\magnification=1200 \baselineskip=14pt plus 1pt minus 1pt \tolerance=1000\hfuzz=1pt \def\printertype{ps: }% default postscript \font\bigfont=cmr10 scaled\magstep3 \def\section#1#2{\vskip32pt plus4pt \goodbreak \noindent{\bf#1. #2} \xdef\currentsec{#1} \global\eqnum=0 \global\thmnum=0} \def\leftheader{} \def\rightheader{} \def\header{\headline={\tenrm\ifnum\pageno>1 \ifodd\pageno\rightheader \else\noindent\leftheader \fi\else\hfil\fi}} \newcount\thmnum \global\thmnum=0 \def\prop#1#2{\global\advance\thmnum by 1 \xdef#1{Proposition \currentsec.\the\thmnum} \bigbreak\noindent{\bf Proposition \currentsec.\the\thmnum.} {\it#2} } \def\defin#1#2{\global\advance\thmnum by 1 \xdef#1{Definition \currentsec.\the\thmnum} \bigbreak\noindent{\bf Definition \currentsec.\the\thmnum.} {\it#2} } \def\lemma#1#2{\global\advance\thmnum by 1 \xdef#1{Lemma \currentsec.\the\thmnum} \bigbreak\noindent{\bf Lemma \currentsec.\the\thmnum.} {\it#2}} \def\thm#1#2{\global\advance\thmnum by 1 \xdef#1{Theorem \currentsec.\the\thmnum} \bigbreak\noindent{\bf Theorem \currentsec.\the\thmnum.} {\it#2} } \def\cor#1#2{\global\advance\thmnum by 1 \xdef#1{Corollary \currentsec.\the\thmnum} \bigbreak\noindent{\bf Corollary \currentsec.\the\thmnum.} {\it#2} } \def\conj#1#2{\global\advance\thmnum by 1 \xdef#1{Conjecture \currentsec.\the\thmnum} \bigbreak\noindent{\bf Conjecture \currentsec.\the\thmnum.} {\it#2} } \def\proof{\medskip\noindent{\it Proof. }} \newcount\eqnum \global\eqnum=0 \def\num{\global\advance\eqnum by 1 \eqno({\rm\currentsec}.\the\eqnum)} \def\eqalignnum{\global\advance\eqnum by 1 ({\rm\currentsec}.\the\eqnum)} \def\ref#1{\num \xdef#1{(\currentsec.\the\eqnum)}} \def\eqalignref#1{\eqalignnum \xdef#1{(\currentsec.\the\eqnum)}} \def\title#1{\centerline{\bf\bigfont#1}} \newcount\subnum \def\Alph#1{\ifcase#1\or A\or B\or C\or D\or E\or F\or G\or H\fi} \def\subsec{\global\advance\subnum by 1 \vskip12pt plus4pt \goodbreak \noindent {\bf \currentsec.\Alph\subnum.} } \def\newsubsec{\global\subnum=1 \vskip6pt\noindent {\bf \currentsec.\Alph\subnum.} } \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi\space\number\day, \number\year} \input amssym.def \input amssym.tex \def\ol{\overline} \def\ul{\underline} \def\smooth#1{C^\infty(#1)} \def\tr{\mathop{\rm tr}\nolimits} \def\str{\mathop{\rm Str}\nolimits} \def\dint{\int^\oplus} \def\bC{{\Bbb C}} \def\bN{{\Bbb N}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bT{{\Bbb T}} \def\bZ{{\Bbb Z}} \def\cA{{\cal A}} \def\cB{{\cal B}} \def\cD{{\cal D}} \def\cF{{\cal F}} \def\cH{{\cal H}} \def\cL{{\cal L}} \def\cM{{\cal M}} \def\cO{{\cal O}} \def\cP{{\cal P}} \def\cS{{\cal S}} \def\cV{{\cal V}} \def\fA{{\frak A}} \def\fK{{\frak K}} \def\fN{{\frak N}} \def\fP{{\frak P}} \def\fR{{\frak R}} \def\fT{{\frak T}} \def\fZ{{\frak Z}} \chardef\o="1C \def\frac#1#2{{#1\over #2}} \header \def\leftheader{\centerline{S. KLIMEK and A. LESNIEWSKI}} \def\rightheader{\centerline{QUANTUM RIEMANN SURFACES}} \def\intsec{I} \def\pionesec{II} \def\autformsec{III} \def\toeplitzsec{IV} \def\modularsec{V} \def\gz{\Gamma_0} \def\ud{{\cal U}} \def\hg{{\cal H}^r(\Gamma,v)} \def\hgz{{\cal H}^r(\gz,v)} \def\hgt{{\cal H}^r(\Gamma,v_\theta)} \def\k#1#2{K^r(#1,#2)} \def\ks#1#2{K^s(#1,#2)} \def\kgz#1#2{K^r_{\gz,v}(#1,#2)} \def\tg#1{T^r_{\gz,v}(#1)} {\baselineskip=12pt \nopagenumbers \line{\hfill \bf HUTMP 95/443} \line{\hfill \today} \vfill \title{Quantum Riemann Surfaces} \vskip 1cm \title{for Arbitrary Planck's Constant} \vskip 1in \centerline{{\bf Slawomir Klimek}$^*$ and {\bf Andrzej Lesniewski}$^{**}$ } \vskip 12pt \centerline{$^*$Department of Mathematics} \centerline{IUPUI} \centerline{Indianapolis, IN 46205, USA} \centerline{Supported in part by the National Science Foundation under grant DMS--9500463} \vskip 12pt \centerline{$^{**}$Lyman Laboratory of Physics} \centerline{Harvard University} \centerline{Cambridge, MA 02138, USA} \centerline{Supported in part by the National Science Foundation under grant DMS-9424344}} \vskip 1in \centerline{\bf Abstract} \vskip 3mm\noindent We continue our study of quantum Riemann surfaces initiated in [1, 2, 3]. We construct a one parameter family of deformations of compact Riemann surfaces of genus $g\geq 2$. Our construction does not require any discretness condition on the value of Planck's constant. It coincides with the construction of [2] in the case when Planck's constant assumes the discrete set of values dictated by geometric quantization. \vfill\eject \section\intsec{Introduction} In a series of papers [1], [2], [3], we studied non-perturbative deformation quantization of Riemann surfaces. Our approach is based on the ideas of [4] (for related developments, see also [5], [6], [7], and references therein.) A satisfactory picture of uniformization of exceptional quantum Riemann surfaces emerged from these investigations. In the case of higher genus $(g\geq 2)$ Riemann surfaces, the uniformization on the quantum level is a more complex issue. In fact, if $M$ is a Riemann surface and $N$ is a covering of $M$, then the quantization of $N$ is a covering of the quantization of $M$ in the sense of [3] only if the fundamental group of the covering $N\to M$ is abelian. Part of the problem is the presence of toplogical sectors (similar to the $\theta$-vacua in gauge theory) in the quantum theory, which is related to the non-simple connectedness of the classical phase space. These sectors are classified by the characters of the fundamental group of the phase space. This does not reflect the non-commutativity of that group. On the other hand, whether the fundamental group is commutative or not seems to be important for the quantum uniformization in the sense of [3]. A similar phenomenon was discussed previously in [8]. Quantization of Riemann surfaces in the framework of geometric quantization (see e.g. [9]) requires a restriction on the allowed values of the deformation parameter $r$ (``the quantization condition''.) However, it is desirable to have a definition of quantum Riemann surfaces for all values of $r$. A definition consistent with geometric quantization was given in [2], for $r=n(2g-2)^{-1}$, where $n\in \bN$. In principle, quantum uniformization allows for a construction of quantum Riemann surfaces for all values of $r$, using the universal covering, the Poincar\'e disk, as the point of departure. Since the fundamental groups of higher genus Riemann surfaces are non-abelian, it is likely, however, that the so defined algebra of quantized functions does not reduce to the algebra of [2], when $r=n(2g-2)^{-1}$. In this paper, we construct quantization of compact Riemann surfaces for arbitrary values of $r>0$ in a manner consistent with [2]. Our starting point is a non-compact covering space $\hat