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\begin{document}
%\title[Poisson transforms and principal series]{Poisson transforms and
%the principal series for $\SL_2(\RR)$}
\title[Function theory related to $\PSL_2(\RR)$] {Function theory
related to the group $\PSL_2(\RR)$}

\author{R.\,Bruggeman} \author{J.\,Lewis} \author{D.\,Zagier}



\maketitle


\tableofcontents

\renewcommand\theequation{\arabic{equation}}

\section*{Introduction} The aim of this article is to discuss some of
the analytic aspects of the group $G=\PSL_2(\RR)$ acting on the
hyperbolic plane and its boundary. Everything we do is related in
some way with the (spherical) principal series representations of the
group~$G$.\change{ \emph{spherical} inserted.
(referee)}

These principal series representations are among the best known and
most basic objects of all of representation theory. In this article,
we will review the standard models used to realize these
representations and then describe a number of new properties and new
models. Some of these are surprising and interesting in their own
right, while others have already proved useful in connection with the
study of cohomological applications of automorphic forms \cite{BLZ-p}
and may potentially have other applications in the future. The
construction of new models may at first sight seem superfluous, since
by definition any two models of the same representation are
equivariantly isomorphic, but nevertheless gives new information
because the isomorphisms between the models are not trivial and also
because each model consists of the global sections of a certain
$G$-equivariant sheaf, and these sheaves are completely different
even if they have isomorphic spaces of global sections.

The principal \alsochanged{YJ}series representations of $G$ are
indexed by a complex number~$s$, called the {\it spectral parameter},
which we will always assume to have real part between~0 and~1. (The
condition $\,\re(s)=\h\,$, corresponding to unitarizability, will
play no role in this paper.) There are two basic realizations. One is
the space~$\V{}s$ of functions on~$\RR$ with the (right) action
of~$G$ given by
\be (\ph\,|\,g)(t)\=|ct+d|^{-2s}\,\ph\bigl(\frac{at+b}{ct+d})
\qquad\qquad\bigl(t\in\RR,\;g=\matc abcd\in G\bigr)\,. \label{iAA}\ee
The other is the space $\E_s$ of functions $u$ on~$\uhp$ (complex
upper half-plane) satisfying
\be \Dt\,u(z)\=s(1-s)\,u(z)\qquad\qquad(z\in\uhp), \label{iAB} \ee
where
$\,\Dt=-y^2\bigl(\frac{\p^2}{\p x^2}+\frac{\p^2}{\p x^2}\bigr)\,$
($z=x+iy\in\uhp$) is the hyperbolic Laplace operator, with the action
$u\mapsto u\circ g$. They are related by Helgason's {\it Poisson
transform} (thus named because it is the analogue of the
corresponding formula given by Poisson for holomorphic functions)
\be \ph(t)\quad\mapsto\quad(\poiss s\ph)(z)
\=\frac1\pi\,\int_{-\infty}^\infty\ph(t)\,R(t;z)^{1-s}\,dt\,,
\label{iAC}\ee
where $\,R(t;z)\,=\, R_t(z) \,=\, \frac y{(z-t)(\bar z-t)}\,$ for
$z=x+iy\in\uhp$ and $t\in\CC$. The three main themes of this paper
are: the explicit inversion of the Poisson transformation, the study
of germs of Laplace eigenfunctions near the boundary
$\proj\RR=\RR\cup\{\infty\}$ of~$\uhp$, and the construction of a new
model of the principal series representation which is a kind of
hybrid of~$\V{}s$ and~$\E_s$. We now describe each of these briefly.


\sssect{Inverse Poisson transform} We would like to describe the
inverse map of $\poiss s$ explicitly. The right-hand side
of~\eqref{iAC} can be interpreted as it stands if $\ph$ is a smooth
vector in~$\V{}s$ (corresponding to a function $\ph(x)$ which is
$C^\infty$ on~$\RR$ and such that $\,t\mapsto|t|^{-2s}\ph(1/t)\,$ is
$C^\infty$ at~$t=0$). To get an isomorphism between~$\V{}s$ and all
of~$\E_s$, one has to allow {\it hyperfunctions} $\ph(t)$. The
precise definition, which is somewhat subtle in the model used
in~(1), will be reviewed in \S1.2; for now we recall only that a
hyperfunction on $I\subset\RR$ is represented by a holomorphic
function on $U\sm I$, where $U$ is a neighborhood $U$ of $I$ in~$\CC$
with $U\cap\RR=I$ and where two holomorphic functions represent the
same hyperfunction if their difference is holomorphic on all of~$U$.
We will show \change{Reference \S\ref{sect-pms} inserted}in
\S\ref{sect-pms} that for $u\in\E_s$ the vector
$\poiss s^{\!\!-1} u\in\V{}s$ can be represented by the hyperfunction
\be h_{z_0}(\z)\=
\begin{cases} u(z_0)\;+\;\int_{z_0}^\z\,
\bigl[u(z),\,\bigl(R_\z(z)/R_\z(z_0)\bigr)^s\bigr] & \text{ if $\z\in
U\cap\uhp$}\\
\qquad\int_{\bar\z}^{z_0}\,\bigl[\bigl(R_\z(z)/R_\z(z_0)\bigr)^s,
\,u(z)\bigr]
&\text{ if $\z\in U\cap\lhp$}\end{cases} \label{iBA} \ee
for any $z_0\in U\cap\uhp$, where $\lhp=\{z=x+iy\in\CC\;:\; y<0\}$
denotes the lower half-plane and $[u(z),v(z)]$ for any functions $u$
and $v$ in $\uhp$ is the {\it Green's form}
\be [u(z),v(z)] \= \frac{\p u(z)}{\p z}\,v(z)\,dz\;+\; u(z)\,\frac{\p
v(z)}{\p\bar z}\,d\bar z\,, \label{iBB} \ee
which is a closed 1-form if $u$ and $v$ both satisfy the Laplace
equation~\eqref{iAB}. The asymmetry in~\eqref{iBA} is necessary
because, although $R(\z;z)^s$ tends to zero at \change{'both'
omitted.} $z=\z$ and $z=\bar\z$, \change{Changes in rest of
sentence.} both its $z$-derivative at $\z$ and its
$\bar z$-derivative at $\bar\z$ become infinite, forcing us to change
the order of the arguments in the Green's form in the two components
of $U\sm I$. That the two different-looking expressions
in~\eqref{iBA} are nevertheless formally the same follows from the
fact that $[u,v]+[v,u]=d(uv)$ for any functions~$u$ and~$v$.


\sssect{Boundary eigenfunctions} If one looks at known examples of
solutions of the Laplace equation~\eqref{iAB}, then it is very
striking that many of these functions decompose into two pieces of
the form $y^sA(z)$ and $y^{1-s}B(z)$ as $z=x+iy$ tends to a point of
$\RR\subset \proj\RR=\p\uhp$, where $A(z)$ and $B(z)$ are functions
which extend analytically across the boundary. For instance, the
eigenfunctions that occur as building blocks in the Fourier
expansions of Maass wave forms for a Fuchsian group $\G\subset G$ are
the functions
\be k_{s,2\pi n}(z)\=y^{1/2}\,K_{s-1/2}(2\pi|n|y)\,e^{2\pi inx}
\qquad(z=x+iy\in\RR,\;n\in\ZZ,\;n\neq 0), \label{iCA}\ee
where $K_{s-1/2}(t)$ is the standard $K$-Bessel function which decays
exponentially as $t\to\infty$. The function $K_\nu(t)$ has the form
$\,\dfrac\pi{\sin\pi\nu}\,\bigl(I_\nu(t)-I_{-\nu}(t)\bigr)$ with
\[ I_\nu(t)
=\sum\limits_{n=0}^\infty\frac{(-1/4)^n\, t^{2n+\nu}}{n!\,\Gf(n+\nu)}
\,,\]
so $k_{s,2\pi n}(z)$ decomposes into two pieces of the form
$y^s\times$(analytic near the boundary) and $y^{1-s}\times$(analytic
near the boundary). The same is true for other elements of~$\E_s$,
involving other special functions like Legendre or hypergeometric
functions, that play a role in the spectral analysis of automorphic
forms. A second main theme of this paper is to understand this
phenomenon. We will show that to every analytic function $\ph$ on an
interval $I\subset\RR$ there is a unique solution $u$ of~\eqref{iAB}
in $U\cap\uhp$ (where $U$ as before is a neighborhood of $I$ in $\CC$
with $U\cap\RR=I$, supposed simply connected and sufficiently small)
such that $u(x+iy)=y^s\,\Phi(x+iy)$ for an analytic function $\Phi$
on~$U$ with restriction $\Phi|_I=\ph$. In
\change{Reference \S\ref{sect-gb}}\S\ref{sect-gb} we will call the
(locally defined)
map $\ph\mapsto u$ the {\it transverse Poisson transform} of~$\ph$ and
will show that it can be described by both a Taylor series in~$y$ and
an integral formula, the latter bearing a striking resemblance to the
original (globally defined) Poisson transform~\eqref{iAC}\,:
\be (\tpoiss s\ph)(z)
\=\frac{-i\,\Gf(s+\h)}{\Gf(s)\Gf(\h)}\, \int_{\bar
z}^z\ph(\z)\,R(\z;z)^{1-s}\,d\z\,, \label{iCB}\ee
where the function $\ph(\z)$ in the integral is the unique holomorphic
extension of~$\ph(t)$ to $U$ and the integral is along any path
connecting~$\bar z$ and~$z$ within~$U$. The transverse Poisson map
produces an eigenfunction $u$ from a real-analytic function $\ph$ on
an interval~$I$ in~$\proj\RR$. We also give an explicit integral
formula representing the holomorphic function~$\ph$ in~$U$ in terms
of the eigenfunction $u=\tpoiss s \ph$.

As an application, we will show in
\change{Reference \S\ref{sect-split}}\S\ref{sect-split} that the
elements of $\E_s$ corresponding under the Poisson transform to
analytic vectors in~$\V{}s$ (which in the model~\eqref{iAA} are
represented by analytic functions $\ph$ on~$\RR$ for which
$t\mapsto|t|^{-2s}\ph(1/t)$ is analytic at~$t=0$) are precisely those
which have a decomposition $u=\tpoiss s\ph_1+\tpoiss{1-s}\ph_2$ near
the boundary of $\uhp$, where $\ph_1$ and $\ph_2$, which are uniquely
determined by~$u$, are analytic functions on~$\proj\RR$.


\sssect{Canonical model} We spoke above of two realizations of the
principal series, as~$\V{}s$
(functions on $\p\uhp=\proj\RR$) and as $\E_s$ (eigenfunctions of the
Laplace operator in~$\uhp$). In fact $\V{}s$ comes in many different
variants, discussed in detail in~\S1, each of which resolves various
of the defects of the others at the expense of introducing new ones.
For instance, the ``line model''~\eqref{iAA} which we have been using
up to now has a very simple description of the group action, but
needs special treatment of the point $\,\infty\in\proj\RR$, as one
could already see several times in the discussion above (e.g.~in the
description of smooth and analytic vectors or in the definition of
hyperfunctions). One can correct this by working on the projective
rather than the real line, but then the description of the group
action becomes very messy, while yet other models (circle model,
plane model, induced representation model, \dots) have other
drawbacks\change{earlier \emph{blemishes}}. In
\change{Reference \S\ref{sect-pms}}\S\ref{sect-pms} we will introduce
a new realization $\can_s$ (``canonical model'') that has many
advantages:
\begin{itemize}
\item all points in hyperbolic space, and all points on its boundary,
are treated in an equal way; %equally; gaf een overvolle hbox
\item the formula for the group action is very simple;
\item its objects are actual functions, not equivalence classes of
functions;
\item the Poisson transformation is given by an extremely simple
formula;
\item the canonical model $\can_s$ coincides with the image of a
canonical inversion formula for the Poisson transformation;
\item the elements of $\can_s$ satisfy differential equations,
discussed below, which lead to a sheaf $\D_s$ that is interesting in
itself;
\item it uses {\it two} variables, one in $\uhp$ and one in
$\proj\CC\sm\proj\RR$, and therefore gives a natural bridge between
the models of the principal series representations as eigenfunctions
in $\uhp$ or as hyperfunctions in a deleted neighborhood of
$\proj\RR$ in $\proj\CC$. \end{itemize}

The elements of the space $\can_s$ are precisely the functions
$(z,z_0)\mapsto h_{z_0}(z)$ arising as in~\eqref{iBA} for some
eigenfunction $u\in\E_s$, but also have several intrinsic
descriptions, of which perhaps the most surprising is a
characterization by a system of two linear differential equations:
%  $$ \frac{\d h}{\d z}\=\frac{-s}{z-\bz}\,\frac{\z-\bz}{\z-z}\,h^*\,,\qquad
%  \frac{\d h^*}{\d\bz}\=\frac s{z-\bz}\,\frac{\z-z}{\z-\bz}\,h\,, \tag\DA$$
% $$ \frac{\d h}{\d z}\=-s\,\frac{\z-\bz}{z-\bz}\,h^*\,,\qquad
%   \frac{\d h^*}{\d\bz}\=s\,\frac{(\z-\bz)\i}{z-\bz}\,h\,, \tag\DA$$
\be \frac{\p h}{\p z}\=-s\,\frac{\z-\bar z}{z-\bar z}\,h^*\,,
\qquad\frac{\p h^*}{\p\bar z}\=\frac s{(\z-\bar z)(z-\bar z)}\,h\,,
\label{iDA}\ee
% $$ (z-\bz)\,\frac{\d h}{\d z}\=-s\,(\z-\bz)\,h^*\,,\qquad
%  (z-\bz)\,\frac{\d h^*}{\d\bz}\=s\,(\z-\bz)\i\,h\,, \tag\DA$$
where $h(\z,z)$ is a function on
$\,\bigl(\proj\CC\sm\proj\RR)\times\uhp$ which is holomorphic in the
first variable and where $h^*(\z,z):=(h(\z,z)-h(z,z))/(\z-z)$. The
``Poisson transform'' in this model is very simple: it simply assigns
to $h(\z,z)$ the function $u(z)=h(z,z)$, which turns out to be an
eigenfunction of the Laplace operator. The name ``canonical model''
refers to the fact that $\V{}s$ consists of hyperfunctions and that
in $\can_s$ we have chosen a family of canonical representatives of
these hyperfunctions, indexed in a $G$-equivariant way by a parameter
in the upper half-plane: $\;h(\dv,z)$ for each $z\in\uhp$ is the
unique representative of the hyperfunction $\ph(t)R(t;z)^{-s}$ on
$\proj\RR$ which is holomorphic in all of $\proj\CC\sm\proj\RR$ and
vanishes at~$\bar z$.

\sssect{Further remarks} The known or potential applications of the
ideas in this paper are to automorphic forms in the upper half plane.
When dealing with such forms, one needs to work with functions of
general weight, not just weight~$0$ as considered here. We expect
that many of our results can be modified to the context of general
weights, where the group $G=\PSL_2(\RR)$ has to be replaced by
$\SL_2(\RR)$ or its universal covering group.
\medskip

Some parts of what we do in this paper are available in the
literature, but often in a different form or with another emphasis.
In \S4 of the introduction of \cite{Hel84}, Helgason gives an
overview of analysis on the upper half plane. One finds there the
Poisson transformation; the injectivity is proved by a polar
decomposition. As far as we know, our approach in
Theorem~\ref{thm-pi} with the Green's form is new, and in
\cite{BLZ-p} it is an essential tool to build cocycles. Helgason
gives also the asymptotic expansion near the boundary of
eigenfunctions of the Laplace operator, from which the results in
Section~\ref{sect-split} may also be derived. For these asymptotic
expansions one may also consult the work of Van den Ban and
Schlichtkrull, \cite{BS}. A more detailed and deeper discussion can
be found in \cite{KKMOOT}, \change{ (referee)}where Section~0
discusses the inverse Poisson transformation in the context of the
upper half-plane. Our presentation stresses the transverse Poisson
transformation, which also seems not to have been treated in the
earlier literature and which we use in \cite{BLZ-p} to recover Maass
wave forms from their associated cocycles. Finally, the hybrid models
in Section~\ref{sect-pms} and the related sheaf $\D_s$ are, as far as
we know, new.\smallskip

The paper ends with an appendix giving a number of explicit formulas,
including descriptions of various eigenfunctions of the Laplace
operator and tables of Poisson transforms and transverse Poisson
transforms.

\medskip\noindent\textbf{Acknowledgments.}
The two first-named authors would like to thank the Max Planck
Institute in Bonn and the Coll\`ege de France in Paris for their
repeated hospitality and for the excellent working conditions they
provided.
\change{Thanks YJ}We thank YoungJu Choie for her comments on an
earlier version.
\bigskip

\noindent\textbf{Conventions and notations.} We work with the Lie
group
%\sssect{Conventions and notations} We work with the Lie group
 \[G=\PSL_2(\RR) \allowbreak= \SL_2(\RR)/\{\pm \Id\}\,.\] We denote
the element $\,\pm\rmatc abcd\,$ of $G$ by $\,\matc abcd\,$. A
maximal compact subgroup is
$K=\PSO(2) = \bigl\{ k(\th)\;:\th\in \RR/\pi\ZZ\bigr\}$, with
\beqabc
\be\label{kd} k(\th) \=
\matr{\cos\th}{\sin\th}{-\sin\th}{\cos\th}\,.\ee
We also use the Borel subgroup $NA$, with the unipotent subgroup
$N=\bigl\{ n(x)\;:\; x\in \RR\bigr\}$ and the torus
$A=\bigl\{ a(y)\;:\; y>0\bigr\}$, with
\be\label{nad} a(y) \= \matc{\sqrt y}00{1/\!\!\sqrt y}\,,\qquad n(x)
\= \matc1x01\,.\ee
\eeqabc

We use $\H$ as a generic letter to denote the hyperbolic plane. We use
two concrete models: the unit disk
$\disk =  \bigl\{ w\in \CC\;:\; |w|<1\bigr\}$, and the upper half
plane $\uhp = \bigl\{z\in \CC\;:\; \im z>0\bigr\}$. We will denote by
$x$ and $y$ the real and imaginary part of $z\in \uhp$ respectively.
The boundary $\p\H$ of the hyperbolic plane is in these models
$\p\uhp = \proj\RR = \RR \cup\{\infty\}$, the real projective line,
and $\p\disk = \uc$, the unit circle. Both models of $\H \cup\p\H$
are contained in $\proj\CC$, on which $G$ acts in the upper half
plane model by $\matc abcd :z \mapsto \frac{az+b}{cz+d}$, and in the
 disk model by $\matc abcd: w\mapsto \frac{ Aw+B}{\bar B w + \bar A}$,
with $\matc AB{\bar B}{\bar A} = \matr1{-i}1i \matc abcd
\matr1{-i}1i^{-1}$.

All the representations that we discuss in the first five sections
depend on $s\in \CC$, the \emph{spectral parameter}; it determines
the eigenvalue $\ld_s=s-s^2$ of the Laplace operator~$\Dt$, which is
given in the upper half plane model by $-y^2\p_x^2-y^2\p_y^2$, and in
the disk model by $-(1-\nobreak |w|^2)^2\, \p_w\,\p_{\bar w}$. We
will always assume $s\notin\ZZ$, and usually restrict to
$0<\re(s)<1$. We work with right representations of $G$, denoted by
$v \mapsto v|_{2s}\, g$ or $v\mapsto v\,|\,g$.
\bigskip



\renewcommand\theequation{\equationstandard}
\section{The principal series representation $\V{}s$}\label{sect-ps}
This section serves to discuss general facts concerning the principal
series representation. Much of this is standard, but quite a lot of
 it is not, and the material presented here will be used extensively
in the rest of the paper. We will therefore give a self-contained and
fairly detailed presentation.

