\[
Q_{\Hils[L]} e_{k,l} \defeq q^{-l} e_{k,l}.
\]
This is a strictly positive operator on~\(\Hils[L]\) with
spectrum~\(q^\Z\cup\{0\}\). It commutes with \(\udrinf\)
and~\(\Nhatdrinf\) and therefore satisfies
\eqref{eq:br_manag_commute_U} and~\eqref{eq:br_manag_commute_V}.
The operator \(Q_{\Hils[L]}\otimes Q_{\Hils[L]}\), mapping \(e_{k,l}
\otimes e_{n,p}\mapsto q^{-(l+p)} e_{k,l} \otimes e_{n,p}\),
commutes with \(X= \Upsilon q^{-2\Nhatdrinf} \otimes \Upsilon^{-1}\)
and therefore with \(\BrMultunit = \brmultunit_q(X)\).
Thus~\eqref{eq:br_manag_commute_F} holds as well.
Finally, we need a unitary \(\widetilde{\BrMultunit}\in
\U(\conj{\Hils[L]}\otimes\Hils[L])\) that
satisfies~\eqref{eq:br_manag}. It suffices to check this if the
vectors \(x,y,u,v\) involved are standard basis vectors
\(x=e_{k,l}\), \(y=e_{n,p}\), \(u=e_{a,b}\), \(v=e_{c,d}\)
for~\(\Hils[L]\). Using our explicit formulas for~\(Z\)
and~\(\widetilde{Z}\), we may rewrite~\eqref{eq:br_manag} as
\[
(e_{k,l} \otimes e_{a-l,b} \mid \BrMultunit \mid e_{n,p} \otimes
e_{c,d}) =
(\conj{e_{n,p}}\otimes q^{-b} e_{a,b} \mid \widetilde{\BrMultunit}
\mid \conj{e_{k,l}} \otimes q^d e_{c+l,d}).
\]
Substituting \(\gamma= c+l\) and \(\BrMultunit = \brmultunit_q(X)\),
this becomes
\begin{equation}
\label{eq:manageable_E2_concrete}
(\conj{e_{n,p}}\otimes e_{a,b} \mid \widetilde{\BrMultunit}
\mid \conj{e_{k,l}} \otimes e_{\gamma,d})
= q^{b-d} (e_{k,l} \otimes e_{a-l,b} \mid \brmultunit_q(X) \mid
e_{n,p} \otimes e_{\gamma-l,d})
\end{equation}
for \(a,b,\gamma,d,k,l,n,p\in\Z\). So the issue is whether the
bilinear form~\(\widetilde{\BrMultunit}\) defined by this equation
is unitary.
To compute the right hand side in~\eqref{eq:manageable_E2_concrete},
we Fourier transform the restrictions of~\(\brmultunit_q\) to the
circles \(\abs{z}=q^n\), \(n\in\Z\), and write
\[
\brmultunit_q(z) = \sum_{m\in\Z} \brmultunit_m(\Mod{z}) \Ph{z}^m,
\]
see~\cite{Baaj:Regular-Representation-E-2} or
\cite{Woronowicz:Quantum_azb}*{Appendix A}. The
scalars~\(\brmultunit_m(q^n)\) for \(m,n\in\Z\) are real and satisfy
\[
\brmultunit_m(q^n) = (-q)^m \brmultunit_{-m}(q^{n-m}).
\]
The vectors \(e_{n,p} \otimes e_{\gamma-l,d}\) and \(e_{k,l} \otimes
e_{a-l,b}\) are eigenvectors of~\(\Mod{X}\) with eigenvalues
\(q^{2(n-\gamma+l) - (p+d)+1}\) and \(q^{2(k-a+l) - (b+l)+1}\),
respectively. And~\(\Ph{X}^m\) acts on these vectors by
\(e_{k,l}\otimes e_{a,b}\mapsto e_{k,l+m}\otimes e_{a,b-m}\). Thus
\begin{align*}
&\phantom{{}={}} q^{b-d} (e_{k,l} \otimes e_{a-l,b} \mid \brmultunit_q(X) \mid
e_{n,p} \otimes e_{\gamma-l,d})\\
&=\sum_{m\in\Z} q^{b-d} (e_{k,l} \otimes e_{a-l,b} \mid
\Ph{X}^m \brmultunit_m(\Mod{X})\mid e_{n,p} \otimes e_{\gamma-l,d})
\\&= \sum_{m\in\Z} q^{b-d} \cdot \brmultunit_m(q^{2(n-\gamma+l) - (p+d)+1})
\delta_{k,n}\delta_{l,p+m} \delta_{a-l,\gamma-l} \delta_{b,d-m}
\\&= \delta_{k,n} \delta_{a,\gamma} \delta_{p,l+b-d}\cdot
q^{b-d} \cdot \brmultunit_{d-b}(q^{2 k- 2 a + l - b+1})
\\&= \delta_{k,n} \delta_{a,\gamma} \delta_{p,l+b-d}\cdot
(-1)^{b-d} \cdot \brmultunit_{b-d}(q^{2 k- 2 a + l - d+1}).