M$ such that the group of cover transformation is the abelian group $\bZ$. Since $\hat M$ is a non-compact Riemann surface, all holomorphic line bundles are holomorphically trivial [10], and geometric quantization does not impose any restrictions on Planck's constant. It was explained in [2] that geometric quantization of Riemann surfaces leads to certain operator algebras on Hilbert spaces of automorphic forms. We define the quantization of $M$ in terms of Toeplitz operators with invariant symbols on a space of automorphic forms on $\hat M$, and prove deformation estimates. The proof of the estimates is based on the methods developed in [2] (see [11] for a different approach.) The paper is organized as follows. In Section \pionesec , we study the fundamental groups of $M$ and $\hat M$. We define and study suitable spaces of automorphic forms on $M$ and $\hat M$ in Section \autformsec . Section \toeplitzsec\ contains proofs of deformation estimates. \section\pionesec{The fundamental groups} In this section, we fix our notation and describe certain group theoretic properties of the fundamental groups of Riemann surfaces which will be useful in later sections. Let $M$ be a compact Riemann surface of genus $g\geq 2$. We pick a basis $\{a_i,b_i\}$, $i=1,..,g,$ of one-cycles on $M$. The fundamental group $\Gamma$ of $M$ is a finitely generated group with generators $\{a_i,b_i\}$, $i=1,\ldots , g,$ obeying the relation (see e.g [12]) $$ \prod_{i=1}^g a_ib_ia_i^{-1}b_i^{-1}=1\ . $$ We denote $a:=a_g$ and $b:=b_g$. An element $\gamma\in\Gamma$ can be represented as $$ \gamma=\prod_{j=1}^\infty\prod_{i=1}^g a_i^{n_{i,j}}b_i^{m_{i,j}}, $$ where almost all integers $n_{i,j}, m_{i,j}$ are equal to zero. Consider now the homomorphism $$ \gamma\to\sum_j m_{g,j}\in\bZ\ . $$ One can easily verify that the above map is well defined, i.e. it does not depend on the way we represent $\gamma$, and is indeed a homomorphism of groups. We denote the kernel of this homomorphism by $\gz$. The group $\gz$ is no longer finitely generated. In fact, it is not difficult to see that the following elements are its generators: $$ a,\ b^na_ib^{-n},\ b^nb_ib^{-n},\ {\rm where}\ n\in \bN,\ i=0,1,\ldots , g-1\ , $$ and there are no relations among them, i.e. $\Gamma_0$ is the free product of infinite cyclic groups generated by the above generators. Let $\hat M$ be a covering of $M$ with no branching points and such that $\pi_1(\hat M)=\gz$. The group of cover transformations of the covering $\hat M\to M$ is then equal to $\bZ$. Intuitively, $\hat M$ is obtained from $M$ by cutting along the $a$ cycle and then continuing $M$ across the cut along the $b$ cycle. One can visualize $\hat M$ as an infinite cylinder with infinite number of handle bodies attached, each handle body having $g-1$ handles. $\hat M$ is a non-compact Riemann surface. The universal covering space of both $M$ and $\hat M$ is the unit disk $\ud$. The groups $\Gamma$ and $\gz$ can be think of as Fuchsian groups on $\ud$. For $$ \gamma=\left[\matrix{a&b\cr \bar a&\bar b\cr}\right]\in\Gamma, $$ we denote $$ \gamma(z)={az+b\over \bar b x+\bar a}\; ,\ \ z\in\ud.\ref{\gammaref} $$ Since the derivative of \gammaref\ is $\gamma'(z)=(\bar bz+\bar a)^{-2}$, we can set: $$ \gamma'(z)^{1/2}:=(\bar bz+\bar a)^{-1}. $$ Let $G$ be either $\Gamma$ or $\gz$, and let $r>0$. A multiplier of weight $r$ for the group $G$ is a map $v:G\to \bC$ such that [12] $$ |v(\gamma)|=1, $$ and $$ \gamma_1'(\gamma_2(z))^{-r/2}\gamma_2'(z)^{-r/2} v(\gamma_1)v(\gamma_2)= (\gamma_1\gamma_2)'(z)^{-r/2} v(\gamma_1\gamma_2)\ .