The principal series representations can be realized in various ways.
One of the aims of this paper is to gain insight by combining several
of these models. Subsection~\ref{sect-models} gives six standard
models for the continuous vectors in the principal series
representation. Subsection~\ref{secthf} presents the larger space of
hyperfunction vectors in some of these models, and in
Subsection~\ref{sectIM} we discuss the isomorphism (for $0<\re s<1$)
between the principal series representations with the values $s$ and
$1-s$ of the spectral parameter.

\subsection{Six models of the principal series
representation}\label{sect-models} In this subsection we look at six
models to realize the principal series representation $\V{}s$, each
of which is the most convenient in certain contexts. Three of these
models are realized on the boundary $\p\H$ of the hyperbolic plane.
Five of the six models have easy algebraic isomorphisms between them.
The sixth has a more subtle isomorphism with the others, but gives
explicit matrix coefficients. In later sections we will describe more
models of $\V{}s$ with a more complicated relation to the models
here. We also describe the duality between $\V{}s$ and $\V{}{1-s}$ in
the various models. ({\it Note}. We will use the letter $\V{}s$
somewhat loosely to denote ``the'' principal series representation in
a generic way, or when the particular space of functions under
consideration plays no role. The spaces $\V\infty s$ and $\V\om s$ of
smooth and analytic vectors, and the spaces $\V{-\infty}s$ and
$\V{-\om}s$ of distributions and hyperfunctions introduced in
\S\ref{secthf}, will be identified by the appropriate superscript.
Other superscripts such as $\prj$ and $\ci$ will be used to
distinguish vectors in the different models when needed.)

\sssect{Line model}This well-known model of the principal series
consists of complex valued functions on $\RR$ with the action of $G$
given by % change: in the sequel variable x replaced by t
\begin{equation} \ph\bigl|_{2s}\,\matc abcd (t) \= |ct+d|^{-2s}\,
\ph\biggl(\frac{at+b}{ct+d}\biggr) \label{trfln}\,.\end{equation}
Since $G$ acts on $\proj\RR=\RR\cup\{\infty\}$, and not on $\RR$, the
point at infinity plays a special role in this model and a more
correct description requires the use of a pair $(\ph,\ph_\infty)$ of
functions $\RR\lra \CC$ related by
$\ph(t) = |t|^{-2s} \ph_\infty(-1/t)$ for $t\neq 0$, and with the
right hand side in \eqref{trfln} replaced by
$|at+b|^{-2s}\ph_\infty\bigl(-\frac{ct+d}{at+b}\bigr)$ if $ct+d$
vanishes, together with the obvious corresponding formula for
$\ph_\infty$. However, we will usually work with $\ph$ alone and
leave the required verification at~$\infty$ to the reader.

The space $\V\infty s$ of \emph{smooth vectors} in this model consists
of the functions $\ph \in C^\infty(\RR)$ with an asymptotic expansion
\be\ph(t)\;\sim\;|t|^{-2s}\;\sum_{n=0}^\infty\,c_n\,t^{-n}\label{asy-p}\ee
as $|t|\to\infty$. Similarly, we define the space $\V\om s$ of
\emph{analytic vectors} as the space of $\ph \in C^\om(\RR)$
(real-analytic functions on $\RR$) for which the series appearing on
the right-hand side of \eqref{asy-p} converges to $\ph(t)$ for
$|t|\ge t_0$ for some $t_0$. Replacing $C^\infty(\RR)$ or
$C^\om(\RR)$ by $C^p(\RR)$ and the expansion \eqref{asy-p} with a
Taylor expansion of order~$p$, we define the space $\V p s$ for
$p\in \NN$.

\sssect{Plane model} The line model has the advantage that the action
\eqref{trfln} of $G$ is very simple and corresponds to the standard
formula for its action on the complex upper half-plane $\uhp$, but
the disadvantage that we have to either cover the boundary
$\RR\cup\{\infty\}$ of $\uhp$ by two charts and work with pairs of
functions or else give a special treatment to the point at infinity,
thus breaking the inherent $G$-symmetry. Each of the next five models
eliminates this problem, at the expense of introducing complexities
elsewhere. The first of these is the plane model, consisting of even
functions $\Phi:\RR^2\sm\{0\}\rightarrow\CC$ satisfying
$\Phi(tx,ty) = |t|^{-2s} \Phi(x,y)$ for $t\neq0$, with the action
\begin{equation} \Phi\bigl|\,\matc abcd (x,y) \= \Phi(ax+by,cx+dy)\,.
\end{equation}
The relation with the line model is
\begin{equation}\label{line-plane}
\begin{aligned} \ph(t)&\=\Phi(t,1)\,,\qquad
\ph_\infty(t)\=\Phi(-1,t)\,,\\
\Phi(x,y)&\=\begin{cases} |y|^{-2s}\,\ph(x/y)&\text{ if }y\neq0\,,\\
|x|^{-2s}\,\ph_\infty(-y/x)&\text{ if }x\neq 0\,,
\end{cases}
\end{aligned}
\end{equation}
and of course the elements in $\V p s$, for $p=0,1,\ldots,\infty,\om$,
are now just given by $\Ph\in C^p(\RR^2\sm\{0\})$. This model has the
advantage of being completely $G$-symmetric, but requires functions
of two variables rather than just one.

\sssect{Projective model} If $(\ph,\ph_\infty)$ represents an element
of the line model, we put
\be\label{line-proj}
 \ph^\prj(t) \= \begin{cases}\;(1+t^2)^s\,\ph(t)
   &\quad\text{if $t\in \proj\RR\sm\{\infty\}\,=\,\RR\,$,}\\
  \;(1+t^{-2})^s\,\ph_\infty(-1/t)
   &\quad\text{if
   $t\in \proj\RR\sm\{0\}\,=\,\RR^*\cup\{\infty\}\,$.}\end{cases} \ee
The functions $\ph^\prj$ form the projective model of $\V{}s$,
consisting of functions $f$ on the real projective line $\proj\RR$
with the action
\begin{equation}\label{projac-p} f\bigl|_{2s}^\prj\,\matc
abcd\bigl(t\bigr)\= \biggl(\frac{t^2+1}{(at+b)^2+(ct+d)^2}
\biggr)^s \,f\biggr(\frac{at+b}{ct+d} \biggl). \end{equation}
Note that the factor
$ \left(\frac{t^2+1}{(at+b)^2+(ct+d)^2}\right)^s$ is real-analytic
on the whole of $\proj\RR$, since the factor in parentheses is
analytic and strictly positive on $\proj\RR$. This model has the
advantage that all points of $\proj\RR$ get equal treatment, but the
disadvantage that the formula for the action is complicated and
unnatural.


\sssect{Circle model} The transformation $\x=\frac{t-i}{t+i}$ in
$\proj\CC\,$, with inverse $t=i\frac{1+\x}{1-\x}\,$, maps $\proj\RR$
isomorphically to the unit circle
$\uc=\bigl\{\x\in\CC\;:\;\,|\x|=1\bigr\}$ in $\CC$, and leads to the
circle model of $\V{}s\,$, related to the three previous models by
\be \label{circle-line-proj}
\ph^\ci(e^{-2i\th})\=
\ph^\prj\bigl(\cot\th\bigr)\=\Ph(\cos\th,\sin\th)
    \=|\sin\th|^{-2s}\,\ph(\cot\th)\,. \ee
The action of $g=\matc abcd \in \PSL_2(\RR)$ is described by
$\tilde g = \matc1{-i}1i\,g\,\matc1{-i}1i^{-1}=\matc AB{\bar B}{\bar A}$
in $\mathrm{PSU}(1,1)\subset\PSL_2(\CC)$, with
$A=\frac12(a+ib-ic+d),\;B=\frac12(a-ib-ic-d)\,$:
\be\label{trfcimo-p} f\bigl|_{2s}^\ci\,g\,\bigl(\x\bigl)
\=|A\x+B|^{-2s} f\left(\frac{A\x+B}{\vphantom{A^A}\bar B\x+\bar
  A}\right)\qquad(|\x|=1)\,.\ee
Since $|A|^2-|B|^2=1$, the factor $|A\x+B|$ is non-zero on the unit
circle.

Note that in both the projective and circle models, the elements in
$\V p s$ are simply the elements of $C^p(\proj\RR)$ or $C^p(\uc)$, so
that as vector spaces these models are independent of $s$.

\sssect{Induced representation model} The principal series is
frequently defined as the induced representation from the Borel group
$NA$ to $G$ of the character $n(x)a(y) \mapsto y^{-s}$, in the
notation in \eqref{nad}. (See for instance Chap.~VII in \cite{Kn}.)
This is the space of functions $F$ on $G$ transforming on the right
according to this character of $AN$, with $G$ acting by left
translation. Identifying $G/N$ with $\RR^2\sm\{(0,0)\}$ leads to the
plane model, via $F\matc abcd=\Phi(a,c)$. On the other hand, the
functions in the induced representation model are determined by their
values on $K$, leading to the relation
$\ph^\ci(e^{2i\th}) = F( k(\th))$ with the circle model, with
$k(\th)$ as in~\eqref{kd}.

We should warn the reader that in defining the induced representation
one often considers functions whose restrictions to $K$ are square
integrable, obtaining a Hilbert space isomorphic to $L^2(K)$. The
action of $G$ in this space is a bounded representation, unitary if
$\re s=\frac12$. Since not all square integrable functions are
continuous, this Hilbert space is larger than $\V0 s$. For
$p\in \NN$, the space of $p$ times differentiable vectors in this
Hilbert space is larger than our $\V p s   $.
(It is between $\V{p-1}s$ and $\V p s$.)
However, $\V\infty s$ and $\V\om s$ coincide with the spaces of
\alsochanged{hyphen}infinitely-often differentiable, respectively
analytic, vectors in this Hilbert space.

\sssect{Sequence model} We define elements $\eph_{s,n} \in \V\om s$,
$n\in \ZZ$, represented in our five models as follows:
\beqabclbl{phisn-def}
\begin{align}\label{phisndef-p}
  \eph_{s,n}(t)&\=(t^2+1)^{-s}\biggl( \frac{t-i}{t+i}\biggr)^n\,,
  \displaybreak[0]\\
  \eph^{\RR^2}_{s,n}(x,y) &\= (x^2+y^2)^{-s}\biggl(
  \frac{x-iy}{x+iy}\biggr)^n\,, \displaybreak[0]\\
  \eph^\prj_{s,n}(t) &\= \biggl( \frac{t-i}{t+i}\biggr)^n\,,
  \displaybreak[0]\\
  \label{phisndef2-p} \eph^\ci_{s,n}(\x) &\=\x^n\,, \displaybreak[0]\\
\eph_{s,n}^{\mathrm{ind\,repr}}\biggl( \matc abcd \biggr) &\=
\bigl(a^2+c^2\bigr)^{-s} \biggl( \frac{a-ic}{a+ic} \biggr)^n\,.
\end{align}\eeqabc
Fourier expansion gives a convergent representation
$\ph^\ci(\xi) = \sum_n c_n \eph_{s,n}(\xi)$ for each element of
$\V0 s$. This gives the sequence model, consisting of the sequences
of coefficients $\bold c=\left(c_n\right)_{n\in \ZZ}\,$. The action
of $G$ is described by \alsochanged{YJ}$\bold c\mapsto \bold c'$ with
$\bold c'_m = \sum_n A_{m,n}(g)c_n$, where the matrix coefficients
$A_{m,n}(g)$ are given (by the binomial theorem) in terms of
$\tilde g=\matc AB{\bar B}{\bar A}$ as
\begin{equation} A_{m,n}(g) \= \frac{(A/B)^m\,(A/\bar B)^n}{|A|^{2s}}
\sum_{l\,\ge\,\max(m,n)}\binom{n-s}{l-m}\;\binom{-n-s}{l-n}\;
\biggl|\frac BA\biggr|^{2l}\,, \end{equation}
which can be written in closed form in terms of hypergeometric
functions as
\be\label{HGF}
\begin{aligned}
 A_{m,n}&(g) \\
 &\=\begin{cases} \,\frac{A^{n+m}\bar B^{m-n}}{|A|^{2s+2m} }
 \,\binom{-s-n}{m-n}\;F\bigl(s-n,s+m;m-n+1;\bigl|\frac
 BA\bigr|^2\,\bigr) &\text{if }m\geq n\,,\\
 \,\frac{A^{n+m}B^{n-m}}{|A|^{2s+2n}}\,\binom{-s+n}{n-m}\;
 F\bigl(s+n,s-m;n-m+1;\bigl|\frac BA\bigr|^2\,\bigr) &\text{if }n\geq
 m\,.
\end{cases}
 \end{aligned}\ee
The description of the smooth and analytic vectors is easy in the
sequence model:
\be\label{coi-p}
\begin{aligned}
\V\om s &\= \left\{ (c_n)\;:\; c_n = \oh\left( e^{-a|n|}\right)\text{
for some }a>0\right\},\\
\V\infty s &\= \left\{ (c_n)\;:\; c_n = \oh\left((1+|n|)^{-a}\right)
 \text{ for all }a\in \RR\right\}.
\end{aligned}\ee
The precise description of $\V p s$ for finite $p\in \NN$ is less
obvious in this model, but at least we have
$(c_n) \in \V p s \Rightarrow c_n=\o(|n|^{-p})$ as
$|n|\rightarrow\infty$, and, conversely, $c_n=\oh(|n|^{-\rho})$ with
$\rho>p+1$ implies $(c_n) \in \V p s$.

\sssect{Duality} There is a duality between $\V0s$ and $\V0{1-s}\,$,
given in the six models by the formulas
\beqabclbl{duality}
\begin{align}
\langle \ph,\ps\rangle&\=\frac1\pi\,\int_\RR \ph(t)\,\ps(t)\,dt\,,
\label{duA} \displaybreak[0]
\\
\langle\Phi,\Psi\rangle&\=\frac1{2\pi}\,\int_0^{2\pi}
\Phi(\cos\th,\sin\th)
\,\Psi(\cos\th,\sin\th)\,d\th\,,\label{duB} \displaybreak[0]\\
\langle\ph^\prj,\ps^\prj\rangle&\=\frac1\pi\int_{\proj\RR}
\ph^\prj(t)\,\ps^\prj(t)\,\frac{dt}{1+t^2}\,, \label{duD}
\displaybreak[0]\\
\langle\ph^\ci,\ps^\ci\rangle&\=\frac1{2\pi i} \int_{\uc}
\ph^\ci(\x)\,\ps^\ci(\x)\,\frac{d\x}\x\,, \label{duC}
\displaybreak[0]\\
\langle F, F_1 \rangle &\= \int_0^\pi F\bigl(k(\th) \bigr) \,
F_1\bigl( k(\th) \bigr)\, \frac{d\th}\pi\,, \displaybreak[0]\\
\bigl\langle \bold c,\bold d \bigr\rangle &\= \sum_n c_n d_{-n} \,.
\end{align}\eeqabc
This bilinear form on $\V0s\times \V0{1-s}\,$ is $G$-invariant:
\be\label{equiv}\langle\ph|_{2s}\,g,\,\ps|_{2-2s}\,g\rangle
  \=\langle\ph,\ps\rangle\qquad(g\in G)\,.\ee
Furthermore we have for $n,m\in \ZZ$:
\be\label{de}
\langle \eph_{1-s,n},\eph_{s,m} \rangle\= \dt_{n,-m}\,. \ee

\sssect{Topology}\change{Reordered and explanation added.} The natural
topology of $\V p s$ with $p\in \NN \cup\{\infty\}$ is given by
seminorms which we define with use of the action
$\ph\mapsto \ph|\WW = \frac d{dt}\ph|e^{t\WW}\bigr|_{t=0}$ where
$\WW=\rmatr0{-1}10$ is in the Lie algebra. The differential operator
$\bold W$ is given by $2i\x \, \p_\x$ in the circle model, by
$(1+\nobreak t^2)\,\p_t$ in the projective model, and by
$(1+\nobreak x^2)\,\p_x+2sx$ in the line model. For $p\in \NN$ the
space $\V p s$ is a Banach space with norm equal to the sum over
$j=0,\ldots,p$ of the seminorms
\be\label{seminorm-p} \|\ph\|_j \= \sup_{x\in \p\H} \; \bigl|
\ph|\bold W^j
(x)
\bigr|\,.\ee
The collection of all seminorms $\|\dv\|_j$, $j\in \NN$, gives the
natural topology of $\V\infty s = \bigcap_{p\in \NN} \V p s$. In
\S\ref{secthf} we shall discuss the natural topology on $\V\om s$.

Although we have strict inclusions
$\V\infty s \subset \cdots \subset \V1s \subset
\V0 s$, all these representation spaces of $G$ are irreducible as
topological $G$-representations, due to our standing assumptions
$0<\re s<1$, which implies $s\not\in \ZZ$.

\sssect{Sheaf aspects}In the line model, the projective model and the
circle model, we can extend the definition of the $G$-equivariant
spaces $\V p s$ for $p=0,1,\ldots,\infty,\om$ of functions on $\p\H$
to $G$-equivariant sheaves on~$\p\H$. For instance, in the circle
model we can define $\V\om s(I)$ for any open subset $I\subset \uc$
as the space of real-analytic functions on~$I$. The action of $G$
induces linear maps $ f \mapsto f\,|\,g$, from $\V\om s(I)$ to
$\V\om s(g^{-1}I)$, so that $I\mapsto \V\om s(I)$ is a
$G$-equivariant sheaf on the $G$-space $\uc$ whose space of global
sections is the representation $\V\om s$ of~$G$. For the line model
and the projective model we proceed similarly.
\subsection{Hyperfunctions}\label{secthf}
So far we have considered $\V{}s$ as a space of functions. We now want
to include generalized functions: distributions and hyperfunctions.
We shall be most interested in hyperfunctions on $\p\H$, in the
projective model and the circle model.

\sssect{$\V\om s$ and holomorphic functions} Before we discuss
hyperfunctions, let us first consider $\V\om s$. In the {\it circle
model}, it is the space $C^\om(\uc)$ of real-analytic functions on
$\uc$, with the action \eqref{trfcimo-p}. Since the restriction of a
holomorphic function on a neighborhood of $\uc$ in $\CC$ to $\uc$ is
real-analytic, and since every real-analytic function on $\uc$ is
such a restriction, $C^\om(\uc)$ can be identified with the space
$\limind {} \hol(U)$, where $U$ in the inductive limit runs over all
open neighborhoods of $\uc$ and $\hol(U)$ denotes the space of
holomorphic functions on~$U$.

\sssect{Hyperfunctions} We can also consider the space
$\HHH(\uc)\=\limind {}\hol(U\sm\uc)$
    %\be \HHH\bigl(\uc\bigr)\=\limind U \hol\bigl(U\sm \uc\bigr)\ee
(with $U$ running over the same sets as before) of germs of
holomorphic functions in deleted neighborhoods of $\uc$ in $\CC$. The
space $\hpf(\uc)$ of \emph{hyperfunctions} on $\uc$ is the quotient
in the exact sequence
  \be\label{hdef}0\lra C^\om(\uc) \lra\HHH(\uc)\lra\hpf(\uc)\lra
  0\,;\ee
see e.g.~\S1.1 of \cite{Skr-p}.
\change{Explanation added.}So $C^{-\om}(\uc)=
\limind U \hol(U\setminus \uc) /\hol(U)$ where $U$ is as above, and
where restriction gives an injective map
$\hol(U) \rightarrow \hol(U\sm \uc)$. Actually, the quotient
$\hol(U\!\sm\!\uc)/\hol(U)$ does not depend on the choice of $U$, so
it gives a model for $\hpf(\uc)$ for any choice of $U$. Intuitively,
a hyperfunction is the jump across $\uc$ of a holomorphic function on
$U\sm \uc$.