\end{align*}
Now we define an unbounded normal operator~\(\widetilde{X}\)
on~\(\conj{\Hils[L]} \otimes \Hils\) with spectrum~\(\cl{\C}_q\) by
\[
\Mod{\widetilde{X}}(\conj{e_{k,l}} \otimes e_{n,p}) =
q^{2(k-n) + l - p+1} \conj{e_{k,l}} \otimes e_{n,p},\quad
\Ph{\widetilde{X}}(\conj{e_{k,l}} \otimes e_{n,p}) =
-\conj{e_{k,l+1}} \otimes e_{n,p+1},
\]
so \(\widetilde{X}(\conj{e_{k,l}} \otimes e_{n,p}) = -1\cdot
q^{2(k-n) + l - p+1} \conj{e_{k,l+1}} \otimes e_{n,p+1}\). We claim
that the unitary \(\widetilde{\BrMultunit} \defeq
\brmultunit_q(\widetilde{X})^*\) will do:
\begin{align*}
(\conj{e_{n,p}}\otimes e_{a,b} \mid \widetilde{\BrMultunit}
\mid \conj{e_{k,l}} \otimes e_{\gamma,d})
&=\sum_{m\in\Z} (\conj{e_{k,l}} \otimes e_{\gamma,d} \mid
\Ph{\widetilde{X}}^m \brmultunit_m(\Mod{\widetilde{X}})
\mid \conj{e_{n,p}}\otimes e_{a,b})
\\&= \sum_{m\in\Z} (-1)^m \brmultunit_m(q^{2(n-a) + p - b+1})
\delta_{k,n}\delta_{l,p+m} \delta_{\gamma,a} \delta_{d,b+m}
\\&= \delta_{k,n} \delta_{a,\gamma} \delta_{p,l+b-d}\cdot
(-1)^{d-b} \cdot \brmultunit_{d-b}(q^{2 k- 2 a + l - d + 1}).
\end{align*}
This is equal to the result of the computation above,
so~\eqref{eq:manageable_E2_concrete} holds. Thus our braided
multiplicative unitary is manageable, and
Theorem~\ref{the:Br_mult_Qnt_pl} is proved.
\appendix
\section{Some Manageability Techniques}
\label{app_sec:mang_bichar}
Let \(\Multunit[A]\in\U(\Hils_A\otimes\Hils_A)\) and
\(\Multunit[B]\in\U(\Hils_B\otimes\Hils_B)\) be manageable
multiplicative unitaries as in
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Definition
1.2}, which generate \(\Cst\)\nb-quantum groups \(\Qgrp{G}{A}\) and
\(\Qgrp{H}{B}\).
Let \(\bichar\in\U(\hat{A}\otimes B)\) be a bicharacter from \(\G\)
to~\(\G[H]\). Let \(\Bichar\in\U(\Hils_A\otimes\Hils_B)\) be the
concrete realisation of~\(\bichar\). Then \(\Bichar\in
\U(\Hils_A\otimes \Hils_B)\) is adapted to~\(\Multunit[B]\) in the
sense of \cite{Woronowicz:Mult_unit_to_Qgrp}*{Definition 1.3} by
\cite{Meyer-Roy-Woronowicz:Homomorphisms}*{Lemma 3.2}. Thus
\(\Bichar\in\U(\Hils_A\otimes\Hils_B)\) is manageable by
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Theorem 1.6}; that is, there is a
unitary \(\widetilde{\Bichar}\in\U(\conj{\Hils}_A\otimes\Hils_B)\)
with
\begin{equation}
\label{eq:second_leg_bichar_adapted}
\bigl(x\otimes u\big| \Bichar\big| z\otimes y\bigr)
= \bigl(\conj{z}\otimes Q_B u\big| \widetilde{\Bichar}\big|
\conj{x}\otimes Q_B^{-1}y\bigr)
\end{equation}
for all \(x,z\in\Hils_A\), \(u\in\dom(Q_B)\) and
\(y\in\dom(Q_B^{-1})\); here \(Q_B\) is one of the operators in the
manageability condition for~\(\Multunit[B]\).