\ref{\multref} $$ In \multref , the standard branch of the logarithm is taken to define the $r$-th power. The existence of multipliers is a cohomological question, see [13] for the modern treatment. Equation \multref\ can be interpreted as the triviality of a 2-cocycle in the group cohomology of $G$. A multiplier is essentially a 1-cochain whose coboundary is that 2-cocycle. It is well-known that multipliers exist for $\Gamma$ if and only if $r=n(2g-2)^{-1}$, $n=1,2,\ldots$. The situation is different for $\gz$. \prop\cohoprop{The second cohomology group $H^2(\gz,\bZ)$ of $\gz$ is trivial.} \medskip\noindent{\it Proof.} Since $\gz=\pi_1(\hat M)$, and the universal covering space of $\hat M$ is contractible, it follows from Eilenberg-MacLane's theorem [14] that $$ H^*(\gz,\bZ)\simeq H^*(\hat M,\bZ)\ . $$ By Poincar\'e duality [15], $H^2(\hat M,\bZ)$ is isomorphic to the compactly supported cohomology group of $\hat M$ in dimension zero. The latter is trivial, as $\hat M$ is non-compact. $\square$ Consequently, multipliers for $\gz$ exist for arbitrary $r$. Let us also remark that the ratio of two multipliers is a character of the group. \section\autformsec{Automorphic forms} As before, let $G$ be either $\Gamma$ or $\gz$, and let $v$ be a multiplier for the group $G$. Recall that a holomorphic function $\phi:\,\ud\to\bC$ is called an automorphic form for $G$ of weight $r>0$ with multiplier $v$, if [12]: $$ \phi(\gamma(z))=v(\gamma)\gamma '(z)^{-r/2}\phi(z)\ ,\ref{\autformref} $$ for each $\gamma\in G$. Automorphic forms for infinitely generated Fuchsian groups like $\gz$ have been studied less extensively than those for finitely generated groups, but there is a fair amount of information available, see [16], [17], and references therein. Let $R$ be a fundamental polygon for $\Gamma$. Then $R_0:=\bigcup_{n\in\bZ}b^nR$ is a fundamental polygon for $\gz$. It has infinitely many sides. Here we use the same symbol $b$ to denote the group element of $\Gamma$ corresponding to the cycle $b=b_g$. For $r=n(2g-2)^{-1}$, we define $\hg$ to be the Hilbert space of automorphic forms for $\Gamma$ equipped with the following scalar product: $$ (\phi,\psi):=\int_R\overline{\phi(z)}\psi(z)\,d\mu^r(z),\ref{\scalprod} $$ where the measure $d\mu^r(z)$ is $$ d\mu^r(z):={r-1\over\pi}(1-|z|^2)^{r-2}\,d^2z\ . $$ For arbitrary $r>0$, let $\hgz$ be the space of automorphic forms for $\gz$ with the scalar product as in \scalprod\ but $R_0$ replacing $R$. If $v$ is a multiplier for $\Gamma$ and $e^{i\theta}\in S^1$, we let $v_\theta$ denote a new multiplier for $\Gamma$ defined by: $$ v_\theta(b)=v(b)e^{i\theta}, $$ and $v_\theta=v$ on all other generators. In particular $v_0=v$. All of the multipliers $v_\theta$ are equal when restricted to the subgroup $\gz$. The restricted multiplier will be again denoted by $v$. \thm\autformthm{With the above definitions, there is a canonical isomorphism $$ \hgz\cong\int_{S^1}^\oplus\hgt\,d\theta,\hskip .5cm r=n(2g-2)^{-1},\ n=1,2,\ldots\ . $$ } \noindent{\it Proof.} For an automorphic form $\phi$ for $\gz$, define $$ U\phi(z)=b'(z)^{r/2}\phi(bz)\ .