\sssect{Embedding} The image of $C^\om(\uc)$ in $C^{-\om}(\uc)$ in
\eqref{hdef} is of course zero. There is an embedding
$C^\om(\uc) \lra C^{-\om}(\uc)$ induced by
\be\label{hfemb}
\bigl( \ph \in \hol(U) \bigr) \mapsto\bigl( \ph_1\in \HHH_s(\uc)
\bigr)\,,\quad \ph_1(w) \= \begin{cases} \ph(w)&\text{ if }w\in U\,,\;
|w|<1\,,\\
0&\text{ if }w\in U\,,\; |w|>1\,.\end{cases}
\ee

\sssect{Pairing} We next define a pairing between hyperfunctions and
analytic functions on $\uc$. We begin with a pairing on
$\HHH(\uc)\times\HHH(\uc)$. Let $\ph,\,\ps\in\HHH(\uc)$ be
represented by $f,\,h\in\hol(U\!\sm\!\uc)$ for some $U$. Let $C_+$
and $C_-$ be closed curves in $U\sm\uc$ which are small deformations
of $\uc$ to the inside and outside, respectively, traversed in the
positive direction, e.g. $C_\pm=\bigl\{|w|=e^{\mp\e}\bigr\}$ with
$\e$ sufficiently small. Then the integral
\be\label{dual} \langle\ph,\,\ps\rangle \= \frac1{2\pi i}\; \intCC
  f(w)\,h(w)\,\frac{dw}w\ee
is independent of the choice of the contours $C_\pm$ and of the
neighborhood $U$. Moreover, if $f$ and $h$ are both in $\hol(U)$,
then Cauchy's theorem gives $\langle \ph,\ps\rangle=0$. Hence if
$\ps\in C^\om(\uc)$, then the right-hand side of \eqref{dual} depends
only on the image (also denoted $\ph$) of $\ph$ in $\hpf(\uc)$ and we
get an induced pairing $\hpf(\uc)\times C^\om(\uc)\to\CC$, which we
also denote by $\langle\,\cdot\,,\,\cdot\,\rangle$.
\alsochanged{Sentence inserted}Similarly,
$\langle\cdot\,,\,\cdot\rangle$ gives a pairing
$C^\om(\uc)\times C^{-\om}(\uc)\rightarrow\CC$. Finally, if $\ph$
belongs to the space $C^\om(\uc)$, embedded into $\hpf(\uc)$ as
explained in the preceding paragraph, then it is easily seen that
$\langle\ph,\ps\rangle$ is the same as the value of the pairing
$C^\om(\uc)\times C^\om(\uc)\to\CC$ already defined in \eqref{duC}.


\sssect{Group action} We now define the action of $G$. We had
identified $\V\om s$ in the circle model with $C^\om(\uc)$ together
with the action \eqref{trfcimo-p} of
$G=\PSL_2(\RR)\cong\mathrm{PSU}(1,1)$. For
$\tilde g=\matc AB{\bar B}{\bar A}$ and $\x\in\uc$ we have
$|A\x+B|^2=(A+B\x^{-1})(\bar A+\bar B\x)$, which is holomorphic and
takes values near the positive real axis for $\xi$ close to $\uc$
(because $|A|>|B|$). So if we rewrite the automorphy factor in
\eqref{trfcimo-p} as $\bigl[(\bar A+\bar B\x)(A+B/\x)\bigr]^{-s}$,
then we see that it extends to a \change{one-valued changed to
single-valued everywhere.}single-valued and holomorphic function on a
neighborhood of $\uc$ (in fact, outside a path from~$0$ to $-B/A$ and
a path from~$\infty$ to $-\bar A/\bar B$). In other words, in the
description of $\V\om s$ as $\limind{}\hol(U)$, the $G$-action
becomes
  \be \ph|_{2s}g\,(w)\=\bigl[(\bar A+\bar Bw)(A+B/w)\bigr]^{-s}
  \,\ph(\tilde g w)\;.\label{trfcihmo} \ee
This description makes sense on $\hol(U\sm \uc)$, and hence also on
$\HHH(\uc)$ and $\hpf(\uc)$. We define $\V{-\om}s$ as $\hpf(\uc)$
together with this $G$-action. It is then easy to check that the
embedding $\V\om s\subset\V{-\om}s$ induced by the embedding
$C^\infty(\uc)\subset\hpf(\uc)$ described above is $G$-equivariant,
and also that the pairing \eqref{dual} satisfies \eqref{equiv} and
hence defines an equivariant pairing
$\V{-\om}s\times\V\om{1-s}\to\CC$ extending the pairing \eqref{duC}
on $\V\om s\times\V\om{1-s}$.

Note also that, if we denote by $\HHH_s$ the space $\HHH(\uc)$
equipped with the action \eqref{trfcihmo}, then \eqref{hdef} becomes
a short exact sequence
  \be\label{hdefG-p}0\lra \V\om s
  \lra\HHH_s\overset\pi{\lra}\V{-\om}s\lra 0\ee
of $G$-modules and \eqref{dual} defines an equivariant pairing
$\HHH_s\times\HHH_{1-s}\to\CC$.



The equivariant duality identifies $\V{-\om}s$ with a space of linear
forms on $\V\om{1-s}$, namely (in the circle model) the space of all
linear forms that are continuous for the inductive limit topology on
$C^{-\om}(\uc)$ induced by the topologies on the spaces $\hol(U)$
given by supremum norms on annuli $1-\e<|w|<1+\e$. Similarly, the
space $\V{-\infty}s$ of distributional vectors in $\V{}s$ can be
defined in the circle model as the space of linear forms on $\V p s$
that are continuous for the topology with supremum norms of all
derivatives as its set of seminorms. We thus have an increasing
sequence of spaces:
\be\label{4spaces}
\begin{aligned}\V\om s \;&(\text{analytic functions})\; \subset
\V\infty s \;(\text{smooth functions})\; \subset \cdots \\
   &\qquad\quad\subset \V{-\infty}s\;(\text{distributions})\; \subset
   \V{-\om}s\;(\text{hyperfunctions})\,,
\end{aligned}
\ee
where all of the inclusions commute with the action of~$G$.

\sssect{Hyperfunctions in other models} The descriptions of the spaces
$\V{-\om}s$ and $\V{-\infty}s$ in the {\em projective model} are
similar. The space of hyperfunctions $\hpf(\proj\RR)$ is defined
similarly to \eqref{hdef}, where we now let $U$ run through
neighborhoods of $\proj\RR$ in $\proj\CC$. The formula
\eqref{projac-p} describing the action of $G$ on functions on
$\proj\RR$ makes sense on a neighborhood of $\proj\RR$ in $\proj\CC$
and can be rewritten
\begin{equation}\label{grotesque-p} f|^\prj_{2s}\, \matc abcd \,(\tau)
\= \bigl(a^2+c^2\bigr)^{-s} \allowbreak \left(
 \frac{\tau-i}{\tau-g^{-1}(i)} \right)^s\left(
\frac{\tau+i}{\tau-g^{-1}(-i)}\right)^s
f\left(\frac{a\tau+b}{c\tau+d}\right) . \end{equation}
where the automorphy factor now makes sense and is holomorphic and
single-valued outside a path from $i$ to $g^{-1}(i)$ and a path from
$-i$ to $g^{-1}(-i)$. The duality in this model is given by
\begin{equation} \label{projpairing} \left\langle \ph,\,\psi
\right\rangle\= \frac1\pi\,\intCC\ph(\tau)\,\psi(\tau)\,
\frac{d\tau}{1+\tau^2}
\end{equation}
where the contour $C_+$ runs in the upper half-plane $\uhp$, slightly
above the real axis in the positive direction, and returning along a
wide half circle in the positive direction and the contour $C_-$ is
defined similarly, but in the lower half-plane $\lhp$, going
clockwise. Everything else goes through exactly as before.

The kernel function
\be k(\z,\tau) \= \frac{(\z+i)(\tau-i)}{2i(\tau-\z)} \ee
can be used to obtain a representative in $\HHH_s$ (in the projective
model)
for any $\al \in \V{-\om}s$: if we think of $\al$ as a linear form on
$\V\om{1-s}$, then
\be\label{proj-repr} g(\z) \= \bigl\langle k(\z,\dv) , \al
\bigr\rangle\ee
is a holomorphic function on $\proj\CC\sm \proj\RR$ such that
$\pi(g) = \al$. Cauchy's theorem implies that $g$ and any
representative $\ps\in \HHH_s$ of $\al$ differ by a holomorphic
function on a neighborhood of $\proj\RR$. The particular
representative $g$ has the nice properties of being holomorphic on
$\uhp\cup\lhp$, and being normalized by $g(-i)=0$.

If one wants to handle hyperfunctions in the line model, one has to
use both hyperfunctions $\ph$ and $\ph_\infty$ on $\RR$, glued by
$\ph(\tau) =
(\tau^2)^{-s} \ph_\infty(-1/\tau)$ on neighborhoods of $(0,\infty)$
and $(-\infty,0)$. For instance, for $\re s<\frac12$ the linear form
$\ph \mapsto \frac1\pi  \int_{-\infty}^\infty \ph(t)\, dt$ on
$\V0{1-s}$ defines a distribution $\bold 1_s \in \V{-\infty}s$. In
\S\ref{sect-Pta} we use \eqref{proj-repr} to describe
$\bold 1_s\in \V{-\om}s$ in the line model. The plane model seems not
to be convenient for working with hyperfunctions.

Finally, in the {\em sequence model} there is the advantage that one
can describe all four of the spaces in \eqref{4spaces} very easily,
since the descriptions in \eqref{coi-p} applied to $\V\om{1-s}$ and
$\V\infty{1-s}$ lead immediately to the descriptions
\be\begin{aligned}\label{chf}
 \V{-\om}s &\= \left\{ (c_n) \;:\; c_n=\oh\bigl(e^{a|n|}\bigr)
   \text{ for all }a>0\right\}\,,\\
\V{-\infty}s&\=\left\{(c_n)\;:\; c_n = \oh \left(
(1+|n|)^a\right)\text{ for some }a\in \RR\right\}
 \end{aligned}\ee
of their dual spaces, where a sequence $\bold c$ corresponds to the
hyperfunction represented by the function which is
$\sum_{n\ge0}c_nw^n$ for $1-\e<|w|<1$ and $-\sum_{n<0}c_nw^n$ for
$1<|w|<1+\e$; the action of $G$ still makes sense here because the
matrix coefficients as given in \eqref{HGF} decay exponentially (like
$\,(|B|/|A|)^{|n|}\,$)
as $|n|\to\infty$ for any $g\in G$. Thus in the sequence model, the
four spaces in \eqref{4spaces} correspond to sequences $\{c_n\}$ of
complex numbers having exponential decay, superpolynomial decay,
polynomial growth, or subexponential growth, respectively.
\alsochanged{reference inserted.} (See \eqref{coi-p}
and~\eqref{chf}.)



\subsection{The intertwining map $\V{-\om}s \to \V{-\om}{1-s}$}
\label{sectIM} The representations $\V{-\om}s$ and $\V{-\om}{1-s}$,
with the same eigenvalue $s(1-s)$ for the Casimir operator, are not
only dual to one another, but are also isomorphic
(for $s\not\in\ZZ$). Suppose first that $F \in C^p(G)$ is in the
induced representation model of $\V p s$ with $\re s>\frac12$ and
$p=0,1,\ldots,\infty$. With $n(x) = \matc 1x01$ as in~\eqref{nad} and
$w=\matr0{-1}10$ we define
\be \label{intertw1}I_s F (g) \= \frac1{\bb(s-\frac12)}
\int_{-\infty}^\infty F(gn(x)w)\, dx\,,
\qquad \bb\bigl(s\bigr) \= \Beta\bigl( s,\frac12) \= \frac{\Gf\bigl(
s\bigr)
\Gf\bigl( \frac12\bigr)}{\Gf(s+\frac12)}\,, \ee
where the gamma factor $\bb\bigl(s-\frac12\bigr)$ is a normalization
the reason of which will become clear later. The shift over $\frac12$
is chosen since we will meet the same gamma factor unshifted in
\S\ref{sect-gb}. {}From
$n(x) w \in  k(-\arccot x) a\bigl( \sqrt{1+x^2}\bigr) \,N$ we find
\[ I_s F (g) \= \bb\bigl(s-\frac12\bigr)^{-1} \int_{-\infty}^\infty
F\bigl( g k(-\arccot x)\bigr)
(1+x^2)^{-s}\,dx\,,\]
which shows that the integral converges absolutely for
$\re s>\frac12$. By differentiating under the integral in
\eqref{intertw1} we see that $I_s F \in C^p(G)$. {}From
$a(y)n(x)w=n(yx)w a(y)^{-1}$ it follows that $I_s F(g a(y)n(x') )
= y^{s-1} F(g)$. The action of $G$ in the induced representation model
is by left translation, hence $I_s$ is an intertwining operator
$\V p s\to \V p{1-s}$:\alsochanged{$|_s$ and $|_{1-s}$ }
\be\label{intertwG}
 (I_s F)\,|_{1-s}\,g_1 \= I_s(F\,|_s\,g_1)\qquad\text{for }g_1\in G\,.
 \ee


\beqabc To describe $I_s$ in the plane model, we choose for a given
$(\x,\eta) \in \RR^2\sm\{0\}$ the element
$g_{\x,\eta}=\matc \x {-\eta/(\x^2+\eta^2)} \eta {
\x/(\x^2+\eta^2)} \in G$ to obtain
\begin{align}
I_s \Ph(\x,\eta) &\= \bb\bigl(s-\frac12\bigr)^{-1}
\int_{-\infty}^\infty \Ph\biggl( g_{\x,\eta} \matr x{-1}10\biggr)\,
dx\nonumber
\\
&\= \bb\bigl(s-\frac12\bigr)^{-1} \int_{-\infty}^\infty \Ph\biggl( x\,
(\x,\eta) + \frac1{\x^2+\eta^2}(-\eta,\x) \biggr)\, dx\,.
\end{align}
By relatively straightforward computations, we find that the formulas
for $I_s$ in the other models (still for $\re s>\frac12$) are given
by:
\begin{align}
\label{Iln}
I_s\ph(t) &\= \bb\bigl(s-\frac12\bigr)^{-1} \int_{-\infty}^{\infty}
|t-x|^{2s-2}\, \ph(x)\, dx\,, \displaybreak[0]\\
I_s \ph^\prj(t) &\= \bb\bigl(s-\frac12\bigr)^{-1} \int_{\proj\RR}
\biggl( \frac{(t-x)^2}{(1+t^2)(1+x^2)} \biggr)^{s-1}\, \ph^\prj(x)\,
\frac{dx}{x^2+1}\,, \displaybreak[0]\\
\label{Ic}
I_s \ph^\ci(\x) &\= \frac{2^{1-2s}}i\, \bb\bigl(s-\frac12\bigr)^{-1}
\int_{\uc}
(1-\x/\eta)^{s-1}(1-\eta/\x)^{s-1} \ph^\ci(\eta)\,
\frac{d\eta}{\eta}\,, \displaybreak[0]\\
\label{Iseq}
(I_s \bold c)_n &\= \frac{\Gf(s)}{\Gf(1-s)} \,
\frac{\Gf(1-s+n)}{\Gf(s+n)}\, c_n \= \frac{(1-s)_{|n|}}{(s)_{|n|}} \,
c_n\,,
\end{align}
\eeqabc
\change{Description Pochhammer symbol inserted.}with in the last line
the Pochhammer symbol given by
$(a)_k = \prod_{j=0}^{k-1} (a+\nobreak j)$ for $k\geq 1$ and
$(a)_0=1$. The factor $(1-\nobreak s)_{|n|}/(s)_{|n|}$ is holomorphic
on $0<\re s<1$. Hence $I_s \eph_{s,n}$ is well defined for these
values of the spectral parameter. The polynomial growth of the factor
shows that $I_s$ extends to a map $I_s :\V p s\to \V p{1-s}$ for
$0<\re s<1$ for $p=\om,\infty,-\infty,-\om$, but for finite $p$ we
have only $I_s\V p s \subset \V{p-1}{1-s}$ if $0<\re s<1$. See the
characterizations \eqref{coi-p} and~\eqref{chf}. The intertwining
property \eqref{intertwG} extends holomorphically. The choice of the
normalization factor in \eqref{intertw1} implies that
$I_{1-s} \circ I_s= \mathrm{Id}$, as is more easily seen from formula
\eqref{Iseq}. {}From this formula we also see that
$\langle I_{1-s}\ph,I_s\al\rangle= \langle\ph,\al\rangle$ for
$\ph \in \V\om{1-s}$, $\al\in \V{-\om}s$, and that
$I_{1/2}=\mathrm{Id}$.

For $\ph\in \V p s$, $p\geq 1$, we have in the line model $\ph'(x) =
\oh(|x|^{-2s-1})$ as $|x|\rightarrow\infty$. For $\re s>\frac12$,
integration by parts gives
\be\label{I-ln-alt} I_s\ph(t) \= \frac{-\Gf(s)}{2\sqrt\pi\,
\Gf(s+\frac12)} \int_{-\infty}^\infty \sign(t-x)\, |t-x|^{2s-1}\,
\ph'(x)\, dx\,,\ee
and this now defines $I_s\ph$ for $\re s>0$, and shows that $I_s
\V1s \subset \V0{1-s}$.

We can describe the operator $I_s : \V{-\om}s \to \V{-\om}{1-s}$ on
the representatives of hyperfunctions in $\HHH_s$ by sending a
Laurent series $\sum_{n\in \ZZ} b_n w^n$ on an annulus $\al<|w|<\bt$
in $\CC^\ast$ to
$\sum_{n\in \ZZ} \frac{(1-s)_{|n|}}{(s)_{|n|}} b_n w^n$ converging on
the same annulus. One can check that this gives an intertwining
operator $I_s :\HHH_s \to \HHH_{1-s}$. (Since $G$ is connected, it
suffices to check this for generators of the Lie algebra, for which
the action on the $\eph_{s,n}$ is relatively simple.
See~\S\ref{sect-Lie}.)

\section{Laplace eigenfunctions and the Poisson
transformation}\label{secthyp}
The principal series representations can also be realized as the space
of eigenfunctions of the Laplace operator $\Dt$ in the hyperbolic
plane $\H$. This model has several advantages: the action of $G$
involves no automorphy factor at all, the model does not give a
preferential treatment to any point, all vectors correspond to actual
functions, with no need to work with distributions or hyperfunctions,
and the values $s$ and $1-s$ of the spectral parameter give the same
space. The isomorphism from the models on the boundary used so far to
the hyperbolic plane model is given by an simple integral transform
(Poisson map). Before discussing this transformation in
Section~\ref{sect-Poiss}, we consider in Section~\ref{LapEF}
eigenfunctions of the Laplace operator on hyperbolic space, and
discuss in Section~\ref{sect-Gf} the Green's form already used
in~\cite{LZ-p}.

Finally, in Section~\ref{sect-secorder} we consider second order
eigenfunctions, i.e., functions on~$\H$ that are annihilated by
$\bigl(\Dt-\nobreak s(1-\nobreak s) \bigr)^2$.

\subsection{The space $\E_s$ and some of its elements} \label{LapEF}
\change{paragraph inserted}We use $\H$ as general notation for the
hyperbolic $2$-space. For computations it is convenient to work in a
realization of~$\H$. In this paper we use the realization as the
complex upper half-plane and a realization as the complex unit disk.