\begin{lemma}
\label{lem:V_tilde_commute_Q}
\(\Bichar (Q_A\otimes Q_B)\Bichar^* = Q_A\otimes Q_B\) and
\(\widetilde{\Bichar} (Q_A^\transpose\otimes
Q_B^{-1})\widetilde{\Bichar}^* = Q_A^\transpose\otimes Q_B^{-1}\).
\end{lemma}
\begin{proof}
The scaling group \(\tau_t^A\colon \R\to\Aut(A)\) of~\(A\) acts
through conjugation by~\(Q_A^{\ima t}\) by
\cite{Woronowicz:Mult_unit_to_Qgrp}*{Theorem 1.5.5}, and similarly
for~\(B\). \cite{Meyer-Roy-Woronowicz:Homomorphisms}*{(20)} says
that~\(V\) is fixed by \(\tau^A_t\otimes \tau^B_t\). This means
that~\(\Bichar\) commutes with \(Q_A\otimes Q_B\), as asserted.
Hence the left-hand side of~\eqref{eq:second_leg_bichar_adapted}
does not change if we replace \(x\), \(u\), \(z\) and~\(y\) by
\(Q^{\ima t}_A(x)\), \(Q^{\ima t}_B(u)\), \(Q^{\ima t}_A(z)\)
and~\(Q^{\ima t}_B(y)\), respectively, for any \(t\in\R\). Thus the
right-hand sides also remain the same, that is,
\[
\bigl(\conj{z}\otimes Q_Bu\big|
\widetilde{\Bichar}\big|\conj{x}\otimes Q_B^{-1}y\bigr)
= \Bigl(\bigl[Q_A^\transpose\bigr]^{-\ima t}\conj{z}\otimes
Q_B^{\ima t}Q_Bu\Big| \widetilde{\Bichar}\Big|
\bigl[Q_A^\transpose\bigr]^{-\ima t}\conj{x}\otimes
Q_B^{\ima t} Q^{-1}_Bu\Bigr).
\]
Hence \(\widetilde{\Bichar} =
\Bigl(\bigl[Q_A^\transpose\bigr]^{\ima t} \otimes Q^{-\ima
t}_B\Bigr)
\widetilde{\Bichar}\Bigl(\bigl[Q_A^\transpose\bigr]^{\ima t}
\otimes Q^{-\ima t}_B\Bigr)\) for all \(t\in\R\). This says
that~\(\widetilde{\Bichar}\) commutes with \(Q_A^\transpose\otimes
Q_B^{-1}\).
\end{proof}
\begin{lemma}
\label{lem:nice_basis_Q}
Let~\(Q\) be a self\nb-adjoint, strictly positive operator on a
Hilbert space~\(\Hils\). There is an orthonormal
basis~\((e_i)_{i\in\N}\) in~\(\Hils\) with
\(e_i\in\dom(Q)\cap\dom(Q^{-1})\) and strong convergence
\begin{equation}
\label{eq:nice_basis_Q}
\sum_{i\in\N} \ket{Q^{-1}e_i}\bra{Q e_i} = \Id_{\Hils}.
\end{equation}
\end{lemma}
\begin{proof}
For \(n\in\Z\) and \(\lambda\in [2^{2n-1},2^{2n+1})\), let
\(f(\lambda)=2^{-2n}\lambda\) and \(g(\lambda) =
\lambda/f(\lambda)\). The function
\[
\R_{>0} \ni \lambda\mapsto f(\lambda)\in [2^{-1},2)
\]
is piecewise linear and bounded with bounded inverse. Hence the
Borel functional calculus for self-adjoint operators gives
\(Q'\defeq f(Q)\), which is self-adjoint and bounded with a bounded
inverse. We also get the self-adjoint operator \(Q'' = g(Q)\),
which has countable spectrum~\(\{2^{2n}\mid n\in\Z\}\).