\ref{\udefref} $$ We claim that \udefref\ is again an automorphic form, and in fact $U$ is a unitary operator on $\hgz$. Let us first verify \autformref : $$\eqalign{ U\phi(\gamma z)&=b'(\gamma z)^{r/2}\phi(b\gamma b^{-1}bz)\cr &=v(b\gamma b^{-1}) b'(\gamma z)^{r/2} (b\gamma b^{-1})'(bz)^{-r/2}\phi(bz).\cr }$$ Using the chain rule and \multref , we obtain \autformref . Unitarity of $U$ is a consequence of the following calculation: $$\eqalign{ (U\phi,U\psi)&=\int\limits_{R_0}\overline{b'(z)^{r/2}\phi(bz)} b'(z)^{r/2}\psi(bz)\,d\mu^r(z)\cr &=\int\limits_{R_0}\overline{\phi(bz)}\psi(bz)\,d\mu^r(bz)=(\phi,\psi),\cr }$$ where we have used the transformation properties of $d\mu^r(z)$ and the fact that $R_0$ is invariant under $b$. The isomorphism of Hilbert spaces that we want to establish is in essence the spectral decomposition of $U$. Explicitly, we define $$ P:\;\hgz\to\int_{S^1}^\oplus\hgt\,d\theta\ $$ by the following formula: $$ P\phi(\zeta,\theta)=\sum_{n\in \bZ}v_\theta(b^{-n}) U^n\phi(z)\ .\ref{\isoref} $$ We need to verify that the right hand side of \isoref\ is in $\hgt$. If $\gamma\in\gz$, this follows from the fact that $U^n\phi(z)$ are automorphic forms for $\gz$, and the fact that $v_\theta|_{\gz}= v|_{\gz}$. If $\gamma=b$, we have $$ U\left(\sum_{n\in \bZ}v_\theta(b^{-n}) U^n\phi(z)\right)=v_\theta(b) \left(\sum_{n\in \bZ}v_\theta(b^{-n}) U^n\phi(z)\right)\ . $$ The general case follows easily. We verify that $P$ is an isometry: $$ \eqalign{ \|P\phi\|^2&=\int\limits_{-\pi}^{\pi} \biggl(\int\limits_R |P\phi(z,\theta)|^2\, d\mu^r(z)\biggr)\,{d\theta\over 2\pi}\cr &=\int\limits_{-\pi}^{\pi} \biggl(\int\limits_R \sum_{n,m} \overline{U^n\phi(z)} U^m\phi(z) v_\theta(b)^{n-m} \, d\mu^r(z)\biggr)\,{d\theta\over 2\pi}\cr &=\int\limits_R \sum_n |U^n\phi(z)|^2\,d\mu^r(z)\cr &=\sum_n \int\limits_{b^nR} |\phi(z)|^2\,d\mu^r(z)\cr &=\int\limits_{R_0} |\phi(z)|^2\,d\mu^r(z)\ .} $$ Similar calculations show that the inverse of $P$ is given by: $$ P^{-1}\psi(z)=\int\limits_{-\pi}^\pi \psi(z,\theta)\,d\theta\ .\quad \square $$ This result implies that the quantization of Riemann surfaces proposed in this paper reduces, when $r=n(2g-2)^{-1}$, to the definition of [2]. \section\toeplitzsec{Toeplitz quantization} In this section, we construct a quantization of the Riemann surface $M$ in terms of Toeplitz operators on $\hgt$. We first recall the relevant definitions. The reproducing kernel $\kgz zw$ for $\hgz$ is given by the following Poincar\'e series: $$ \kgz zw =\sum_{\gamma\in\gz}v(\gamma)^{-1}\gamma '(z)^{r/2} \k {\gamma(z)}w,\ref{\kernelref} $$ where $$ \k zw =(1-z\bar w)^{-1}\ . $$ Let $C_\Gamma(\ud)$ be the $\bC^*$-algebra of bounded continuous functions on $\ud$ which are invariant under $\Gamma$, so that $C_\Gamma(\ud)\cong C(M)$. For $f\in C_\Gamma(\ud)$, we define the Toeplitz operator $\tg f$ on $\hgz$ with symbol $f$ by: $$ (\tg f \phi)(z)=\int_{\gz}\kgz zw f(w)\phi(w)\,d\mu^r(w)\ . \ref{\toeplitzref} $$ The goal of this section is to prove that the correspondence $f\mapsto \tg f$ is a quantization of $M$. This means that for $f\in C_\Gamma(\ud)$ we have the norm limit $$ \lim_{r\to \infty} \| \tg f \| = \| f \|_{\infty},\ref{\normestref} $$ where $||\cdot||$ denotes the operator norm, and where $||\cdot||_\infty$ denotes the sup-norm. If, moreover, $f,g$ are smooth, then $$ \lim_{r\to \infty}\|r\bigl[\tg f,\tg g\bigr]+\tg {i\{f,g\}}\|=0, \ref{\comestref} $$ where $\{f,g\}$ is the usual Poisson bracket, $$ \{f,g\}(z)=i(1-|z|^2)^2[\partial f(z)\bar\partial g(z)- \partial g(z)\bar\partial f(z)]\ .