The upper half-plane model of $\H$ is $\uhp=\{z=x+iy\,:\,y>0\}$, with
boundary $\proj \RR$. Lengths of curves in $\uhp$ are determined by
integration of $\,y^{-1}\sqrt{(dx)^2+(dy)^2}$. To this metric are
associated the Laplace operator
$\Dt=-y^2\bigl(\p_x^2+\p_y^2\bigr)=(z-\bar z)^2\p_z\p_{\bar z}$ and
the volume element $d\mu=\frac{dx\,dy}{y^2}$. The hyperbolic distance
$\dist(z,z')$ between two points $z,\,z'\in\H$ is given in the upper
half plane model by
\be \cosh
 \disth(z,z')\=\rho^\uhp(z,z')\=1\,+\,\frac{|z-z'|^2}{2yy'}\qquad
(z,z'\in\uhp)\;. \label{Hdist} \ee
The isometry group of $\uhp$ is the group $G=\PSL_2(\RR)$, acting as
usual by fractional linear transformations
$z\mapsto\frac{az+b}{cz+d}$. The subgroup leaving fixed $i$ is
$K=\PSO(2)$. So $G/K \cong \uhp$. The action of~$G$ leaves invariant
the metric and the volume element and commutes with~$\Dt$.

We use also the disk model $\disk=\left\{ w\in \CC\;:\; |w|<1\right\}$
of $\H$, with boundary $\ci^1$. It is related to the upper half-plane
model by $w=\frac{z-i}{z+i}$, $z=i\frac{1+w}{1-w}$. The corresponding
metric is $\frac{2\sqrt{(d\,\re w)^2+(d\,\im w)^2}}{1-|w|^2}$, and
 the Laplace operator $\Dt=-\bigl(1-|w|^2\bigr)^2\p_w\p_{\bar w}$. 
The
formula for hyperbolic distance becomes
 \be \cosh \distd(w,w')\=\rho^\disk(w,w')
 \=1\,+\,\frac{2|w-w'|^2}{\bigl(1-|w|^2\bigr)
    \bigl(1-|w'|^2\bigr)}\;. \label{Ddist} \ee
Here the group of isometries, still denoted $G$, is the group
$\mathrm{PSU}(1,1)$ of matrices $\matc AB{\bar B}{\bar A}$
($A,\,B\in\CC$, $|A|^2-|B|^2=1$), again acting via fractional linear
transformations.

By $\E_s$ we denote the space of solutions of $\Dt u = \ld_s\,u$ in
$\H$, where $\ld_s=s(1-s)$. Since $\Dt$ is an elliptic differential
operator with real analytic coefficients, all elements of $\E_s$ are
real-analytic functions. The group $G$ acts by $(u|g)(z)=u(gz)$. (We
will use $z$ to denote the coordinate in both $\uhp$ and $\disk$ when
we make statements applying to both models of $\H$.) Obviously,
$\E_s=\E_{1-s}$. If $U$ is an open subset of $\H$, we denote by
$\E_s(U)$ the space of solutions of $\Dt u=\ld_su$ on $U$.


There are a number of special elements of $\E_s$ which we will use in
the sequel. Each of these elements is invariant or transforms with
some character under the action of a one-parameter subgroup
$H\subset G$. The simplest are $z=x+iy \mapsto y^s$ and
$z\mapsto y^{1-s}$, which are invariant under
$N=\bigl\{ \matc 1x01\;:\; x\in \RR\bigr\}$, and transform according
to a character of $A= 
\bigl\{ \matc {\sqrt y}00{1/\!\!\sqrt y}\;:\; y>0\bigr\}$. More
generally, the functions in $\E_s$ transforming according to
non-trivial characters of $N$ are written in terms of Bessel
functions. These are important in describing Maass forms with respect
to a discrete subgroup of $G$ that contains $\matc 1101$. The
functions transforming according to a character of $A$ are described
in terms of hypergeometric functions. (The details, and properties of
all special functions used, are given in \S\ref{sect-spf}.)

If we choose the subgroup $H$ to be $K=\PSO(2)$, we are led to the
functions $P_{\!s,n}$ described in the disk model with polar
coordinates $w=re^{i\th}$ by\alsochanged{$:=$}
\be\label{LegP}
P_{\!s,n}(re^{i\th}) \;:=\;
P_{\!s-1}^{\,n}\biggl(\frac{1+r^2}{1-r^2}\biggr)\;e^{in\th}
\qquad(n\in\ZZ)\,, \ee
where $P_{\!s-1}^n$ denotes the Legendre function of the first kind.
Note the shift of the spectral parameter in $P_{\!s-1}^n$ and
$P_{\!s,n}$. If $n=0$ one usually writes $P_{\!s-1}$ instead of
$P^0_{\!s-1}$; but to avoid confusion we will not omit the $0$ in
$P_{\!s,0}$.


Every function in $\E_s$ can be described in terms of the $P_{\!s,n}$:
if we write the Fourier expansion of $u\in\E_s$ as
$u(re^{i\th})= \sum_{n\in \ZZ} A_n(r)e^{in\th}$, \alsochanged{YJ}then
$A_n(r)$ has the form
$a_n\,P^n_{\!s-1}\bigl( \frac{1+r^2}{1-r^2}\bigr)$ for some
$a_n\in \CC$, so we have an expansion
\be \label{u-exp} u(w) \= \sum_{n\in \ZZ} a_n\,
P_{\!\!s,n}(w)\,,\qquad a_n\in\CC\,. \ee


Sometimes it will be convenient to consider also subgroups of $G$
conjugate to~$K$. For a given $z'=x'+iy'\in \uhp$ we choose $g_{z'} 
= \matc {\sqrt {y'}}{x'/\!\!\sqrt {y'}}0{1/\!\!\sqrt {y'}} 
\in NA \subset G$ to obtain an automorphism of $\uhp$ sending $i$ to
$z'$. If we combine this with our standard identification of $\uhp$
and $\disk$, we get a new identification sending the chosen point
$z'$ to $0\in\disk$, and the function $P_{\!s,n}$ on $\disk$ becomes
the following function on $\uhp\times\uhp\,$:\alsochanged{$:=$}
\be \label{psn-def-p}
 p_{s,n}(z,z') \;:=\; P_{\!s,n}\biggl(
 \frac{z-z'}{z-\overline{z'}}\biggr)\,. \ee

This definition of $P_{\!s,n}$ depends in general on the choice of
$g_{z'} $ in the coset $g_{z'}K$. In the case $n=0$ the choice has no
influence, and we obtain the the very important \emph{point pair
invariant} $p_s(z,z')$, defined, in either the disk or the upper half
plane, by the formula\change{$\H$ explicitly indicated.}
\be\label{pppi} p_s(z,z') \;:=\; p_{s,0}(z,z') \=
P_{\!s-1}\bigl(\rho^\H(z,z')\bigr)
\qquad(z,z'\text{ in }\H)\,, \ee
with the argument $\rho(z,z')=\cosh d(z,z')$ of the Legendre function
$P_{\!s-1}=P^0_{\!s-1}$ being given algebraically in terms of the
coordinates of $z$ and $z'$ by formulas \eqref{Hdist} or
\eqref{Ddist}, respectively. This function is defined on the product
$\H\times\H$, is invariant with respect to the diagonal action of $G$
on this product, and satisfies the Laplace equation with respect to
each variable separately.\smallskip

The Legendre function $Q^n_{s-1}$ \alsochanged{YJ}in
equation~\eqref{Leg-d} in the appendix provides elements of
$\E_s(\disk\sm\{0\})$:
\be Q_{s,n}\bigl(re^{i\th}\bigr)\=
 Q_{s-1}^{\,n}\biggl(\frac{1+r^2}{1-r^2}\biggr)
   \,e^{in\th} \qquad(n\in\ZZ)\,.\label{LegQ} \ee
The corresponding point-pair invariant with
$Q^0_{s-1}=Q_{s-1}$\change{ $\H$ explicitly mentioned.}
\be q_s(z,z')\=Q_{s-1}\bigl(\rho^\H(z,z')\bigr) \qquad
(z,z'\text{ in }\H)\label{QQQ} \ee
is the well-known Green's function for $\Dt$ (integral kernel function
of $(\Dt-\nobreak\ld_s)^{-1}$), has a logarithmic singularity as
$z\to z'$ and grows like the $s$-th power of the Euclidean distance
(in the disk model)
from $z$ to the boundary as $z\to\p\H$ with $z'$ fixed. This latter
property will be crucial in \S\ref{sect-gb}, where we will study a
space $\W\om s$ of germs of eigenfunctions near $\p\H$ having
precisely this boundary behavior.

The eigenfunction $R(t;\dv) ^s $, given in the $\uhp$-model by
\be \label{Rtsz}
R(t;z)^s \= \frac{y^s}{|t-z|^{2s} } \qquad(t\in \RR\,,\; z=x+iy\in
\uhp)\,, \ee
is the image under the action of $\matr01{-1}t\in G$ on the
eigenfunction $z\mapsto y^s$. This function was already used
extensively in \cite{LZ-p} (\S2 and \S5 of Chapter~II). For fixed
$t\in \RR$, the functions $R(t;\dv)^s$ and $R(t;\dv)^{1-s}$ are both
in $\E_s$. For fixed $z\in \uhp$, we have $R(\dv;z)^s$ in the line
model of $\V\om s$. The basic invariance property
\be \label{Requiv-p}
|ct+d|^{-2s}R(gt;gz)^s \= R(t,z)^s\qquad\bigl( g=\matc abcd\in
G\bigr)\ee
may be viewed as the statement that $(t,z) \mapsto R(t;z)^s$ belongs
to $(\V\om s\otimes \E_s)^G$. The function $R(\dv;\dv)^{1-s}$ is the
kernel function of the Poisson transform in~\S\ref{sect-Poiss}.

We may allow $t$ to move off $\RR$, in such a way that $R(t,z)^s$
becomes holomorphic in this variable:
\be \label{Rext} R(\z;z)^s \= \Bigl( \frac y{(\z-z)(\z-\bar
z)}\Bigr)^s
   \qquad(\z\in \CC\,,\; z=x+iy\in \uhp)\,. \ee
However, this not only has singularities at $z=\z$ or $z=\bar\z$, but
is also many-valued. To make a well-defined function we have to
choose a path $C$ from $\z$ to $\bar\z$, in which case
\alsochanged{Old version: $R(\dv;z)^s$}$R(\z;\dv)^s$ becomes
single-valued on $U=\uhp\sm C$ and lies in $\E_s(U)$.
(Cf.~\cite{LZ-p}, Chap.~II, \S1.)
Sometimes it is convenient to write $R_\z^s$ instead of $R(\z;\dv)^s$.

Occasionally, we will choose other branches of the multivalued
function $R(\dv;\dv)^s$. We have
\be\label{R-diff}
\p_z R(\z;z)^s \= \frac{s}{z-\bar z} \, \frac{\z-\bar z}{\z-z}\,
R(\z;z)^s\,,\qquad \p_{\bar z} R(\z;z)^s \= - \frac s{z-\bar z}\,
\frac{\z-z}{\z-\bar z}\, R(\z;z)^s\,, \ee
provided we use the same branch on the left and the right.


\subsection{The Green's form and a Cauchy formula for
$\E_s$}\label{sect-Gf}

Next we recall the bracket operation from \cite{LZ-p}, which
associates to a pair of eigenfunctions of $\Dt$ with the same
eigenvalue\footnote{More generally, for any differentiable 
functions $u$ and $v$ on $\H$ we have
\[  d \{u,v\}\;=\; 2i\, d[u,v] \;=\;
\Bigl( (\Dt u)\, v - u \, (\Dt v) \Bigr)\,d\mu\,,\]
where $d\mu$ ($\,=\, y^{-2}\, dx\,dy$ in the upper half-plane model) is the
invariant measure
in~$\H$.} %\change{Footnote inserted}
a closed 1-form (Green's form). It comes in two versions,
differing by an exact form:
\be\label{Grf} [u,v]\=u_z\,v\,dz\,+\,u\,v_{\bar z}\,d\bar
z\,,\qquad\{u,v\}\=2i\,[u,v]\,-\,i\,d(uv)\,.\ee
Because \alsochanged{$u|g$ instead of
$u\circ g$}$[u| g,v| g]=[u,v]\circ g$ for any locally defined
holomorphic map~$g$ (cf.~\cite{LZ-p}, lemma in \S2 of Chapter~II),
these formulas make sense and define the same 1-form whether we use
the $\uhp$- or $\disk$-model of $\H$, and define $G$-equivariant maps
$\E_s\times\E_s\to\Omega^1(\H)$
(or $\E_s(U)\times\E_s(U)\to\Omega^1(U)$ for any open subset $U$ of
$\H$). The $\{u,v\}$-version of the bracket, which is antisymmetric,
is given in $(x,y)$-coordinates $z=x+iy\in\uhp$ by
\be \{u,\,v\}\=\left|\,\begin{matrix}
u&u_x&u_y\\v&v_x&v_y\\0&dx&dy\end{matrix}\,\right| \ee
and in $(r,\th)$-coordinates $w=re^{i\th}\in\disk$ by
\be \label{D1form} \{u,\,v\}
 \=\left|\,\begin{matrix}u&r\,u_r&u_\th\\v&r\,v_r&v_\th\\
 0&dr/r&d\th\end{matrix}\,\right|\,.\ee

We can apply the Green's form in particular to any two of the special
functions discussed above, and in some cases the resulting closed
form can be written as the total differential of an explicit
function. A trivial example is
$2i\,[y^s,y^{1-s}]=s\,dz-(1-s)\,d\bar z$,
$\{y^s,y^{1-s}\}=(2s-1)\,dx$. A less obvious example is
\be \label{goodlemma} [R_a^s,\,R_b^{1-s}](z)\=\frac1{b-a}\;
  d\,\biggl(\frac{(\bar z-a)(z-b)}{z-\bar
  z}\,R_a^s(z)\,R_b^{1-s}(z)\biggr)\,,\ee
where $a$ and $b$ are either distinct real numbers or distinct complex
numbers and $z\not\in \{a,b,\bar a,\bar b\}$. On both sides we take
the same branches of $R_a^s$ and $R_b^{1-s}$. This formula, which can
be verified by direct computation, can be used to prove the Poisson
inversion formula discussed below (cf.~Remark~1, \S\ref{sect-Pi}).
Some other examples are given in~\S\ref{sect-pot}.

We can also consider the brackets of any function $u\in\E_s$ with the
point-pair invariants $p_s(z,z')$ or $q_s(z,z')$. The latter is
especially useful since it gives us the following $\E_s$-analogue of
Cauchy's formula:
\begin{thm}\label{thm-psC-p} Let $C$ be a piecewise smooth simple
closed curve in~$\H$ and $u$ an element of $\E_s(U)$, where
$U\subset\H$ is some open set containing $C$ and its interior. Then
for $w\in\H\sm C$ we have
\be\label{psCc-p} \frac1{\pi i} \int_C\left[ u, q_s(\dv,w)\right] \=
\begin{cases}
\;u(w)&\text{ if $\,w\,$ is inside $C$},\\ \;0&\text{ if $\,w\,$ is
outside $C$,} \end{cases}\ee
where the curve $C$ is traversed is the positive direction. \end{thm}
\begin{proof} Since $[u,q_s(\dv,w)]$ is a closed form, the value of
the integral in \eqref{psCc-p} does not change if we deform the path
$C$, so long as we avoid the point $w$ where the form becomes
singular. The vanishing of the integral when $w$ is outside of $C$ is
therefore clear, since we can simply contract $C$ to a point. If $w$
is inside $C$, then we can deform $C$ to a small hyperbolic circle
around $w$. We can use the $G$-equivariance to put $w=0$, so that
this hyperbolic circle is also a Euclidean one, say $z=\e e^{i\th}$.
We can also replace $[u,q_s(\dv,0)]$ by $\{u,q_s(\dv,0)\}/2i$, since
their difference is exact. {}From \eqref{D1form} and the asymptotic
result \eqref{qas0} we find that the closed form
$-\frac i2 \{ u, q_s(\dv,0)\}$ equals
$\bigl( \frac i2 u(0) + \oh(\e\log\e) \bigr)\, d\th$ on the circle.
The result follows. \end{proof}

The method of the proof just given can also be used to check that for
a contour $C$ in $\disk$ encircling $0$ once in positive direction we
have for all $n\in \ZZ$
\be\label{pq-br} \int_C [ P_{\!s,n},\,Q_{s,m}] \= \pi i \,(-1)^n
\,\dt_{n,-m} \,.\ee
Combining this formula with the expansion~\eqref{u-exp}, we arrive at
the following generalization of the standard formula for the Taylor
expansion of holomorphic functions:
\begin{prop}For each $u\in \E_s$:
\be \label{p-exp}
u(w) \= \sum_{n\in \ZZ} \, \frac{(-1)^n}{\pi i} \,P_{\!s,n}(w) \,
\int_C [ u,Q_{s,-n}] \,. \ee
\end{prop}

If $u\in \E_s(A)$, where $A$ is some annulus of the form
$r_1< |w| < r_2$ in $\disk$, there is a more complicated expansion of
the form
\be\label{ann-exp} u(w) \= \sum_{n\in \ZZ} \bigl( a_n P_{\!s,n}(w) +
b_n Q_{s,n}(w)\bigr)\,.\ee

For fixed $w'\in \disk$ the function $w\mapsto q_s(w,w')$ has only one
singularity, at $w=w'$. So both on the disk $|w|<|w'|$ as on the
annulus $|w|>|w'|$ the function $q_s(\dv,w')$ has a polar Fourier
expansion, which can be given explicitly:
\begin{prop}For $w,w'\in\disk$ with $|w|\neq |w'|$:
\be q_s(w,w') \= \begin{cases} \sum_{n\in \ZZ} (-1)^n
\,P_{\!s,-n}(w')\,Q_{s,n}(w)& \text{ if }|w|>|w'|\,,\\
 \sum_{n\in \ZZ} (-1)^n \,P_{\!s,-n}(w)\,Q_{s,n}(w')& \text{ if
}|w|<|w'|\,.
\end{cases}
\ee
\end{prop}
\begin{proof}Apply \eqref{ann-exp} to $q_s(\dv,w')$ on the annulus
$A=\{w\in \disk\;:\; |w'|<|w|\}$. Since $q_s(\dv,w')$ represents an
element of $\W\om s$ the expansion becomes
\[ q_s(w,w') \= \sum_{n\in \ZZ} b_n(w')\, Q_{s,n}(w)\qquad
(|w|>|w'|)\,.\]
{}From $q_s(e^{i\th}w,e^{i\th}w')=q_s(w,w')$ it follows that
$b_n(e^{i\th}w')=e^{-in\th}b_n(w)$. For $w\in \disk\sm\{0\}$ we have
$q_s(w,\dv)\in \E_s(B)$ with $B=\{w'\in\disk\;:\; |w'|<|w|\}$. Then
the coefficients $b_n$ are also in $\E_s(B)$. Since $Q_{s,-n}$ has a
singularity at~$0\in \disk$, the coefficients have the form
$b_n(w')=c_n \, P_{\!s,-n}(w')$. Now we apply \eqref{psCc-p} and
\eqref{pq-br} to obtain with a path $C$ inside the region~$A$:
\begin{align*}
P_{\!s,m}(w')&\= \frac1{\pi i}\int_C\bigl[ P_{\!s,m},q_s(\dv,w')\bigr]
\= \frac1{\pi i} \sum_{n\in \ZZ} c_n P_{\!s,-n}(w') \,\int_C\bigl[
P_{\!s,m},Q_{s,n}\bigr]\\
& \= c_{-m}\, P_{\!s,m}(w')\, (-1)^m\,.
\end{align*}
Hence $c_m=(-1)^m$, and the proposition follows, with the symmetry
of~$q_s$.
\end{proof}



\subsection{The Poisson transformation}\label{sect-Poiss}
There is a well-known isomorphism $\poiss s$ from $\V{-\om}s$ to
$\E_s$. This enables us to view $\E_s$ as a model of the principal
series. We first describe $\poiss s $ abstractly and then more
explicitly in various models of $\V{-\om}s$. In Section~\ref{sect-Pi}
we will describe the inverse isomorphism from $\E_s$ to~$\V{-\om}s$.