Thus~\(\Hils\) is the orthogonal direct sum of the
\(2^{2n}\)-eigenspaces of~\(\Hils\). We choose orthonormal bases
for all these eigenspaces and put them together to an orthonormal
basis~\((e_i)_{i\in\N}\) of~\(\Hils\). We have \(Q=Q'Q''\) and
\(Q^{-1} = (Q')^{-1}(Q'')^{-1}\) by functional calculus. Since the
operators~\((Q')^{\pm1}\) are bounded and self-adjoint, \(Q\)
and~\(Q''\) have the same domain. Thus \(e_i\in \dom(Q)\cap
\dom(Q^{-1})\) because it is an eigenvector of~\(Q''\) with some
positive eigenvalue~\(2^{2n}\) for some \(n\in\Z\) depending
on~\(i\). Since~\((Q')^{\pm1}\) are bounded and \(Q''e_i = 2^{2n}
e_i\), we may rewrite
\begin{multline*}
\ket{Q^{-1}e_i}\bra{Q e_i}
= \ket{(Q')^{-1} (Q'')^{-1}e_i} \bra{Q' Q''e_i}
= \ket{(Q')^{-1} 2^{-2n} e_i} \bra{Q' 2^{2n} e_i}
\\= (Q')^{-1} \ket{e_i} \bra{e_i} Q'.
\end{multline*}
The sum \(\sum_{i\in\N} \ket{e_i}\bra{e_i}\) converges strongly to
the identity on~\(\Hils\) because~\((e_i)_{i\in\N}\) is an
orthonormal basis for~\(\Hils\). Since~\((Q')^{\pm1}\) are
bounded, the sum over \((Q')^{-1} \ket{e_i} \bra{e_i} Q'\)
converges strongly to \((Q')^{-1} \cdot 1 \cdot Q' = 1\).
\end{proof}
The following lemma completes the proof of
Theorem~\ref{the:mang_mult_from_braided_and_standard}.
\begin{lemma}
\label{lem:mang_mult_from_braided_and_standard}
In the situation of the proof of
Theorem~\textup{\ref{the:mang_mult_from_braided_and_standard}},
\(\Multunit[C]\), \(Q\) and~\(\widetilde{\Multunit}{}^C\)
verify~\eqref{eq:Multunit_manageable}.
\end{lemma}
\begin{proof}
We continue in the notation of the proof of
Theorem~\ref{the:mang_mult_from_braided_and_standard}. It suffices
to check~\eqref{eq:Multunit_manageable} when \(x,z,u,y\) are tensor
monomials: \(x=x_1\otimes x_2\), \(z=z_1\otimes z_2\),
\(u=u_1\otimes u_2\), \(y=y_1\otimes y_2\) with \(x_1,z_1\in\Hils\),
\(x_2,z_2\in\Hils[L]\), \(u_1\in\dom(Q)\), \(u_2\in\dom(Q_{\Hils[L]})\),
\(y_1\in\dom(Q^{-1})\), \(y_2\in\dom(Q_{\Hils[L]}^{-1})\). This implies the
assertion for all \(x,y,z,u\).
Equation~\eqref{eq:braiding} is equivalent to
\(\Corep{U}_{23}\DuCorep{V}_{34}^* = \DuCorep{V}_{34}^*
\Corep{U}_{23} Z_{13}^*\), which gives
\begin{equation}
\label{eq:W_1234_equiv_form}
\Multunit[C]_{1234} = \Multunit_{13} \DuCorep{V}^*_{34}
\Corep{U}_{23} (Z^*_{24} \BrMultunit_{24}) \DuCorep{V}_{34}.