\ref{\bracketref} $$ \medskip \thm\defquantthm{With the above definitions, the correspondence $$ f\mapsto \tg f $$ is a quantization of $M$. } \medskip\noindent{\it Proof.} The details of analogous estimates were explained in [1] and [2]. Here we follow [2], where the estimates were proved for the quantization based on automorphic forms of $\Gamma$. However, the compactness of the fundamental domain of $\Gamma$ was used in an essential way in several places, so that the results cannot be applied to the case of $\gz$ (as $R_0$ is not compact in $\ud$). The main difference is that ``transfer of regularity" argument has to be done more carefully in the present case. To prove \normestref , we consider the vectors $$ \phi_w(z):=\kgz ww^{-1/2}\kgz zw $$ in $\hgz$, and verify that: $$ \sup_{x\in R}|f(w)-(\phi_w,\tg f\phi_w)|\to 0,\ \ \ {\rm as}\ r\to\infty , \ref{\phiwestref} $$ in a way analogous to [2]. The proof there was based on Lemmas 4.1 and 4.2 which are also valid for $\gz\subset\Gamma$. Since $f$ is invariant under $\Gamma$, and not just $\gz$, and since the supremum in \phiwestref\ is taken over a compact set, \phiwestref\ follows exactly as in [2]. The estimate \comestref\ is a consequence of $$ ||r\left(\tg f\tg g-\tg {fg}\right) +\tg {(1-|z|)^2\partial f\bar\partial g}||\to 0,\ \ \ {\rm as}\ r\to\infty, \ref{\prodestref} $$ for $f,g\in C_\Gamma^\infty$. To prove \prodestref , one expands $(\phi,\tg f\tg g\psi)$ in a Taylor series as in [2] formula (5.6). The first three terms in that formula combine to give the second and third terms in \prodestref . The analog of the fourth term of [2], formula (5.6), is $O(r^{-2})$ as in [1]. It remains to estimate the remainder. The technique developed in [1] for estimating the remainder terms can be applied to our case with one modification. The integral $\int_\ud |\psi(w)|^2\,d\mu^r(w)$ is infinite if $\psi\in\hgz$, and one needs to transfer a power of $1-|w|^2$ to make it convergent. This ``transfer of regularity" trick was explained in detail on the last two pages of [2] for $\Gamma$-automorphic forms. However, the compactness of the fundamental domain of $\Gamma$ was used in an essential way. We show below that a modification of the argument used in [2] can be used to in our case. \lemma\firstlemma{ Let $\psi\in\hgz$ and $s>1$. Then we have: $$ \int_\ud|\psi(w)|^2(1-|w|^2)^s\,d\mu^r(w)=O(1)||\psi||^2\ . $$ } \lemma\secondlemma{Let $b\in SU(1,1)$ be hyperbolic, let $K$ be a compact set in $\ud$, and let $t>0$. Then we have: $$ \sup_{z\in K}\;\sum_{n\in\bZ}\left|(b^n)'(z)\right|^t=O(1)\ . $$ } We will prove these lemmas after we have completed the main line of the argument. We now use the above lemmas to estimate the term in formula (5.16) of [2]. This will conclude the proof of \defquantthm . That term reads: $$ \int_{R_0\times\ud}|\phi(z)|(1-|z|^2)^{1-r/2}|\psi(\gamma_z(w))| |\gamma_z'(w)|^{r/2}{|w|^2\over (1-|w|^2)^{11}}\,d\mu^r(z)\,d\mu^r(z), \ref{\atermref} $$ where $\gamma_z(w)=(w+z)/(\bar zw+1)$. Let $0<\epsilon<1/2$. We multiply and divide the integrand by $(1-|\gamma_z(w)|^2)^{1-\epsilon}$, and use the following elementary bound: $$ {1\over (1-|\gamma_z(w)|^2)^{1-\epsilon}}\leq{O(1)\over (1-|z|^2)^{1-\epsilon}(1-|w|^2)}\ . $$ The integral in \atermref\ is consequently less then $$ \eqalign{ O(1)&\int_{R_0\times\ud}|\phi(z)|(1-|z|^2)^{\epsilon-r/2}| \psi(\gamma_z(w))|(1-|\gamma_z(w)|^2)^{1-\epsilon}|\gamma_z'(w)|^{r/2}\cr &\times{|w|^2\over (1-|w|^2)^{12}}\,d\mu^r(z)\,d\mu^r(z)\ .