For $\al\in \V{-\om}s$ and $g\in G$
\be \label{Potr}
(\poiss s \al)(g) \= \langle\al|_{2s}\,g , \,\eph_{1-s,0} \rangle \=
\langle \al, \,\eph_{1-s,0}|_{2-2s}\,g^{-1}\rangle \ee
describes a function on $G$ that is $K$-invariant on the right. Hence
it is a function on $G/K \cong\H$. The center of the enveloping
algebra is generated by the Casimir operator. It gives rise to a
differential operator on $G$ that gives, suitably normalized, the
Laplace operator $\Dt$ on the right-$K$-invariant functions. Since
the Casimir operator acts on \alsochanged{}$\V{-\om}{1-s}$ as
multiplication by $\ld_s = (1-s)s$, the function $\poiss s \al$
defines an element of~$\E_s$. We write in the upper half plane model
\be\label{Pd} \poiss s \al(z) = \poiss s \al
\bigl(n(x)a(y)\bigr)\,,\ee
with the notation in~\eqref{nad}. The definition in~\eqref{Potr}
implies that the Poisson transformation is $G$-equivariant:
\be\label{P-G}
\poiss s(\al|_{2s}g)(z) \= \poiss s\al(gz)\,. \ee
The fact that the intertwining operator
$I_s:\V{-\om}s\to\V{-\om}{1-s}$ preserves the duality implies that
the following diagram commutes:
\be \label{P-I-p}
\xymatrix@=.3cm{ \V{-\om}s \ar[dd]^{I_s} \ar[rrd] ^{\poiss s}
\\
&& \E_s = \E_{1-s}
\\
\V{-\om}{1-s} \ar[rru]^{\poiss{1-s}} } \ee


If $\al \in \V0 s$, we can describe $\poiss s \al$ by a simple
integral formula. In the line model this takes the
form\alsochanged{corrections and intermediate step}
\be\label{poiss-l}
\begin{aligned}\poiss s\al(z) &\= \bigl\langle \eph_{s,0}, \al|_{2s}
n(x)a(y) \bigr\rangle \= \frac1\pi \int_{-\infty}^\infty
\eph_{s,0}|_{2-2s}\,\bigl(n(x)a(y)\bigr)^{-1} (t)
\, \al(t)\, dt \\
&\= \frac1\pi \int_{-\infty}^\infty y^{-1+s} \bigl( (\frac{t-x}y)^2
+1\bigr)^{s-1}\, \al(t)\, dt \= \frac1\pi \int_{-\infty}^\infty
R(t;z)^{1-s} \al(t)\, dt\,,
\end{aligned}
\ee
so that $R^{1-s}$ is the kernel of the Poisson transformation in the
line model. If $\al$ is a hyperfunction, the pairing in \eqref{Potr}
has to be interpreted as discussed in \S\ref{secthf} as the
difference of two integrals over contours close to and on opposite
sides of $\p\H$ (equation~\eqref{dual} in the circle model), with
$R(\dv;z)^{1-s}$ extended analytically to a neighborhood of $\p\H$.

In the projective model and the circle model we find:
\beqabclbl{Po-pr-ci}
\be \poiss s \al (z) \= \langle R(\dv;z)^{1-s},\al \rangle\,,\ee
with $R(\dv;z)^{1-s}$ in the various models given by
\begin{align}
\label{Rproj-p}
R^\prj(\z;z)^{1-s} &\= y^{s-1} \biggl( \frac{\z-i}{\z-z}\biggr)^{1-s}
 \biggl( \frac{\z+i}{\z-\bar z} \biggr)^{1-s} \= \biggl(
 \frac{R(\z;z)}{R(\z;i)}\biggr)^{1-s} \,,\\
\label{Rcirc-p}
R^\ci(\x;w)^{1-s} &\= \biggl( \frac{ 1-|w|^2}{(1-w/\x)(1-\bar w\x)}
\biggr)^{1-s}\,.\alsochanged{period added. (referee)}
\end{align}
\eeqabc
By $R(\dv,\dv)^{1-s}$, without superscript on the $R$, we denote the
Poisson kernel in the line model (as in \eqref{Rext}). We take the
branch for which $\arg R(\z;z)=0$ for $\z$ on~$\RR$.

In the circle model we have for each $\al\in \V{-\om}s$:
\be\label{poiss-ci} \poiss s \al (w) \= \frac{(1-|w|^2)^{1-s}}{2\pi i}
\intCC g(\x) \,\bigl(
(1-w/\x)(1-\bar w \x)\bigr)^{s-1}\, \frac{d\x}\x\,, \ee
with $C_+$ and $C_-$ as in \eqref{dual}, adapted to the domain of the
representative $g\in \HHH_s$ of the hyperfunction~$\al$.\medskip

For the values of $s$ we are interested in, Helgason has shown that
the Poisson transformation is an isomorphism:
\begin{thm}\label{thmHel-p} \text{\rm(Theorem 4.3 in \cite{Hel-p})}.
The Poisson map $\poiss s:\V{-\om}s \to \E_s$ is a bijection for
$0<\re s<1$.
\end{thm}
The usual proof of this uses the $K$-Fourier expansion, where $K$
($\,\cong\PSO(2)\,$) is the standard maximal compact subgroup of~$G$.
One first checks by explicit integration the formula
\be \poiss s\eph_{s,m}\= (-1)^m \,
\frac{\Gf(s)}{\Gf(s+m)}\,P_{\!s,m}\qquad(n\in\ZZ), \label{Poisofe}\ee
with $\eph_{s,m}$ and $P_{\!s,m}$ as defined in \eqref{phisndef2-p}
and \eqref{LegP} respectively. (Indeed, with~\eqref{Rproj-p} and
\eqref{poiss-ci} we obtain the Poisson integral
\[ \poiss s \eph_{s,m}(w) \= \frac{(1-|w|^2)^{1-s}}{2\pi i}
\int_{|\x|=1} \x^m\,\bigl(
(1-w/\x)(1-\bar w \x)\bigr)^{s-1}\, \frac{d\x}\x\,.\]
Since $|w/\x|<1$ and $|\bar w\x|<1$ this leads to the expansion
\begin{align*} (1-&|w|^2)^{1-s} \sum_{n_1,n_2\geq 0,\, -n_1+n_2=m}
\frac{(1-s)_{n_1}\,(1-s)_{n_2}}{n_1!\;n_2!}\, w^{n_1}\, \bar
w^{n_2}\\
&\=(1-|w|^2)^{1-s}\,\frac{(1-s)_{|m|}}{|m|!} \sum_{n\geq 0}
\frac{(1-s)_n\,
(1-s+|m|)_n}{(1+|m|)_n\,n!}\, |w|^{2n}\cdot\begin{cases}
w^m&\text{ if }m\geq 0\,,\\
\bar w^{-m}&\text{ if }m\leq 0\,.
\end{cases}
\end{align*}
This is $(-1)^m \Gf(s)/\Gf(s+\nobreak m)$ times $P_{s,m}$ as defined
in \eqref{Leg-d} and~\eqref{Leg}.) Then one uses the fact that the
elements of $\V{-\om}s$ are given by sums $\sum c_n\,\eph_{s,n}$ with
coefficients $c_n$ of subexponential growth
(eq.~\eqref{chf}), and shows that the coefficients in the expansion
\eqref{p-exp} also have subexponential growth for each $u\in \E_s$.
 This is the analogue of the fact that a holomorphic function in the
unit disk has Taylor coefficients at~0 of subexponential growth, and
can be proved the same way. An alternative proof of
Theorem~\ref{thmHel-p} will follow from the results of
\S\ref{sect-Pi}, where we shall give an explicit inverse map
for~$\poiss s$.\medskip

Thus, $\E_s$ is a model of the principal series representation
$\V{-\om}s$, and also of $\V{-\om}{1-s}$ , that does not change under
the transformation $s\mapsto 1-s$ of the spectral parameter. It is
completely $G$-equivariant. The action of $G$ is simply given by
$u\,|\,g = u \circ g$.

As discussed in \S1, the space $\V{-\om}s$
(hyperfunctions on $\p\H$) contains three canonical subspaces
$\V{-\infty}s$ (distributions), $\V\infty s$ (smooth functions)
and $\V\om s$ (analytic functions on $\p\H$), and we can ask whether
there is an intrinsic characterization of the corresponding subspaces
$\E_s^{-\infty}$, $\E_s^\infty$, and $\E_s^\om$ of $\E_s$. For
$\E_s^{-\infty}$ the answer is simple and depends only on the
asymptotic properties of the eigenfunctions near the boundary,
namely:
\begin{thm}\label{thm-John}{\rm( \cite{Le-p}, Theorem 4.1 and Theorem
5.3.})
Let $0<\re s<1$. The space
$\E_s^{-\infty}=\poiss s\bigl(\V{-\infty}s\bigr)$ consists of the
functions in $\E_s$ having at most polynomial growth near the
boundary.
\\
\noindent{\rm (``At most polynomial growth near the boundary'' means
$\ll\bigl(1-|w|^2\bigr)^{-C}$ for some $C$ in the disk model and
$\ll\bigl(\bigl(|z+i|^2\bigr)/y\bigr)^C$ in the upper half-plane
model.)}
\end{thm}

The corresponding theorems for the spaces $\E^\infty_s$ and
$\E^\om_s$, which do not only involve estimates of the speed of
growth of functions near~$\p\H$, are considerably more complicated.
We will return to the description of these spaces in
Section~\ref{sect-split}.

\sssect{Explicit examples}One example is given in \eqref{Poisofe}.
Another example is
\be\label{dtsi}
\poiss s \dt_{s,\infty} (z) \= R^\prj(\infty;z)^{1-s} \= y^{1-s}\,,\ee
where $\dt_{s,\infty}\in \V{-\infty}s$ is the distribution associating
to $\ph \in \V\infty{1-s}$, in the projective model, its value at
$\infty$. As a third example we consider the element
$R(\dv;z_0)^s \in \V\om s$. For convenience we use the circle model.
Then \alsochanged{exponent $s$ inserted}$a(w,w') 
= \bigl(\poiss s R^\ci(\dv;w')^s \bigr)(w)$ satisfies the relation
$a(gw',gw)=a(w',w)$ for all $g\in G$, by equivariance of the Poisson
transform and of the function $R^s$. So $a$ is a point-pair
invariant. Since $a(w,\dv)\in \E_s$, it has to be a multiple of
$p_s$. We compute the factor by taking $w'=w=0\in \disk$:
\begin{align*}
\poiss s R^\ci(\dv;0)^s(0) &\= \frac1{2\pi i}\int_\uc R^\ci(\x;0)^s
R^\ci(\x;0)^{1-s}\, \frac{d\x}\x\\
&\= \frac1{2\pi i}\int_\uc 1\cdot 1 \,\frac{d\x}\x \= 1\,.
\end{align*}
Thus we have
\be\label{P-R} \poiss s R(\dv;w') ^s (w) \= p_s(w',w)\,.\ee
With \eqref{P-I-p} and the fact that $P_{\!1-s,0}=P_{\!s,0}$ this
implies
\be \label{IR} I_s \,R(\dv;w')^s \= R(\dv;w')^{1-s}\,.\ee

\subsection{Second order eigenfunctions}\label{sect-secorder}
The Poisson transformation allows us to prove results concerning the
space
\be \E_s'\;:=\;\ker\bigl( (\Dt-\ld_s)^2: C^\infty(\H)\lra C^\infty(\H)
\bigr)\,. \ee
\begin{prop}\label{prop-Eex}
The following sequence is exact:
\be \label{E-ex}
 0 \lra \E_s \lra \E_s' \stackrel{\Dt-\ld_s}\lra \E_s \lra 0\,. \ee
\end{prop}
\begin{proof}Only the surjectivity of $\E_s'\lra \E_s$ is not
 immediately clear.

Let $0<\re s_0<1$. Suppose we have a family $s\mapsto f_s$ on a
neighborhood of $s_0$ such that $f_s \in \E_s$ for all $s$
near~$s_0$, and suppose that this family is $C^\infty$ in $(s,z)$ and
holomorphic in~$s$. Then
\[ (\Dt-\ld_{s_0}) \bigl( \p_s f_s |_{s=s_0} \bigr ) - (1-2s_0)
f_{s_0}=0\,.\]
For $s_0\neq \frac12$, this gives an element of $\E_{s_0}'$ that is
mapped to $f_{s_0}$ by $\Dt-\ld_{s_0}$. If $s_0=\frac12$, we replace
$f_s$ by $\frac12(f_s+f_{1-s})$, and differentiate twice.\smallskip

To produce such a family, we use the Poisson transformation. By
Theorem~\ref{thmHel-p} there is a unique $\al\in \V{-\om}{s_0}$ such
that $f= \poiss {s_0}\al$. We fix a representative
$g\in \hol(U\sm \uc)$ of $\al$ in the circle model, which represents
a hyperfunction $\al_s$ for all $s\in \CC$. (The projective model
works as well.) We put
\[ f_s(w) \= \poiss s \al_s (w) \= \frac1{2\pi i} \;\biggl(
\int_{C_+}-\int_{C_-}\biggr)\; R^\ci(\z;w)^{1-s} g(\z)\,
\frac{d\z}\z\,. \]
The contours $C_+$ and $C_-$ have to be adapted to $w$, but can stay
the same when $w$ varies through a compact subset of~$\uhp$.
Differentiating this family provides us with a lift of $f$ in
$\E_{s_0}'$.
\end{proof}
This proof gives an explicit element
\be \tilde f(w) \= \frac{-1}{2\pi i} \intCC R^\ci(\z;w)^{1-s_0}\;(
\log R^\ci(\z;w))\; g(\z) \, \frac{d\z}\z\ee
of $\E_s'$ with $(\Dt-\nobreak\ld_s) \tilde f=(1-\nobreak 2s)f$. Note
that for $s=\frac12$ the function $\tilde f$ belongs to $\E_{1/2}$,
giving an interesting map $\E_{1/2}\lra \E_{1/2}$. As an example, if
$f(z) = y^{1/2}$, then we can take $g(\z) = \frac{\z}{2i}$ as the
representative of the hyperfunction $\al=\dt_{1/2,\infty}$ with
$\poiss {1/2}\al=h$, and by deforming the contours $C_+$ and $C_-$
into one circle $|\z|=R$ with $R$ large we obtain (in the projective
model):
\be
\begin{aligned}
\tilde f(z) &\= \frac{-1}\pi \int_{|\z|=R} R^\prj(\z;z)^{1/2} \left(
\log y + \log \frac{\z^2+1}{(\z-z)(\z-\bar z)} \right) \,
\frac{\z}{2i}\, \frac{d\z}{\z^2+1}\\
&\= - y^{1/2} \log y \,.
\end{aligned}
\ee
In part~C of Table~\ref{tab-Pta} in \S\ref{sect-Pta} we describe the
distribution in $\V{-\infty}{1/2}$ corresponding to this element of
$\E_{1/2}\,$.
\medskip

Theorem~\ref{thm-John} shows that the subspace $\E^{-\infty}_s$
corresponding to $\V{-\infty}s$ under the Poisson transformation is
the space of elements of $\E_s$ with polynomial growth. We define
$(\E_s')^{-\infty}$ as the subspace of $\E'_s$ of elements with
polynomial growth. The following proposition, including the somewhat
technical second statement, is needed in Chapter~V of~\cite{BLZ-p}.
\begin{prop}\label{prop-E'ex}
The sequence
\be\label{E'-ex}
0 \lra \E_s^{-\infty} \lra (\E'_s)^{-\infty} \stackrel{\Dt-\ld_s}\lra
\E_s^{-\infty}\lra 0 \ee
is exact. All derivatives $\p_w^l \p_{\bar w}^m f(w)$, $l,m\geq 0$, of
$f\in (\E'_s)^{-\infty}$, in the disk model, have polynomial growth.
\end{prop}
\begin{proof}We use the construction in the proof of
Proposition~\ref{prop-Eex}. We use
\[ f_s(w) \= \langle R^\ci(\dv;w)^{1-s},\al \rangle\,,\]
with $\al\in \V{-\infty}{s_0}$. For $\z\in \uc$ we obtain by
differentiating the expression for $R^\ci$ in~\eqref{Rcirc-p}
\[ (\z\p\z)^n \p_w^l\p_{\bar w}^m \, R^\ci(\z;w)^{1-s} \;\ll_{n,l,m}\;
(1-|w|^2)^{ s-l-m-n}\,.\]
With the seminorm $\|\dv\|_n$ in \eqref{seminorm-p} we can reformulate
this as
\be\label{nmest} \|\p_w^l \p_{\bar w}^m R^\ci(\dv;w)^{1-s}\|_n
\;\ll_{n,l,m}\; (1-|w|^2)^{ s-l-m-n}\,.\ee
Since $\al$ determines a continuous linear form on $\V p s$ for some
$p\in \NN$, this gives an estimate
\[ \p_w^l\p_{\bar w}^m f(w) \;\ll_{\al,l,m}\; (1-|w|^2)^{\re
s-1-l-m-p}\]
for $f\in \E_{s_0}^{-\infty}$.

Differentiating $R^\ci(\dv;w)^{1-s}$ once or twice with respect to~$s$
 multiplies the estimate in \eqref{nmest} with at most a factor
 $\bigl| \log(1-|w|^2) \bigr|^2$. The lift $\tilde f\in \E'_{s_0}$ of
$f_{s_0}$ in the proof of Proposition~\ref{prop-Eex} satisfies
\[ \p_w^l\p_{\bar w}^m \tilde f(w) \;\ll_{\al,l,m,\e}\; (1-|w|^2)^{\re
s-1-l-m-p-\e}\]
for each $\e>0$.
\end{proof}
\medskip

\section{Hybrid models for the principal series
representation}\label{sect-pms}
In this section we introduce the canonical model of the principal
series, discussed in the introduction. In Subsection~\ref{sect-pm} we
define first two other models of $\V{}s$ in functions or
hyperfunctions on $\p\H \times \H$, which we call {\it hybrid
models}, since they mix the properties of the model of $\V{}s$ in
eigenfunctions, as discussed in Section~\ref{secthyp}, with the
models discussed in Section~\ref{sect-ps}. The second of these,
called the flabby hybrid model, contains the {\it canonical model} as
a special subspace. The advantage of the canonical model becomes very
clear in Subsection~\ref{sect-Pi}, where we give an explicit inverse
for the Poisson transformation whose image coincides exactly with the
canonical model.

\alsochanged{}In Subsection~\ref{sect-diff-can} we will characterize
the canonical model as a space of functions on
$(\proj\CC\sm\proj\RR)\times\uhp$ satisfying a certain system of
differential equations. We use these differential equations to define
a sheaf~$\D_s$ on~$\proj\CC\times \uhp$, the {\it sheaf of mixed
eigenfunctions}. The properties of this sheaf and of its sections
over other natural subsets of~$\proj\CC\times \uhp$ are studied in
the remainder of the subsection and in more detail in
Section~\ref{sect-conjectures}.