\end{equation}
We first concentrate on the part \(\Corep{U}_{23} (Z^*_{24}
\BrMultunit_{24})\) in~\eqref{eq:W_1234_equiv_form}. Let
\((e_i)_{i\in\N}\) be an orthonormal basis of~\(\Hils[L]\). Then
\eqref{eq:Corep_U_tilde} and~\eqref{eq:br_manag} give
\begin{align*}
&\phantom{{}={}}\left<x_2\otimes u_1\otimes u_2 \middle| \Corep{U}_{12}
Z^*_{13}\BrMultunit_{13} \middle| z_2\otimes y_1\otimes y_2\right>
\\&= \sum \left<x_2\otimes u_1\otimes u_2 \middle| \Corep{U}_{12}
\cdot \bigl(\ket{e_i}\bra{e_i} \otimes 1\otimes 1\bigr)
\cdot Z^*_{13}\BrMultunit_{13} \middle| z_2\otimes y_1\otimes y_2\right>
\\&= \sum
\left< x_2\otimes u_1 \middle| \Corep{U} \middle|
e_i\otimes y_1\right> \cdot
\left<e_i\otimes u_2\middle| Z^*\BrMultunit \middle| z_2\otimes y_2\right>
\\&= \sum
\left<\conj{z_2}\otimes Q_{\Hils[L]}(u_2) \middle| \widetilde{\BrMultunit} \middle|
\conj{e_i}\otimes Q_{\Hils[L]}^{-1}(y_2)\right>
\left< \conj{e_i}\otimes Q(u_1) \middle| \widetilde{\Corep{U}}
\middle| \conj{x_2} \otimes Q^{-1}(y_1)\right>
\\&= \left<\conj{z_2}\otimes Q(u_1) \otimes Q_{\Hils[L]}(u_2) \middle|
\widetilde{\BrMultunit}_{13} \widetilde{\Corep{U}}_{12} \middle|
\conj{x_2}\otimes Q^{-1}(y_1)\otimes Q_{\Hils[L]}^{-1}(y_2)\right>
\end{align*}
Since~\(\DuCorep{V}\) commutes with~\(Q\otimes Q_{\Hils[L]}\), it also
preserves the domains of \((Q \otimes Q_{\Hils[L]})^{-1}\), and we get an
equivalent statement if we replace \(u_1\otimes u_2\) and
\(y_1\otimes y_2\) above by \(\DuCorep{V}(u_1\otimes u_2)\) and
\(\DuCorep{V}(y_1\otimes y_2)\), respectively. This gives
\begin{multline}
\label{eq:aux_step_manag_W_1234_2}
\left<x_2\otimes u_1\otimes u_2\middle| \DuCorep{V}_{23}^*
\Corep{U}_{12} Z^*_{13} \BrMultunit_{13} \DuCorep{V}_{23}
\middle| z_2\otimes y_1\otimes y_2\right>
\\= \left<\conj{z_2}\otimes Q(u_1) \otimes Q_{\Hils[L]}(u_2) \middle|
\DuCorep{V}_{23}^* \widetilde{\BrMultunit}_{13}
\widetilde{\Corep{U}}_{12} \DuCorep{V}_{23} \middle|
\conj{x_2}\otimes Q^{-1}(y_1)\otimes Q_{\Hils[L]}^{-1}(y_2)\right>.
\end{multline}
Now let \((\epsilon_j)_{j\in\N}\) be a basis of~\(\Hils\) as in
Lemma~\ref{lem:nice_basis_Q}, that is,
\[
\sum_j \ket{Q^{-1}(\epsilon_j)}\bra{Q(\epsilon_j)} = \Id_{\Hils}.