\cr }\ref{\btermref} $$ \noindent Using the Schwarz inequality and changing variables in the $\psi$ term, we get the following bound: $$ \eqalign{ O(1)\;||\phi||\;&\left(\int_{R_0}(1-|z|^2)^{2\epsilon-r}\, d\mu^r(z)\right)^{1/2}\left(\int_\ud|\psi(w)|^2(1-|w|^2)^{2-2\epsilon}\, d\mu^r(w)\right)^{1/2}\cr &\times\left(\int_\ud{|w|^4\over (1-|w|^2)^{24}}\, d\mu^r(w)\right)^{1/2}\ .\cr}\ref{\ctermref} $$ With our choice $0<\epsilon<1/2$, the exponent $2-2\epsilon$ in the third factor is greater than $1$, and so we can apply \firstlemma\ to it, and conclude that it is $O(1)||\psi||^2$. The fourth factor is $O(r^{-1})$ by [2], formula (5.18). One can analyze the second factor in \ctermref\ as follows: $$ \eqalign{ \int_{R_0}(1-|z|^2)^{2\epsilon-r}\,d\mu^r(z)&= O(r)\int_{R_0}(1-|z|^2)^{2\epsilon}\,d\mu_P(z)\cr &=O(r)\sum_{n\in\bZ}\int_{b^{-n}R}(1-|z|^2)^{2\epsilon}\,d\mu_P(z)\cr &=O(r)\sum_{n\in\bZ}\int_{R}(1-|b^nz|^2)^{2\epsilon}\,d\mu_P(z)\cr &=O(r)\int_R\sum_{n\in\bZ}\left|(b^n)'(z)\right|^{2\epsilon} (1-|z|^2)^{2\epsilon}\,d\mu_P(z)\cr &\leq O(r)\sup_{z\in R}\;\sum_{n\in\bZ}\left|(b^n)'(z) \right|^{2\epsilon}\ .\cr }\ref{\dtermref} $$ In \dtermref , $d\mu_P(z)$ is the Poincar\'e measure on $\ud$, and we have used the fact that $R$ is compact. Since $2\epsilon>0$ it follows from \secondlemma\ that \dtermref\ is $O(r)$. This concludes the transfer of regularity argument. \noindent {\it Proof of \firstlemma\ :} We decompose $\ud$ into the translates of $R_0$: $$ \eqalign{ &\int_\ud|\psi(w)|^2(1-|w|^2)^s\,d\mu^r(w)= \sum_{\gamma\in\gz}\int_{\gamma^{-1}R_0}|\psi(w)|^2(1-|w|^2)^s \,d\mu^r(w)\cr &=\sum_{\gamma\in\gz}\int_{R_0}|\psi(w)|^2(1-|\gamma(w)|^2)^s \,d\mu^r(w)\leq\left(\sup_{w\in\ud}\sum_{\gamma\in\gz} (1-|\gamma(w)|^2)^s\right)||\psi||^2\ .\cr } $$ To estimate the supremum factor, we proceed as in the proof of [2] lemma 4.2: $$ \eqalign{ 1&=(1-|w|^2)^s\int_\ud |\ks wz|^2\,d\mu^s(z)\cr &=(1-|w|^2)^s\sum_{\gamma\in\gz}\int_{\gamma^{-1}R_0} |\ks wz|^2\,d\mu^s(z)\cr &=(1-|w|^2)^s\sum_{\gamma\in\gz}|\gamma '(w)|^s\int_{R_0} |\ks {\gamma(w)}z|^2\,d\mu^s(z)\cr &=\sum_{\gamma\in\gz}(1-|\gamma(w)|^2)^s\int_{R_0} |\ks {\gamma(w)}z|^2\,d\mu^s(z)\cr &\geq\sum_{\gamma\in\gz}(1-|\gamma(w)|^2)^s{1\over 2^{2s}}\int_{R_0} \,d\mu^s(z)\ . } $$ Hence, $\sup_{w\in\ud}\sum_{\gamma\in\gz} (1-|\gamma(w)|^2)^s=O(1)$, if $s>1$. This concludes the proof of \firstlemma . \noindent {\it Proof of \secondlemma\ :} Since $b$ is a hyperbolic element of $SU(1,1)$, it has two real eigenvalues $\lambda,\ 1/\lambda$ with $|\lambda|>1$. Letting $$ b^n=\left[\matrix{\alpha_n&\beta_n\cr \bar\beta_n&\bar\alpha_n\cr}\right], $$ we have $\alpha_n=O(|\lambda|^{|n|})$. Furthermore, we have the following bound: $$ \eqalign{ |(b^n)'(z)|&=|\bar\beta_nz+\bar\alpha_n|^{-2}= |\alpha_n|^{-2}|1+{\bar\beta_n\over\bar\alpha_N}z|^{-2}\cr &\leq|\alpha_n|^{-2}(1-|z|)^{-2}.\cr } $$ Since $z$ varies over a compact set and $t>0$, it follows that the series $\sum_{n\in\bZ}\left|(b^n)'(z)\right|^t$ is bounded, uniformly in $z$, by a convergent geometric series. 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