\subsection{The hybrid models and the canonical
model}\label{sect-pm}The line model of principal series
representations is based on giving~$\infty \in \p\uhp$ a special
role. The projective model eliminated the special role of the point
at infinity in the line model, at the expense of a more complicated
description of the action of $G=\PSL_2(\RR)$, but it also broke the
$G$-symmetry in a different way by singling out the point $i\in\uhp$.
The corresponding point $0\in \disk$ plays a special role in the
circle model. \alsochanged{`Also' omitted.} The sequence model is
based on the characters of the \alsochanged{'specific' inserted.}
specific maximal compact subgroup $K=\PSO(2)\subset G$ and not of its
conjugates, again breaking the $G$-symmetry. The induced
representation model depends on the choice of the Borel group $NA$.
Thus none of the one-variable % change: one-variable inserted
models for $\V{}s$ discussed in
Section~\ref{sect-ps} reflects fully the intrinsic symmetry under the
action of~$G$.

To remedy these defects, we will replace our previous functions $\ph$
on~$\p\H$ by functions $\pmd \ph$ on $\p\H \times \H$, where the
second variable plays the role of a base point, with
$\pmd \ph(\dv,i)$ being equal to the function $\ph^\prj$ of the
projective model. This has the disadvantage of replacing functions of
one variable by functions of two, but gives a very simple formula for
the $G$-action, is completely symmetric, and will also turn out to be
very convenient for the Poisson transform. Explicitly, given
$(\ph,\ph_\infty)$ in the line model, we define
$\,\pmd\ph\,:\,\proj\RR\times\uhp\to\CC\,$ by
\be \label{pairdef} \pmd\ph(t,z)\=
\begin{cases}\;\biggl(\dfrac{|z-t|^2}y\biggr)^s \,\ph(t)
    &\text{if $t\in \proj\RR\sm\{\infty\}\,,$}\\
\;\biggl(\dfrac{|1+z/t|^2}y\biggr)^s\,\ph_\infty\biggl(-\dfrac1t\biggr)
     &\text{if $t\in\proj\RR\sm\{0\}$} \end{cases} \ee
(here $y=\img(z)$ as usual), generalizing equation \eqref{line-proj}
for $z=i$. The function $\pmd\ph$ then satisfies
\be \label{pair}
\pmd\ph(t,z_1)\=
\biggl(\frac{|z_1-t|^2/y_1}{|z_2-t|^2/y_2}\biggr)^s\,\pmd\ph(t,z_2)
\= \biggl( \frac{R(t;z_2)}{R(t;z_1)}\biggr)^s \pmd\ph(t,z_2)\ee
for $t\in \proj\RR$ and $z_1,z_2\in \uhp$. A short calculation, with
use of \eqref{Requiv-p}, shows that the action of $G$ becomes simply
\be\label{hybr-Gact}
\pmd{\ph|g}(t,z)\=\pmd\ph(gt,\,gz)\qquad(t\in\proj\RR,\;z\in\uhp,\;g\in
G)\ee
in this model. {}From equations \eqref{pairdef}, \eqref{line-proj} and
\eqref{pair}, we find
\be\label{pm-prj}
\ph^\prj(t)\=\ph(t,i)\,,\qquad \pmd \ph(t, z) \= \biggl(
\frac{(t-z)(t-\bar z)}{(t^2+1)\, y }\biggr)^s\, \ph^\prj(t)\,, \ee
giving the relation between the new model and the projective model.
And we see that only the complicated factor relating $\pmd\ph$ to
$\ph^\prj$ is responsible for the complicated action of $G$ in the
projective model.

We define the \emph{rigid hybrid model} to be the space of functions
$h:\proj\RR\times\uhp\to\CC$ satisfying \eqref{pair} with $\pmd\ph$
replaced by $h$. The $G$-action is given by $F\mapsto F\circ g$,
where $G$ acts diagonally on $\proj\RR\times\uhp$. The smooth
(resp.~analytic) vectors are those for which $F(\dv,z)$ is smooth
(resp.~analytic) on $\proj\RR$ for any $z\in\uhp$; this is independent
of the choice of $z$ because the expression in parentheses in
\eqref{pair} is analytic and strictly positive on $\proj\RR\,$. These
spaces are models for $\V\infty s$ and $\V\om s$, respectively, but
when needed will be denoted $\vinfrig s$ and $\vomrig s$ to avoid
confusion. We may view the elements of the rigid hybrid model as a
family of functions $t\mapsto \pmd\ph(t,z)$ in projective models with
a varying special point $z\in \uhp$. The isomorphism relating the
rigid hybrid model and the line (respectively projective) model is
then given by \eqref{pairdef} (respectively \eqref{pm-prj}).

In the case of $\vomrig s$ we can replace $t$ in \eqref{pair} or
\eqref{pm-prj} by a variable $\z$ on a neighborhood of $\proj\RR$. We
\alsochanged{recall}observe that, although $R(\z;z)^s$ is multivalued
in $\z$, the quotient 
$\Bigl({\displaystyle\frac{R(\z;z_1)}{R(\z;z)}}\Bigr)^s$
in~\eqref{pair} is holomorphic in $\z$ on a neighborhood (depending
on~$z$ and $z_1$) of $\proj\RR$ in~$\proj\CC$. In the rigid hybrid
model, the space $\hrig_s$ consists of germs of functions $h$ on a
deleted neighborhood $U \sm(\proj\RR \times\uhp) $ which are
holomorphic in the first variable and satisfy
\be\label{pairhpf}
h(\z,z_1) \= \biggl( \frac{ R(\z;z_2)}{R(\z;z_1)} \biggr)^s
h(\z,z_2)\qquad(z_1,z_2\in \uhp\,,\; \z\text{ near }\proj\RR)\,, \ee
where ``near $\proj\RR$'' means that $\z$ is sufficiently far in the
hyperbolic metric from the geodesic joining $z_1$ and $z_2$. This
condition ensures that $(\z,z_1)$ and $(\z,z_2)$ belong to $U$ and
the multiplicative factor in~\eqref{pairhpf} is a power of a complex
number not in $(-\infty,0]$ and is therefore well defined. The action
of $G$ on $\hrig_s$ is given by $h(\z,z) \mapsto h(g\z,gz)$. In this
model $\V{-\om}s$ is represented as $\hrig_s/\vomrig s$. The pairing
between hyperfunctions and test functions in this model is given by
\begin{equation} \label{pairhpfpairing} \bigl\langle h,\,\pmd \ps
\bigr\rangle\= \frac1{\pi}\,\intCC h(\z,z) \; \pmd\psi(\z,z) \;
R(\z;z)\; d\z \end{equation}
with the contours $C_+$ and $C_-$ as in \eqref{projpairing}. Provided
we adapt the contours to~$z$, we can use any $z\in\uhp$ in this
formula for the pairing.\medskip

The rigid hybrid model, as described above, solves all of the problems
of the various models of $\V{}s$ as function spaces on $\p\H$, but it
is in some sense artificial, since the elements $h$ depend in a fixed
way on the second variable, and the use of this variable is therefore
in principle superfluous. \change{This and the two following
paragraphs reorganized.}We address the remaining artificiality by
replacing the rigid hybrid model by another model. The intuition is
 to replace functions satisfying~\eqref{pairhpf} by hyperfunctions
satisfying this relation.

Specifically, we define the \emph{flabby hybrid model} as
\[ \vhflab_s \;:=\; \hflab_s /\vomflab_s\,, \]
where $\hflab_s$ is the space of functions\footnote{Here one has the
choice to impose any desired regularity conditions
($C^0$, $C^\infty$, $C^\om$, \dots) in the second variable or in both
variables jointly. We do not fix any such choice since none of our
considerations depend on which choice is made and since in any case
the most interesting elements of this space, like the canonical
representative introduced below, are analytic in both variables.}
$h(\z,z)$ that are defined on
$U\sm\bigl( \proj\RR\times\nobreak\uhp\bigr)$ for some neighborhood
$U$ of $\proj\RR\times\uhp$ in $\proj\CC \times \uhp$, are
holomorphic in $\z$, and satisfy
\be\label{pairg} \z\mapsto h(\z,z_1) - \biggl(
\frac{R(\z;z_2)}{R(\z;z_1)} \biggr)^s h(\z,z_2)
\;\in\; \hol(U_{z_1,z_2})\quad \text{ for all }z_1,z_2\in \uhp\,, \ee
where
$U_{z_1,z_2} = \{ \z\in \proj\CC\;:\; (\z,z_1),(\z,z_2) \in U\}$,
while $\vomflab_s$ consists of functions defined on a neighborhood
$U$ of $\proj\RR\times\uhp$ in $\proj\CC\times\uhp$ and holomorphic
in the first variable. The action of $G$ in $\hflab_s$ is by
$h\,|\,g\,(\z,z)=h(g\z,gz)$. The pairing between hyperfunctions and
analytic functions is given by the same formula
\eqref{pairhpfpairing} as in the rigid hybrid model.


An element $h\in \hflab_s$ can thus be viewed as a family
$\bigl\{h(\dv,z)\bigr\}_{z\in \uhp}$ of representatives of
hyperfunctions parametrized by~$\uhp$. Adding an element of
$\vomflab_s$ does not change this family of hyperfunctions. The
requirement \eqref{pairg} on~$h$ means that the family of
hyperfunctions satisfies~\eqref{pairhpf} in hyperfunction
sense.\medskip


Finally, we describe a subspace $\can_s \subset \hflab_s$ which maps
isomorphically to $\V{-\om}s$ under the projection
$\hflab_s \twoheadrightarrow \V{-\om}s$ and hence gives a canonical
choice of representatives of the hyperfunctions in $\vhflab_s$. We
will call $\can_s$ the \emph{canonical hybrid model}, or simply the
\emph{canonical model} for the principal series representation
$\V{-\om}s$. To define $\can_s$ we recall that any hyperfunction on
$\proj\RR$ can be represented by a holomorphic function on
$\proj\CC\sm\proj\RR$ with the freedom only of an additive constant.
One usually fixes the constant by requiring $h(i)=0$ or
$h(i)+h(-i)=0$, which is of course not $G$-equivariant. Here we can
exploit the fact that we have two variables to make the normalization
in a $G$-equivariant way, by requiring that
\be\label{norm} h(\bar z,z)\=0\,.\ee
We thus define $\can_s$ as the space of functions on
$\left( \proj\CC \sm \proj\RR\right) \times \uhp$ that are
holomorphic in the first variable and satisfy \eqref{pairg}
and~\eqref{norm}. We will see below (Theorem~\ref{thm-pi}) that the
Poisson transform $\poiss s :\V{-\om}s \lra \E_s$ becomes extremely
simple when restricted to $\can_s$, and also that $\can_s$ coincides
with the image of a canonical lifting of the inverse Poisson map
$\poiss s^{-1}:\E_s \lra\V{-\om}s$ from the space of hyperfunctions
to the space of hyperfunction representatives.\medskip

\noindent\emph{Remark. } We will also occasionally use the slightly
larger space $\can_s^+$ (no longer mapped injectively to $\E_s$ by
$\poiss s$) consisting of functions in $\hflab_s$ that that are
defined on all of $\bigl(\proj\CC\sm\proj\RR\bigr)\times\uhp$,
without the requirement~\eqref{norm}. Functions in this space will be
called \emph{semicanonical} representatives of the hyperfunctions
they represent. The decomposition
$\,h(\z,z)=\bigl(h(\z,z)-h(\bar z,z)\bigr)
\,+\,h(\bar z,z)\,$ gives a canonical and $G$-equivariant splitting of
$\can_s^+$ as the direct sum of $\can_s$ and the space of functions
on~$\uhp$, so that there is no new content here, but specific
hyperfunctions sometimes have a particularly simple semicanonical
representative (an example is given below), and it is not always
natural to require~\eqref{norm}.

\sssect{Summary} We have introduced a ``rigid'', a ``flabby'', and a
``canonical'' hybrid model, related by
\be \V{-\om}s \;\cong\; \hrig_s/\vomrig s \;\cong\; \vhflab_s \=
\hflab_s/\vomflab_s \;\cong\; \can_s\;\subset\;\hflab_s\,. \ee

In the flabby hybrid model, the space $\hflab_s$ consists of functions
on a deleted neighborhood $U\sm
(\proj\RR\times\uhp)$ that may depend on the function, holomorphic in
the first variable, and satisfying \eqref{pairg}. The subspace
$\vomflab_s$ are functions on the whole of some neighborhood $U$ of
$\proj\RR\times \uhp$, holomorphic in the first variable.

\alsochanged{Paragraph breaks.}For the elements of $\hflab_s$ and
$\vomflab_s\subset\hflab_s$ we do not require any regularity in the
second variable. In the rigid hybrid model the spaces
$\hrig_s\subset \hflab_s$ and $\vomrig s \subset \vomflab_s$ are
characterized by the condition in \eqref{pairhpf}, which forces a
strong regularity in the second variable. The canonical hybrid model
$\can_s$ consists of a specific element from each class of
$\hflab_s/\vomflab_s$ that is defined on
$(\proj\CC\times\uhp) \sm(\proj\RR \times \uhp)$ and is normalized
by~\eqref{norm}. In \S\ref{sect-diff-can} we will see that this
implies analyticity in both variables jointly.


\sssect{Examples} As an example we represent the distribution
$\dt_{s,\infty}$ in all three hybrid models. This distribution, which
was defined by $\ph^\prj\mapsto \ph^\prj(\infty)$ in the projective
model (cf.~\eqref{dtsi}), is represented in the projective model by
$h^\prj(\z)=\frac1{2i} \z$, and hence, by \eqref{pm-prj}, by
\be \label{pair-rigid}
\pmd h(\z,z)
\= \frac\z{2i}\, y^{-s} \, \biggl( \frac{(\z-z)(\z-\bar
z)}{(\z-i)(\z+i)} \biggr)^s\ee
in the rigid hybrid model. Since the difference
$\frac\z{2iy^s} \biggl( \bigl( \frac{1-z/\z}{1-i/z}\bigr)^s
\bigl(\frac{1-\bar z/\z}{1+i\z}\bigr)^s-1\biggr)$ is holomorphic in
$\z$ on a neighborhood of $\proj\RR$ in~$\proj\CC$ for each $z$, we
obtain the much simpler semicanonical representative
\be\label{dtrepr-H} h_\text{s}(\z,z) \= \frac\z{2i \, y^s}\,, \ee
%\be\label{dtrepr-H} (\z,z) \;\mapsto\; \frac\z{2i \, y^s}\,, \ee
of $\dt_{s,\infty}$ in the flabby hybrid model. Finally, subtracting
$h_{\text{s}}(\bar z,z)$ we obtain the (unique) representative of
$\dt_{s,\infty}$ in the canonical hybrid model:
\be\label{canrep-y-s} h_\text{c}(\z,z) \= \frac{\z-\bar z}{2i}
\,y^{-s}\,. \ee
We obtain other elements of $\can _s$ by the action of~$G$. For
$g\in G$ with $g\,\infty=a\in\RR$ we get
\be\label{can-r1} h_\text{c}|g^{-1} (\z,z) \= \frac{\z-\bar z}{z-\bar
z}\; \frac{z-a}{\z-a}\;R(a;z)^{1-s}\,. \ee
Here property \eqref{norm} is obvious, and \eqref{pairg} holds because
the only singularity of \eqref{can-r1} on $\proj\RR$ is a simple pole
of residue $(i/2)\,R(a;z)^{-s}$ at $\z=a$.

\sssect{Duality and Poisson transform}From \eqref{projpairing} we find
that if $h\in \hflab_s$ and $f \in \vomflab_{1-s} $ are defined on
$U \sm (\proj\RR\times \nobreak  \uhp)$, respectively $U$, for the
same neighborhood $U$ of $\proj\RR\times \uhp$, then
\be\label{flab-pairing}
\langle f, h \rangle \= \frac1\pi\,\intCC f(\z,z)\, h(\z,z)\,
R(\z;z)\, d\z\,, \ee
where $C_+$ and $C_-$ are contours encircling $z$ in $\uhp$ and
$\bar z$ in $\lhp$, respectively, such that $C_+\times \{z\}$ and
$C_-\times\{z\}$ are contained in $U$. The result
$\langle f,h\rangle$ does not change if we replace $h$ by another
element of $h+\vomflab_s \subset \hflab_s$.

We apply this to the Poisson kernel
$f_z^\prj(\z) = R^\prj(\z;z)^{1-s} = \bigl(
\frac{R(\z;z)}{R(\z;i)}\bigr)^{1-s}$, for $z\in \uhp$. The
corresponding element in the rigid pair model is \[\tilde f_z(\z,z_1)
\= \biggl( \frac{R(\z;z)}{R(\z;i)}\biggr)^{1-s} \, \biggl(
\frac{R(\z;i)}{R(\z;z_1)}\biggr)^{1-s} \= \biggl(
\frac{R(\z;z)}{R(\z,z_1)}\biggr)^{1-s}\,.\]
Applying \eqref{flab-pairing} we find for $z,z_1\in \uhp$:
\be\label{Poiss-flab}
\begin{aligned}
\poiss s h (z) &\= \frac1\pi\,\intCC \biggl(
\frac{R(\z;z)}{R(\z;z_1)}\biggr)^{1-s}\, h(\z,z_1) R(\z;z_1)\, d\z\\
&\= \frac1\pi\,\intCC \biggl( \frac{R(\z;z_1)}{R(\z;z)}\biggr)^s \,
h(\z,z_1) \, R(\z;z)\, d\z\,,
\end{aligned}
\ee
where $C_+$ encircles $z$ and $z_1$, and $C_-$ encircles $\bar z$
and~$\overline{z_1}$. Since this does not depend on $z_1$, we can
choose $z_1=z$ to get
\be \label{poiss-gpm}
\begin{aligned}
\poiss s h(z) &\= \frac1\pi\,\intCC \,h(\z,z) \,\,R(\z;z)\; d\z\\
&\= \frac1{2\pi i}\,\intCC\, h(\z,z)\, \frac{(z-\bar
z)}{(\z-z)(\z-\bar z)}\, d\z\,.
\end{aligned} \ee

The representation of the Poisson transformation given by formula
\eqref{poiss-gpm} has a very simple form. The dependence on the
spectral parameter $s$ is provided by the model, not by the Poisson
kernel. But a really amazing simplification occurs if we assume that
the function $h\in\hflab_s$ belongs to the subspace $\can_s$ of
canonical hyperfunction representatives. In that case $h(\z,z)$ is
holomorphic in $\z$ in all of $\CC\sm\RR$, so we can evaluate the
integral by Cauchy's theorem. In the lower half-plane there is no
pole, since $h(\bar z,z)$ vanishes, so the integral over $C_-$
vanishes. In the upper half-plane there is a simple pole of residue
$h(z,z)$ at $\z=z$. Hence we obtain:
\begin{prop}\label{prop-pr} The Poisson transform of a function
$h\in\can_s$ is the function
\be \label{Rd-def} \Rd s h (z) \= h(z,z)\,,\ee
defined by restriction to the diagonal.
\end{prop}

As examples of the proposition we set $\z=z$ in
equations~\eqref{canrep-y-s} and~\eqref{can-r1} to get
\be\label{can-r2}
\begin{aligned}
&u(z)= y^{1-s}&&\Rightarrow&
&\bigl( \poiss s^{-1} u\bigr)_{\text{can}}(\z,z) \,= \, \frac{\z-\bar
z}{2i} \,y^{-s}\,\\
&u(z)=R(a;z)^{1-s}&&\Rightarrow&
&\bigl( \poiss s^{-1} u\bigr)_{\text{can}}(\z,z) \,= \, \frac{\z-\bar
z}{z-\bar z}\; \frac{z-a}{\z-a}\;u(z)\,.
\end{aligned}
\ee

Finally, we remark that on the larger space $\can_s^+$ introduced in
the Remark above, we have \emph{two} restriction maps
\be\label{Rdpm} \Rd s^+h(z)\=h(z,z)\,,\qquad \Rd s^-h(z)\=h(\bar z,z)
\ee
to the space of functions on~$\uhp$. The analogue of the proposition
just given is then that the restriction of $\poiss s$ to $\can_s^+$
equals the difference $\Rd s=\Rd s^+-\Rd s^-\,$.