\]
We compute
\begin{align*}
&\phantom{{}={}}
\left<x_1\otimes x_2\otimes u_1\otimes u_2 \middle| \Multunit_{13}
\DuCorep{V}_{34}^* \Corep{U}_{23} Z^*_{24}\BrMultunit_{24}
\DuCorep{V}_{34} \middle| z_1\otimes z_2\otimes y_1\otimes y_2\right>
\\&=
\begin{multlined}[t][.96\linewidth]
\sum_j
\bigl<x_1\otimes x_2\otimes u_1\otimes u_2 \big|
\Multunit_{13}\cdot
\bigl(1\otimes 1\otimes \big|
\ket{\epsilon_j}\bra{\epsilon_j} \otimes 1\bigr)
\cdot \DuCorep{V}_{34}^* \Corep{U}_{23} Z^*_{24}
\BrMultunit_{24} \DuCorep{V}_{34} \big|
\\ \big| z_1\otimes z_2\otimes y_1\otimes y_2\bigr>
\end{multlined}
\\&= \sum_j
\left<x_1\otimes u_1 \middle| \Multunit \middle|
z_1\otimes \epsilon_i\right> \cdot
\left<x_2\otimes \epsilon_i\otimes u_2 \middle|
\DuCorep{V}_{23}^* \Corep{U}_{12} (Z^*\BrMultunit)_{13}
\DuCorep{V}_{23} \middle| z_2\otimes y_1\otimes y_2\right>
\\&=
\begin{multlined}[t][.96\linewidth]
\sum_j
\left<\conj{z_1}\otimes Q(u_1) \middle| \widetilde{\Multunit}
\middle| \conj{x_1}\otimes Q^{-1}(\epsilon_i)\right>
\\\cdot
\left<\conj{z_2}\otimes Q(\epsilon_i) \otimes Q_{\Hils[L]}(u_2) \middle|
\DuCorep{V}_{23}^* \widetilde{\BrMultunit}_{13}
\widetilde{\Corep{U}}_{12} \DuCorep{V}_{23}\middle|
\conj{x_2}\otimes Q^{-1}(y_1)\otimes Q_{\Hils[L]}^{-1}(y_2)\right>
\end{multlined}
\\&=
\begin{multlined}[t][.96\linewidth]
\Bigl<\conj{z_1} \otimes \conj{z_2} \otimes Q(u_1) \otimes Q_{\Hils[L]}(u_2)\Big|
\widetilde{\Multunit}_{13} \DuCorep{V}_{34}^*
\widetilde{\BrMultunit}_{24} \widetilde{\Corep{U}}_{23}
\DuCorep{V}_{34}\Big|
\\ \Big|\conj{x_1} \otimes \conj{x_2} \otimes Q^{-1}(y_1)
\otimes Q_{\Hils[L]}^{-1}(y_2)\Bigr>.
\end{multlined}
\end{align*}
Thus~\eqref{eq:Multunit_manageable} holds for \(\Multunit[C]\),
\(Q\) and~\(\widetilde{\Multunit}{}^C\).
\end{proof}
\begin{lemma}
\label{lemm:brmanag_analysis}
The unitary
\(\widetilde{\BrMultunit}\in\U(\conj{\Hils[L]}\otimes\Hils[L])\)
defined by~\eqref{eq:mang-br-analysis} satisfies the manageability
condition~\eqref{eq:br_manag} for
\(\BrMultunit\in\U(\Hils[L]\otimes\Hils[L])\)
in Theorem~\textup{\ref{the:analysis_qg_projection}}.
\end{lemma}
\begin{proof}
It suffices to check~\eqref{eq:br_manag} when~\(x,u,y,v\)
are tensor monomials: \(x=\bar{x}_{1}\otimes x_{2}\),
\(u=\bar{u}_{1}\otimes u_{2}\),
\(y=\bar{y}_{1}\otimes y_{2}\),
\(v=\bar{v}_{1}\otimes v_{2}\),
with \(x_{1}, x_{2}, y_{1}, y_{2}\in\Hils\),
\(v_{1}, u_{2}\in\dom(Q_{C})\)
and \(u_{1}, v_{2}\in \dom(Q_{C}^{-1})\).
First we focus on the part
\(\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\ProjBichar^{\transpose\otimes\transpose}_{13}\).
Let \((e_i)_{i\in\N}\)
be an orthonormal basis of~\(\Hils\)
in Lemma~\ref{lem:nice_basis_Q}. Then \((\bar{e}_i)_{i\in\N}\)
is an orthonormal basis of~\(\conj{\Hils}\) with
\begin{equation}
\label{eq:nice_conj_basis}
\sum_{i\in\N} \ket{\conj{Q e_i}}\bra{\conj{Q^{-1}e_i}} = \Id_{\conj{\Hils}}.