\subsection{Poisson inversion and the canonical
model}\label{sect-Pi}The canonical model is particularly suitable to
give an integral formula for the inverse Poisson transformation, as
we see in the main result of this subsection, Theorem~\ref{thm-pi}.
In Proposition~\ref{prop-can-repr} we give an integral formula for
the canonical representative of a hyperfunction in terms of an
arbitrary representative in~$\hflab_s$. Proposition~\ref{prop-canc}
relates, for $u\in \E_s$, the Taylor expansions in the upper and
lower half plane of the canonical representative of $\poiss s^{-1} u$
to the polar expansion of $u$ with the functions $p_{s,n}$.
\medskip

To determine the image $\poiss s^{-1}u$ under the inverse Poisson
transform for a given $u\in \E_s$, we have to construct a
hyperfunction on $\p\H$ which maps under $\poiss s$ to~$u$. A first
attempt, based on \cite{LZ-p}, Chap.~II, \S2, would be (in the line
model) to integrate the Green's form $\{u,R(\z;\dv)^s\}$ from some
base point to~$\z$. This does not make sense at $\infty$ since
$R(\z;\dv)$ has a singularity there and one cannot take a
well-defined $s$-th power of it, so we should renormalize by dividing
$R(\z;\dv)^s$ by $R(\z;i)^s$, or better, to avoid destroying the
$G$-equivariance of the construction, by $R(\z;z)^s$ with a variable
point $z\in \uhp$. This suggests the formula
\be\label{firsttry}
h(\z,z) \= \begin{cases}
\int_{z_0}^\z \bigl\{ u, (R_\z(\dv)/R_\z(z))^s\bigr\}&\text{ if }\z\in
\uhp\,,\smallskip\\
\int_{z_0}^{\bar \z} \bigl\{ u, (R_\z(\dv)/R_\z(z))^s\bigr\}&\text{ if
}\z\in \lhp\,,
\end{cases}
\ee
in the hybrid model, where $z_0\in \uhp$ is a base point, as a second
attempt. This almost works: the fact that the Green's form is closed
implies that the integrals are independent of the path of
integration, and changing the base-point $z_0$ changes $h(\dv,z)$ by
a function holomorphic near~$\proj\RR$ and hence does not change the
hyperfunction it represents. The problem is that both integrals in
\eqref{firsttry} diverge, because $ R(\z;z')^s$ has a singularity
like $(\z-\nobreak z')^{-s}$ near $\z$ and like
$(\z-\nobreak\overline{z'})^{-s}$ near $\bar \z$ and the
differentiation implicit in the bracket $\{\dv,\dv\}$ turns these
into singularities like $(\z-\nobreak z')^{-s-1}$ and
$(\z-\nobreak\overline{z'})^{-s-1}$ which are no longer integrable at
$z'=\z$ or $z'=\bar\z$, respectively. To remedy this in the upper
half-plane, we replace $\{u,(R_\z(\dv)/R_\z(z))^s\}$ by
$[u, (R_\z(\dv)/R_\z(z))^s]$, which differs from it by a harmless
exact 1-form but is now integrable at $\z$. (The same trick was
already used in \S2, Chap.~II of \cite{LZ-p}, where $z_0$ was
$\infty\,$.) In the lower half-plane, $[u,(R_\z(\dv)/R_\z(z)^s]$ is
not small near $z'=\bar\z$, so here we must replace the differential
form $\{u,(R_\z(\dv)/R_\z(z))^s\} =-\{(R_\z(\dv)/R_\z(z))^s,u\}$ by
$-[(R_\z(\dv)/R_\z(z))^s,u]$ instead. (We recall that $\{\dv,\dv\}$
is antisymmetric but $[\dv,\dv]$ is not.) However, since the
differential forms $[u,(R_\z(\dv)/R_\z(z))^s]$ and
$-[ (R_\z(\dv)/R_\z(z))^s,u]$ differ by the exact form
$d(u \,(R_\z(\dv)/\allowbreak R_\z(z))^s)$, this change requires
correcting the formula in one of the half-planes. (We choose the
upper half plane.) This gives the formula
\be h(\z;z) \= \begin{cases}
u(z_0)\bigl(\frac{R(\z;z_0)}{R(\z;z)}\bigr)^s + \int_{z_0}^\z \bigl[
u, \bigl(\frac{R(\z;\dv)}{R(\z;z)}\bigr)^s\bigr]&\text{ if }\z\in
\uhp\,,\\
\hphantom{u(z_0)\bigl(\frac{R(\z;z_0)}{R(\z;z)}\bigr)^s + } \int_{\bar
\z}^{z_0} \bigl[\bigl(\frac{R(\z;\dv)}{R(\z;z)}\bigr)^s, u
\bigr]&\text{ if }\z\in \lhp\,.
\end{cases}
\ee
We note that in this formula, $h(\overline{z_0},z)=0$. So we can
satisfy \eqref{norm} by choosing $z_0=z$, at the same time restoring
the $G$-symmetry which was broken by the choice of a base point
$z_0$. We can then choose the continuous branch of
$\bigl(R_\z/R_\z(z)\bigr)^s$ that equals~$1$ at the end-point $z$ of
the path of integration. Thus we have arrived at the following
Poisson inversion formula, already given in the Introduction
(eq.~\eqref{iBA}):

\begin{thm}\label{thm-pi} Let $u\in \E_s$. Then the function
$\ipoiss s u\in\hflab_s$ defined by
\be\label{reprf}
(\ipoiss s u)(\z,z) \= \begin{cases}
u(z) + \int_z^\z\bigl[ u, \bigl(R_\z/R_\z(z)\bigr)^s \bigr]&\text{ if
}\z\in\uhp\,,\\
\hphantom{u(z)+ \hbox{} } \int_{\bar \z}^z \bigl[ \bigl(
R_\z/R_\z(z)\bigr)^s ,u\bigr]&\text{ if }\z\in \lhp \end{cases}
\ee
along any piecewise $C^1$-path of integration in $\uhp\sm\{\z\}$,
respectively $\uhp\sm\{\bar\z\}$, with the branch of $\bigl(
R_\z/R_\z(z)\bigr)^s$ chosen to be~$1$ at the end-point~$z$, belongs
to $\can_s$ and is a representative of the hyperfunction
$\poiss s^{-1}u \in \vhflab_s = \hflab_s/\vomflab_s$.
\end{thm}
\begin{cor} The maps $\ipoiss s:\,\E_s\to\can_s$ and
$\Rd s:\,\can_s\to\E_s$ defined by \eqref{reprf} and \eqref{Rd-def}
are inverse isomorphisms, and we have a commutative diagram
\be\label{dia-poiss-R} \xymatrix@R=.8cm@C=1.2cm{ \Pm_s \ar@{->>}[r]&
\V{-\om}s \ar[d]^{\poiss s}\\
\can_s \ar@{^{(}->}[u] \ar@<-.4ex>[r]_{\;\Rd s} \ar[ru]^\cong
& \E_s \ar@<-.4ex>[l]_{\;\ipoiss s} } \ee
\end{cor}
\begin{proof}
Let $u\in \E_s$. First we check that $h=\ipoiss s u$ is well defined
and determines an element of $\can_s$. The convergence of the
integrals in \eqref{reprf} requires an estimate of the integrand at
the boundaries. For $\z\in \uhp\sm\{z\}$ we use
\be\label{hke}
\bigl[ u(z'), \bigl(R_\z(z')/R_\z(z) \bigr)^s\bigr]_{z'} \=
\biggl(\frac{R(\z;z')}{R(\z;z)} \biggr)^s \biggl( u_z(z')\, dz' +
\frac{is}{2y'} u(z')\,
\frac{\z-z'}{\z-\overline{z'}}\,d\overline{z'}\biggr)\,. \ee
The factor in front is $1$ for $z'=z$ and
$\oh\bigl((\z-z')^{-s}\bigr)$ for $z'$ near~$\z$. The other
contributions stay finite, so the integral for $\z\in \uhp\sm\{z\}$
converges. (Recall that $\,\re(s)\,$ is always supposed to be $<1$.)
For $\z\in \lhp \sm\{\bar z\}$ we use in a similar way
\[\bigl[\bigl(R_\z(z')/R_\z(z)
\bigr)^s ,u(z')\bigr]_{z'} \=\biggl(\frac{R(\z;z')}{R(\z;z)} \biggr)^s
\biggl( \frac{is}{2 y'} u(z')
\frac{\z-\overline{z'}}{\z-z'}\, dz' + u_{\bar z}(z') \,
d\overline{z'} \biggr)\,. \]

We have normalized the branch of $\bigl( R(\z;z')/R(\z;z) \bigr)^s$ by
prescribing the value $1$ at~$z'=z$. This choice fixes
$\bigl( R(\z;z')/R(\z;z)
\bigr)^s$ as a continuous function on the paths of integration. The
result of the integration does not depend on the path, since the
differential form is closed and since we have convergence at the
other end point $\z$ or~$\bar \z$. Any continuous deformation of the
path within $\uhp\sm\{\z\}$ or $\uhp\sm\{\bar\z\}$ is allowed, even
if the path intersects itself with different values of
$\bigl( R(\z;z')/R(\z;z) \bigr)^s$ at the intersection point.
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If we choose the geodesic path from $z$ to~$\z$, and if $\z$ is very
near the real line, then the branch of
$\bigl(R_\z(z')/R_\z(z)\bigr)^s$ near $z'=\z$ is the principal one
(argument between $-\pi$ and~$\pi$).\smallskip

The holomorphy in $\z$ follows from a reasoning already present in
\cite{LZ-p}, Chap.~II, \S2, and hence given here in a condensed form.
Since the form~\eqref{hke} is holomorphic in $\z$, a contribution to
$\p_{\bar \z} h$ could only come from the upper limit of integration,
but in fact vanishes since
$\oh\bigl( (\z-z')^{-s}\bigr) \,(\z-z')=\text{o}(1)$ as $\z\to z'$.
Hence $h(\dv,z)$ is holomorphic on $\uhp\sm\{z\}$. For $\z$ near~$z$
we integrate a quantity $\oh\bigl((\z-z')^{-s}\bigr)$ from $z$ to
$\z$, which results in an integral estimated by
$\oh\bigl( (\z-z)^{1-s} \bigr)$. So $(\ipoiss s u)(\z,z)$ is bounded
for $\z$ near $z$. Hence $h(\z,z)=(\ipoiss s u)(\z,z)$ is holomorphic
at $\z=z$ as well. For the holomorphy on $\lhp $ we proceed
similarly. This also shows that $h(\bar z,z)=0$, which is condition
\eqref{norm} in the definition of $\can_s$.

For condition~\eqref{pairg} we note that
$$ h(\z;z) \,-\, \biggl(
\frac{R(\z;z_1)}{R(\z;z)}\biggr)^sh(\z;z_1)\= u(z)\,-\,u(z_1)\biggl(
\frac{R(\z;z_1)}{R(\z;z)}\biggr)^s + \int_z^{z_1} \biggl[ u,\,\biggl(
\frac{R(\z;\dv)}{R(\z;z)} \biggr)^s\biggr]$$
if $\z\in \uhp\sm\{z,z_1\}$ and
\[ h(\z;z)\,-\, \biggl( \frac{R(\z;z_1)}{R(\z;z)}\biggr)^sh(\z;z_1)\=
\int_{z_1}^z \biggl[ \biggl(
\frac{R(\z;\dv)}{R(\z;z)}\biggr)^s,\,u\biggr] \]
if $\z\in\lhp\sm\{\bar z,\bar z_1\}$. The right-hand sides both have
holomorphic extensions in $\z$ to a neighborhood of $\proj\RR$, and
the difference of these two extensions is seen, using~\eqref{Grf} and
the antisymmetry of $\{\dv,\dv\}$, to be equal to
\[u(z) - u(z_1)\biggl( \frac{R(\z;z_1)}{R(\z;z)}\biggr)^s +
\int_z^{z_1} d\biggl( u(z') \biggl( \frac{R(\z;z')}{R(\z;z)}\biggr)^s
\biggr) \=0\,. \]

In summary, the function $\ipoiss s u$ belongs to $\hflab_s$, is
defined in all of $\bigl(\proj\CC\sm\proj\RR\bigr)\times\uhp$, and
vanishes on the anti-diagonal, so $\ipoiss s u \in \can_s $, which is
the first statement of the theorem. The second follows immediately
from Proposition~\ref{prop-pr}, since it is obvious from
\eqref{reprf} that $\Rd s\ipoiss s u = u$ and the proposition says
that $\Rd s$ is the restriction of $\poiss s$ to $\can_s\,$.
\end{proof}

The corollary follows immediately from the theorem if we use
Helgason's result (Theorem~\ref{thmHel-p}) that the Poisson
transformation is an isomorphism. However, given that we have now
constructed an explicit inverse map for the Poisson transformation,
we should be able to give a more direct proof of this result, not
based on polar expansions, and indeed this is the case. Since
$\poiss s\ipoiss s u=\Rd s\ipoiss s u=u$, it suffices to show that
$\Rd s$ is injective. To see this, assume that $h\in \can_s$
satisfies $h(z,z)=0$ for all $z\in \uhp$. For fixed $z_1,z_2\in \uhp$
let $c(\z)$ denote the difference in \eqref{pairg}. This function is
holomorphic near $\proj\RR$, and extends to $\proj\CC$ in a
multi-valued way with branch-points of mild growth,
$(\z-\z_0)^{\pm s}$ with $0<\re s<1$, at $z_1$, $\overline{z_1}$,
$z_2$ and $\overline{z_2}$. Moreover, $c(\z)$ tends to $0$ as $\z$
tends to $z_1$ or $\overline{z_1}$ (because $\re s>0$) and also as
$\z$ tends to $z_2$ or $\overline{z_2}$ (because $
h(z_2,z_2)=h(\overline{z_2},z_2)=0$ and $\re s<1$). Suppose that $c$
is not identically zero. The differential form $d\log c(\z)$ is
meromorphic on all of $\proj\CC$ and its residues at $\z_0 \in
\{z_z,z_2,\overline{z_1},\overline{z_2}\}$ have positive real part.
Since $c$ is finite elsewhere on $\proj\CC$, any other residue is
non-negative. This contradicts the fact that the sum of all residues
of a meromorphic differential on $\proj\CC$ is zero. Hence we
conclude that $c=0$. Then the local behavior of
$h(\z,z_1) = h(\z,z_2) \allowbreak \left(
R_\z(z_2)/R_\z(z_1) \right)^s$ at the branch points shows that both
$h(\dv,z_1)$ and $h(\dv,z_2)$ vanish identically.

\begin{remarks}\emph{Remarks.} \rmknohd \nopagebreak
\rmk It is also possible to prove that $\ipoiss s \poiss s \ph = \ph$
and $\poiss s \ipoiss s u =u$ by using complex contour integration
and equation~\eqref{goodlemma}, and our original proof that
$\ipoiss s = \poiss s^{-1}$ went this way, but the above proof using
the canonical space $\can_s$ is much simpler.

\rmk Taking $z=i$ in formula \eqref{reprf} gives a representative for
$\poiss s^{-1}u$ in the projective model, and using the various
isomorphisms discussed in Section~1, we can also adapt it to the
other $\p\H$ models of the principal representation.
\end{remarks}


We know that each element of $\V{-\om}s$ has a unique canonical
representative lying in~$\can_s$. The following proposition, in which
$k(\tau,\z;z)$ denotes the kernel function
\be k(\tau,\z;z) \= \frac1{2i\,(\tau-\z)}\,\frac{\z-\bar z}{\tau-\bar
z}\,,\ee
tells us how to determine it starting from an arbitrary
representative.

\begin{prop}\label{prop-can-repr}Suppose that $g\in \hflab_s$
represents $\al\in \V{-\om}s$. The canonical representative
$g_c\in \can_s$ of $\al$ is given, for each $z_0\in\uhp$ by
\be\label{can-repr}
g_c(\z,z) \= \frac1\pi \intCC \, g(\tau,z_0)\biggl(
\frac{R(\tau;z_0)}{R(\tau;z)} \biggr)^s\, k(\tau,\z;z)\, d\tau\,,\ee
with contours $C_+$ and $C_-$ homotopic to $\proj\RR$ inside the
domain of~$g$, encircling $z$ and $z_0$, respectively $\bar z$ and
$\overline{z_0}$, with $C_+$ positively oriented in $\uhp$ and $C_-$
negatively oriented in~$\lhp $, and $\z$ inside $C_+$ or
inside~$C_-$.
\end{prop}
Note that this can be applied when a representative $g_0$ of $\al$ in
the projective model is given: simply apply the proposition to the
corresponding representative in the rigid hybrid model as given by
\eqref{pm-prj}.

\begin{proof}Consider $k(\dv,\z;z)$ as an element of $\V\om s$ in the
projective model. Then $g_c(\z,z) = \langle\al, k(\dv,\z;z) \rangle$.
Adapting the contours, we see that $g_c(\dv,z)$ is holomorphic on
$\uhp\cup\lhp $.