\end{equation}
Equation~\eqref{eq:Multunit_manageable} for \(\ProjBichar\)
and~\(\DuProjBichar\) gives
\begin{align*}
& \left<\bar{x}_{1}\otimes x_{2}\otimes \bar{u}_{1}\middle|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23} \ProjBichar^{\transpose\otimes\transpose}_{13}
\middle| \bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\right>\\
&=\sum \left<\bar{x}_{1}\otimes x_{2}\otimes \bar{u}_{1}\middle|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\cdot \bigl(1\otimes 1\otimes \ket{e_i}\bra{e_i} \bigr) \cdot
\ProjBichar^{\transpose\otimes\transpose}_{13}
\middle| \bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\right>\\
&=\sum \left<x_{2}\otimes\bar{u}_{1}\middle|
\Flip\widetilde{\DuProjBichar}\Flip\middle|
y_{2}\otimes\bar{e}_{i}\right>
\left<\bar{x}_{1}\otimes\bar{e}_{i}\middle|
\ProjBichar^{\transpose\otimes\transpose}\middle|
\bar{y}_{1}\otimes \bar{v}_{1}\right>\\
&=\sum \left<\bar{u}_{1}\otimes x_{2}\middle|
\widetilde{\DuProjBichar}\middle|
\bar{e}_{i}\otimes y_{2}\right>
\left<y_{1}\otimes v_{1}\middle|
\ProjBichar \middle|
x_{1}\otimes e_{i}\right>\\
&=\sum \left<e_{i}\otimes Q_{C}^{-1}(x_{2})\middle|
\DuProjBichar\middle|
u_{1}\otimes Q_{C}(y_{2})\right>
\left<\bar{x}_{1}\otimes Q_{C}(v_{1})\middle|
\widetilde{\ProjBichar}\middle|
\bar{y}_{1}\otimes Q_{C}^{-1}(e_{i})\right>.
\end{align*}
Lemma~\ref{lem:V_tilde_commute_Q} shows that~\(Q_C\otimes C_C\)
commutes with~\(\DuProjBichar\). Hence
\begin{align*}
& \left<\bar{x}_{1}\otimes x_{2}\otimes \bar{u}_{1}\middle|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23} \ProjBichar^{\transpose\otimes\transpose}_{13}
\middle| \bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\right>\\
&=\sum \left<Q_{C}(e_{i})\otimes x_{2}\middle|
\DuProjBichar\middle|
Q_{C}^{-1}(u_{1})\otimes y_{2}\right>
\left<\bar{x}_{1}\otimes Q_{C}(v_{1})\middle|
\widetilde{\ProjBichar}\middle|
\bar{y}_{1}\otimes Q_{C}^{-1}(e_{i})\right>\\
&=\sum \left<x_{2}\otimes Q_{C}(e_{i})\middle|
\ProjBichar^*\middle|
y_{2}\otimes Q_{C}^{-1}(u_{1})\right>
\left< y_{1}\otimes \conj{Q_{C}^{-1}(e_{i})}\middle|
\widetilde{\ProjBichar}{}^{\transpose\otimes\transpose}\middle|
x_{1}\otimes \conj{Q_{C}(v_{1})}\right>\\
&=\sum \left<\bar{y}_{2}\otimes \conj{Q_{C}^{-1}(u_{1})}\middle|
(\ProjBichar^*)^{\transpose\otimes\transpose} \middle|
\bar{x}_{2}\otimes \conj{Q_{C}(e_{i})}\right>
\left< y_{1}\otimes \conj{Q_{C}^{-1}(e_{i})}\middle|
\widetilde{\ProjBichar}{}^{\transpose\otimes\transpose}\middle|
x_{1}\otimes \conj{Q_{C}(v_{1})}\right>.
\end{align*}
Now~\eqref{eq:nice_conj_basis} gives
\begin{multline}
\label{eq:br-man_aux1}
\left<\bar{x}_{1}\otimes x_{2}\otimes \bar{u}_{1}\middle|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23} \ProjBichar^{\transpose\otimes\transpose}_{13}
\middle| \bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\right>
\\= \left<y_{1}\otimes \bar{y}_{2}\otimes \conj{Q_{C}^{-1}(u_{1})}
\middle|
(\ProjBichar^*)^{\transpose\otimes\transpose}_{23}
\widetilde{\ProjBichar}{}^{\transpose\otimes\transpose}_{13}
\middle|
x_{1}\otimes \bar{x}_{2}\otimes \conj{Q_{C}(v_{1})}\right>.
\end{multline}
A similar computation gives
\begin{multline}
\label{eq:br-man_aux2}
\left<\bar{x}_{1}\otimes x_{2}\otimes u_{2}\middle|
\Multunit[C]_{23}(\widetilde{\Multunit}{}^C_{13})^*\middle|
\bar{y}_{1}\otimes y_{2}\otimes v_{2}\right>
\\= \left<y_{1}\otimes \bar{y}_{2}\otimes Q_{C}(u_{2})\middle|
\widetilde{\Multunit}{}^C_{23} (\Multunit[C]_{13})^*\middle|
x_{1}\otimes \bar{x}_{2}\otimes Q_{C}^{-1}(v_{2})\right>.
\end{multline}
Let \((e_{j})_{j\in\N}\)
be an orthonormal basis of~\(\Hils\).
Equations \eqref{eq:br-man_aux1} and~\eqref{eq:br-man_aux2} imply
\begin{align*}
&\phantom{{}={}}
\left<\bar{x}_{1}\otimes x_{2}\otimes\bar{u}_{1}\otimes u_{2}\middle|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\ProjBichar^{\transpose\otimes\transpose}_{13} \Multunit[C]_{24}
(\widetilde{\Multunit}{}^{C}_{14})^* \middle|
\bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\otimes v_{2}\right>
\\&=
\begin{multlined}[t][.96\linewidth]
\sum_{j,k} \Bigl<\bar{x}_{1}\otimes x_{2}\otimes\bar{u}_{1}\otimes
u_{2}\Big|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\ProjBichar^{\transpose\otimes\transpose}_{13}
\cdot \bigl(\ket{\bar{e}_{j}}\bra{\bar{e}_{j}}\otimes \ket{e_{k}}\bra{e_{k}}\otimes
1 \otimes 1\bigr)
\\\cdot
\Multunit[C]_{24} (\widetilde{\Multunit}{}^{C}_{14})^* \Big|
\bar{y}_{1}\otimes y_{2}\otimes \bar{v}_{1}\otimes v_{2}\Bigr>
\end{multlined}
\\&=
\begin{multlined}[t][.96\linewidth]
\sum_{j,k} \Bigl<\bar{x}_{1}\otimes x_{2}\otimes\bar{u}_{1}\Big|
\Flip_{23}\widetilde{\DuProjBichar}_{23}\Flip_{23}
\ProjBichar^{\transpose\otimes\transpose}_{13} \Big|
\bar{e}_{j}\otimes e_{k}\otimes \bar{v}_{1}\Bigr>
\\
\Bigl<\bar{e}_{j}\otimes e_{k}\otimes u_{2} \Big|
\Multunit[C]_{23} (\widetilde{\Multunit}{}^{C}_{13})^* \Big|
\bar{y}_{1}\otimes y_{2}\otimes v_{2}\Bigr>
\end{multlined}
\\&=\begin{multlined}[t][.96\linewidth]
\sum_{j,k} \left<y_{1}\otimes \bar{y}_{2}\otimes Q_{C}(u_{2})\middle|
\widetilde{\Multunit}{}^{C}_{23} (\Multunit[C]_{13})^*\middle|
e_{j}\otimes \bar{e}_{k}\otimes Q_{C}^{-1}(v_{2})\right>
\\
\left<e_{j}\otimes \bar{e}_{k}\otimes \conj{Q_{C}^{-1}(u_{1})}
\middle|
(\ProjBichar^*)^{\transpose\otimes\transpose}_{23}
\widetilde{\ProjBichar}{}^{\transpose\otimes\transpose}_{13}
\middle|
x_{1}\otimes \bar{x}_{2}\otimes \conj{Q_{C}(v_{1})}\right>
\end{multlined}
\\&=
\begin{multlined}[t][.96\linewidth]
\Bigl< y_{1}\otimes \bar{y}_{2}\otimes \conj{Q_{C}^{-1}(u_{1})}
\otimes Q_{C}(u_{2})
\Bigm|
\widetilde{\Multunit}{}^{C}_{24} (\Multunit[C]_{14})^*
(\ProjBichar^*)^{\transpose\otimes\transpose}_{23}
\widetilde{\ProjBichar}{}^{\transpose\otimes\transpose}_{13}
\Bigm|\\
x_{1}\otimes\bar{x}_{2}\otimes \conj{Q_{C}(v_{1})}\otimes
Q_{C}^{-1}(v_{2}) \Bigr>
\end{multlined}
\end{align*}
This is the equation we have to check.
\end{proof}
\end{document}
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