For a fixed $\z\in \dom g$ we deform the contours such that $\z$ is
between the new contours. This gives a term $ g(\z,z)$ plus the same
integral, but now representing a holomorphic function in $\z$ on the
region between $C_+$ and $C_-$, which is a neighborhood of
$\proj\RR$. So $ g$ and $g_c$ represent the same hyperfunction.
Condition~\eqref{norm} follows from $k(\tau,\bar z;z)=0$.
\end{proof}
Choosing $z_0=z$ in \eqref{can-repr} gives the simpler formula
\be\label{can-repr2}
g_c(\z,z) \= \frac1\pi \intCC \, g(\tau,z)\, k(\tau,\z;z)\,
d\tau\,,\ee
(which is, of courses, identical to \eqref{can-repr} if $g$ belongs to
$\vomrig s$). In terms of $\al \in \V{-\om}s$ as a linear form on
$\V\om{1-s}$ we can write this as
\be \label{can-repr1}
g_c(\z,z) \= \bigl\langle f_\z,\al\bigr\rangle\quad\text{ with}\quad
f_\z(\tau,z) \= \frac{(\z-\bar z)(\tau-z)}{(z-\bar z)(\tau-\z)}\,.
\ee

The integral representation \eqref{can-repr} has the following
consequence:
\begin{cor}\label{cor-can-anal}
All elements of $\can_s$ are real-analytic on
$(\proj\CC\sm\proj\RR)\times\uhp$.
\end{cor}

\sssect{Expansions in the canonical model} For $u\in \E_s$ the polar
expansion \eqref{u-exp} can be generalized, with the shifted
functions $p_{s,n}$ in \eqref{psn-def-p}, to an arbitrary central
point:
\be \label{u-expa}
u(z) \= \sum_{n\in \ZZ} a_n(u,z') \,
p_{s,n}(z,z')\qquad(z'\in\uhp\text{ arbitrary}).\,. \ee

Let $h=\ipoiss s u \in \can_s$ be the canonical representative of
$\poiss s^{-1}u\in \V{-\om}s$. For $z'\in \uhp$ fixed, $h(\z,z')$ is
a holomorphic function of $\z\in \CC\sm\RR$ and has Taylor expansions
in $\frac{\z-z'}{\z-\overline{z'}}$ on $\uhp$ and in
$\frac{\z-\overline{z'}}{\z-z'}$ on $\lhp$. Since $h(\bar z,z)=0$,
the constant term in the expansion on $\lhp$ vanishes. Thus there are
$A_n(h,z') \in \CC$ such that
\be \label{h-expa}
h(\z,z') \= \begin{cases}{\displaystyle
\sum_{n\geq 0} A_n(h,z') \, \biggl(
\frac{\z-z'}{\z-\overline{z'}}\biggr)^n}
&\text{ for }\z\in \uhp\,,\\
{\displaystyle-\sum_{n<0}A_n(h,z') \, \biggl(
\frac{\z-z'}{\z-\overline{z'}}\biggr)^n}
&\text{ for }\z\in \lhp\,.
\end{cases}
\ee
(We use a minus sign in the expansion on $\lhp$ because then
\[ (\z,z')\mapsto \sum_{n\geq n_0} A_n(h,z') \, \biggl(
\frac{\z-z'}{\z-\overline{z'}}\biggr)^n \text{ for }\z\in
\uhp\,,\quad
-\sum_{n<n_0}A_n(h,z') \, \biggl(
\frac{\z-z'}{\z-\overline{z'}}\biggr)^n \text{ for }\z\in \lhp\,, \]
represents the same hyperfunction $\poiss s^{-1} u$ for any choice of
$n_0\in \ZZ$.)
{}From
$\frac{g\z-gz'}{g\z-g\overline{z'}} = \frac{c\overline{z'}+d}{cz'+d}\, 
\frac{\z-z'}{\z-\overline{z'}}$ for $g=\matc abcd\in G$ it follows
that
\be \label{An-g}
A_n(h\,|\,g,z') \= \biggl( \frac{c\overline{z'}+d}{cz'+d}\biggr)^n\,
A_n(h,gz')\,. \ee
Similarly, we have from \eqref{psn-def-p} and~\eqref{LegP}:
\be \label{an-g}
a_n(u\,|\,g,z') \= \biggl(
\frac{c\overline{z'}+d}{cz'+d}\biggr)^n\,a_n(u,gz')\,. \ee
In fact, the coefficients $A_n(\ )$ and $a_n(\ )$ are proportional:
\begin{prop}\label{prop-canc}For $u\in \E_s$ and
$h=\ipoiss s u\in \can_s$, the coefficients in the expansions
\eqref{u-expa} and \eqref{h-expa} are related by
\be\label{a=A} a_n(u,z') \= (-1)^n \frac{\Gf(s)}{\Gf(s+n)}\,
A_n(h,z')\,.\ee
\end{prop}
\begin{proof}
The expansion \eqref{h-expa} for $z'=i$ shows that the hyperfunction
$\poiss s^{-1}u$ has the expansion $\sum_n A_n(h,i) \, \eph_{s,n}$ in
the basis functions in~\eqref{phisn-def}. Then \eqref{poiss-ci} gives
\[ u(z) \= \sum_{n\in \ZZ} (-1)^n \frac{\Gf(s)}{\Gf(s+n)}\,A_n(h,i)\,
P_{\!s,n}(z)\,.\]
This gives the relation in the proposition if $z'=i$, and the general
case follows from the transformation rules \eqref{An-g}
and~\eqref{an-g}.
\end{proof}

The transition $s\leftrightarrow 1-s$ does not change $\E_s=\E_{1-s}$
or $p_{s,n}=p_{1-s,n}$. So the coefficients $a_n(u,z')$ stay the same
under $s\mapsto 1-s$. With the commutative diagram \eqref{P-I-p} we
get:
\begin{cor}\label{cor-canI}
The operator $\can_s \rightarrow \can_{1-s}$ corresponding to
$I_s:\V{-\om}s\rightarrow \V{-\om}{1-s}$ acts on the coefficients in
\eqref{h-expa} by
\[ A_n(I_s h,\,z') \= \frac{(1-s)_{|n|}}{(s)_{|n|}}\, A_n(h,\,z')\,.\]
\end{cor}

We remark that Proposition \ref{prop-canc} can also be used to give an
alternative proof of Corollary~\ref{cor-can-anal},
using~\eqref{p-exp} with $u$ replaced by $u\circ g_{z'}$ to obtain
the analyticity of $a_n(u,z')$ in~$z'$ and then~\eqref{a=A} to
control the speed of convergence in~\eqref{h-expa}.

\subsection{Differential equations for the canonical model and the
sheaf of mixed eigenfunctions}\label{sect-diff-can} The canonical
model provides us with an isomorphic copy $\can_s$ of
$\V{-\om}s\cong\E_s$ inside the flabby hybrid model $\hflab_s$. We
now show that the elements of the canonical model are real-analytic
in both variables jointly, and satisfy first order differential
equations in the variable $z\in \uhp$ with $\z$ as a parameter.

The same differential equations can be used to define a sheaf $\D_s$
on $\proj\CC\times \uhp$. In Proposition~\ref{prop-Dp} and
Theorem~\ref{thm-DKEexs} we describe the local structure of this
sheaf. It turns out that we can identify the space $\vomrig s$ of the
rigid hybrid model with a space of sections of this sheaf of a
special kind. There is a sheaf morphism that relates $\D_s$ to the
sheaf $\E_s:U\mapsto\E_s(U)$ of $\ld_s$-eigenfunctions on~$\uhp$. For
elements of the full space $\E_s=\E_s(\uhp)$ the canonical model
gives sections of $\D_s$ over~$(\proj\CC\sm\proj\RR)\times\uhp$.

\begin{thm}\label{thm-can-DE}Each $h\in \can_s$ and its corresponding
eigenfunction $u=\poiss s h=\Rd sh \in \E_s$ satisfy, for
$\z\in \proj \CC\sm\proj \RR$, $z\in \uhp$, $\z\not\in \{z,\bar z\}$,
the differential equations
\beqabclbl{A-de-glob}
\begin{align}
\label{A-de1-glob}
(z-\bar z)\,\p_z h(\z,z) \;+\;s \,\frac{\z-\bar z}{\z-z}\, \bigl(
h(\z,z) - u(z)
\bigr) &\=0\,, \\
\label{A-de2-glob}
(z-\bar z)\, \p_{\bar z }\bigl( h(\z,z) - u(z) \bigr) \;-\; s\,
\frac{\z-z}{\z-\bar z}\, h(\z,z) &\=0\,.
\end{align}
\eeqabc
Conversely, any continuous function $h$ on
$(\proj\CC\sm\proj\RR)\times\uhp$ that is holomorphic in the first
variable, continuously differentiable in the second variable, and
satisfies the differential equations \eqref{A-de-glob} for some
$u\in C^1(\uhp)$ belongs to $\can_s$, and $u$ is $\poiss sh$.
\end{thm}
The differential equations \eqref{A-de-glob} look complicated, but in
fact are just the $\,dz$- and $\,d\bar z$-components of the identity
\be\label{br-pot} \bigl[ R(\z;\dv)^s,\,u(\z;\dv)\bigr] \= d\bigl(\,
R(\z;\dv)^sh(\z;\dv)\,\bigr)\ \ee
between 1-forms, as one checks easily.
(The function $R(\z;z)^s$ is multivalued, but if we take the same
branch on both sides of the equality, then it makes sense locally.)

\smallskip
\noindent\emph{Proof of Theorem \ref{thm-can-DE}.} The remark just
made shows almost immediately that the function
$h=\ipoiss s u\in\can_s$ defined by \eqref{reprf} satisfies the
differential equations \eqref{A-de-glob}: differentiating
\eqref{reprf} in~$z$ gives
$$ d_z\bigl(h(\z,z)R(\z;s)^s\bigr)\=\begin{cases}
d\bigl(u(z)R(\z;z)^s\bigr)\,-\,[u(z),R(\zeta;z)^s]_z 
&\text{ if $z\in\uhp$,}\\
\phantom{d\bigl(u(z)R(\z;z)^s\bigr)\,}+\,[R(\zeta;z)^s,u(z)]_z 
&\text{ if $z\in\lhp$,}\end{cases} $$
and the right-hand side equals $[R(\zeta;z)^s,u(z)]$ in both cases by
virtue of~\eqref{Grf}.

An alternative approach, not using the explicit Poisson inversion
formula~\eqref{reprf}, is to differentiate~\eqref{pairg} with respect
to~$z_1$
(resp.~$\overline{z_1}$) and then set $z_1=z_2=z$ to see that the
expression on the left-hand side of~\eqref{A-de1-glob}
(resp.~\eqref{A-de2-glob}) is holomorphic in $\z$ near $\proj\RR$.
(Here that we use the result proved above that elements of the
canonical model are analytical in both variables jointly.)
The equations $h(z,z)=u(z)$, $h(\bar z,z)=0$ then show that the
expressions in \eqref{A-de-glob}, for $z$ fixed, are holomorphic
in~$\z$ on all of $\proj\CC$ and hence constant. To see that both
constants vanish, we set $\z=\bar z$ in \eqref{A-de1-glob}
(resp.~$\z=z$ in \eqref{A-de2-glob}) and use
$$ \p_z\bigl(h(\z,z)\bigr)|_{\z=\bar z}=
\p_z\bigl(h(\bar z,z)\bigr)=\p_z(0)=0\,,\qquad
 \p_{\bar z}\bigl(h(\z,z)\bigr)|_{\z=z}=
\p_{\bar z}\bigl(h(z,z)\bigr)=\p_{\bar z}u(z)\,.$$

This proves the forward statement of Theorem~\ref{thm-can-DE}. Instead
of proving the converse immediately, we first observe that the
property of satisfying the differential equations in the theorem is a
purely local one and therefore defines a sheaf of functions.

We now give a formal definition of this sheaf and then prove some
general statements about its local sections that include the second
part of Theorem~\ref{thm-can-DE}.

We note that the differential equations \eqref{A-de-glob} make sense,
not only on $(\proj\CC\sm\proj\RR)\times\uhp$, but on all of
$\proj\CC\times\uhp$, with singularities on the ``diagonal'' and
``antidiagonal'' defined by
\be\label{Dtpm}
 \Dt^+ \= \{\,(z,z)\;:\; z\in \uhp\,\}\,,\qquad\Dt^-\= \{\,(\bar z,z)
 \;:\; z\in \uhp\,\}\,. \ee
We therefore define our sheaf on open subsets of this larger space.
\begin{defn}\label{D-def}For every open subset
$U \subset\proj\CC\times \uhp$ we define $\D_s(U)$ as the space of
pairs $(h,u)$ of functions on $U$ such that
\begin{enumerate}
\item[a)] $h$ and $u$ are continuous on $U$,
\item[b)] $h$ is holomorphic in its first variable,
\item[c)] locally $u$ is independent of the first variable,
\item[d)] $h$ and $u$ are continuously differentiable in the second
variable and satisfy the differential equations \eqref{A-de-glob} on
$U\sm(\Dt^+\cup\Dt^-)$, with $u(z)$ replaced by $u(\z,z)$.
\end{enumerate}
\end{defn}
This defines $\D_s$ as a sheaf of pairs of functions on
$\proj\CC\times\uhp$, the sheaf of \emph{mixed eigenfunctions}. In
this language, the content of Theorem~\ref{thm-can-DE} is that
$\can_s$ can be identified via $h\mapsto(h,\Rd sh)$ with the space of
global sections of
$\D_s\bigl( (\proj\CC\sm\proj\RR)\times\uhp\bigr)$. The following
proposition gives a number of properties of the local sections.

\begin{prop}\label{prop-Dp} Let $(h,u)\in \D_s(U)$ for some open set
$U\subset \proj\CC\times\uhp$. Then
\begin{enumerate}
\item[i)] The functions $h$ and $u$ are real analytic on $U$. The
function $u$ is determined by~$h$ and satisfies
$\Dt u = s(1-\nobreak s) u$.
\item[ii)] If $U$ intersects $\Dt^+\cup\Dt^-$, then we have $h=u$ on
$U\cap\Dt^+$ and $h=0$ on $U\cap\Dt^-$.
\item[iii)] If $u=0$, then the function $h$ locally has the form
$h(\z,z)=\ph(\z)\,R(\z;z)^{-s}$ for some branch of $R(\z;z)^{-s}$,
with $\ph$ holomorphic.
\item[iv)] The function $h$ is determined by $u$ on each connected
component of $U$ that intersects $\Dt^+\cup\Dt^-$.
\end{enumerate}
\end{prop}
\begin{proof}The continuity of $h$ and $u$ allows us to consider them
and their derivatives as distributions. We obtain from
\eqref{A-de-glob} the following equalities of distributions on
$U\sm(\Dt^+\cup\Dt^-)$:
\begin{alignat*}2 \p_z\p_{\bar z} h &\= \p_z \biggl( \p_{ \bar z} u +
\frac s{z-\bar z}\, \frac{\z-z}{\z-\bar z} \, h \biggr) &&\=
\p_z\p_{\bar z} u - \frac s{(z-\bar z)^2}\,h - \frac{s^2}{(z-\bar
z)^2}\,(h-u)\,,\\
\p_{\bar z}\p_z h&\= \p_{\bar z} \biggl( \frac{-s}{z-\bar z}\,
\frac{\z-\bar z}{\z-z}\,(h-u) \biggr)&&\= \frac{-s}{(z-\bar z)^2}\,
(h-u) - \frac{s^2}{(z-\bar z)^2}\, h\,.
\end{alignat*}
The differential operators $\p_z$ and $\p_{\bar z}$ on distributions
commute. In terms of the hyperbolic Laplace operator
$\Dt=(z-\bar z)^2 \, \p_z\p_{\bar z}$, we have in distribution sense
\be \label{Dt-h-u}
\bigl( \Dt -\ld_{s+1}\bigr) \, h \= (\Dt+s^2) u \= s u\,. \ee
Since $u$ is an eigenfunction of the elliptic differential operator
$\Dt-\ld_s$ with real analytic coefficients, $u$ and also $h$ are
real analytic function in the second variable. To conclude that $h$
is real-analytic in both variables jointly, we note that it is also a
solution of the following elliptic differential equation with
analytic coefficients
\[ \bigl(-\p_\z \p_{\bar \z } + \Dt-\ld_{s+1} \bigr) h \= su\,.\]
Near $\infty\in \proj\CC$ we replace $\z$ by $\upsilon=1/\z$ in the
last step.

Since $u$ is locally independent of $\z$, we conclude that $u$ is
real-analytic on the whole of $U$, and satisfies
$\Dt u=s(1-\nobreak s)u$ on $U$. Then \eqref{Dt-h-u} gives the
analyticity of $h$ on $U$. Now we use \eqref{A-de1-glob} to obtain
\[ u(\z,z) \= h(\z,z) + \frac{z-\bar z}s\, \frac{\z-z}{\z-\bar z}
\,\p_z h (\z,z)\,.\]
So $h$ determines $u$ on $U\sm \Dt^-$, and then by continuity on the
whole of~$U$. Furthermore $u=h$ on $\Dt^+$. Similarly,
\eqref{A-de2-glob} implies $h=0$ on $U\cap\Dt^-$. This proves
parts~i) and~ii) of the proposition.


Under the assumption $u=0$ in part~iii) the differential
equations~\eqref{A-de-glob} become homogeneous in~$h$. For fixed $\z$
the solutions are multiples of $z\mapsto R(\z;z)^{-s}$, as is clear
from~\eqref{br-pot}. Hence $h$ locally has the form
$h(\z,z)=\ph(\z)\, R(\z;z)^{-s}$, where $\ph$ is holomorphic by
condition~b) in the definition of~$\D_s$. It also follows that $h$
vanishes on any connected component of $U$ on which $R(\z;z)^{-s}$ is
multi-valued, and in particular on any component that intersects
$\Dt^+\cup\Dt^-$. Part~iv) now follows by linearity.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-can-DE}, converse direction]
Functions $h$ and $u$ with the properties assumed in the second part
of the theorem determine a section
$(h,u) \in \D_s\bigl( (\proj\CC\sm\proj\RR)\times\uhp\bigr)$.
Proposition~\ref{prop-Dp} shows that $u\in \E_s$. By the first part
of the theorem, we have
$(\ipoiss su,u)\in\D_s\bigl((\proj\CC\sm\proj\RR)\times\uhp\bigr)$.
Since this has the same second component as $(h,u)$, part~iv) of the
proposition shows that $h=\ipoiss su\in\can_s$, and then part~ii)
gives $u=\Rd s h=\poiss sh$.
\end{proof}

\sssect{Local description of $h$ near the diagonal} Part~iv) of
Proposition~\ref{prop-Dp} says that the first component of a section
$(h,u)$ of $\D_s$ near the diagonal or antidiagonal is completely
determined by the second component, but does not tell us explicitly
how. We would like to make this explicit. We can do this in two ways,
in terms of Taylor expansions or by an integral formula. We will use
this in~\S\ref{sect-conjectures}

We first consider an arbitrary real-analytic function $u$ in a
neighborhood of a point $z_0\in\uhp$ and a real-analytic solution $h$
of~\eqref{A-de1-glob} near $(z_0,z_0)$ which is holomorphic in the
first variable. Then $h$ has a power series expansion
$ h(\z,z) \= \sum_{n=0}^\infty h_n(z) (\z-z)^n $ in a neighborhood of
$(z_0,z_0)$, and~\eqref{A-de1-glob} is equivalent to the recursive
formulas
$$ h_n(z) \= \begin{cases} u(z) &\text{ if }n=0\,,\\
\dfrac1{1-s}\,\dfrac{\p h_0(z)}{\p z}&\text{ if }n=1\,,\smallskip\\
\dfrac1{n-s}\,\biggl(\dfrac{\p h_{n-1}(z)}{\p z} \,+\, 
\dfrac s{z-\bar z}\,h_{n-1}(z)\biggr)&\text{ if }n\geq 2\,,
\end{cases} $$
which we can solve to get the expansion
\be\label{DD} h(\z,z)\= u(z)\,+\,y^{-s}\,\sum_{n=1}^\infty
\frac{\p^{n-1}}{\p z^{n-1}}\biggl(y^s\,\frac{\p u}{\p
z}\biggr)\;\frac{(\z-z)^n}{(1-s)_n}\;,\ee
where $(1-s)_n=(1-s)(2-s)\cdots(n-s)$ as usual is the Pochhammer
symbol. Conversely, for any real-analytic function $u(z)$ in a
neighborhood of~$z_0$, the series in~\eqref{DD} converges and defines
a solution of~\eqref{A-de1-glob} near $(z_0,z_0)$. Thus there is a
bijection between germs of real-analytic functions $u$ near $z_0$ and
germs of real-analytic solutions of~\eqref{A-de1-glob}, holomorphic
in~$\z$, near~$(z_0,z_0)$. If $u$ further satisfies $\Dt u=\ld_su$,
then a short calculation shows that the function defined
by~\eqref{DD} satisfies~\eqref{A-de2-glob}, so we get a bijection
between germs of $\ld_s$-eigenfunctions $u$ near $z_0$ and the stalk
of $\D_s$ at~$(z_0,z_0)$. An exactly similar argument gives, for any
$\ld_s$-eigenfunction~$u$ near~$z_0$, a unique solution
\be\label{DD-} h(\z,z)\= -\,y^{-s}\,\sum_{n=1}^\infty
\frac{\p^{n-1}}{\p\bar z^{n-1}}\biggl(y^s\,\frac{\p u}{\p\bar
z}\biggr)\; \frac{(\z-\bar z)^n}{(1-s)_n}\,,\ee
of equations~\eqref{A-de1-glob} and~\eqref{A-de2-glob} near the point
$(\bar z_0,z_0)\in\Dt^-$. This proves:
\begin{prop}\label{utoh1} Let $u\in\E_s(U)$ for some open set
$U\subset\uhp$. Then there is a unique section $(h,u)$ of~$\D_s$ in a
neighborhood of
$\{\,(z,z)\mid z\in U\}\cup\{\,(\bar z,z)\mid z\in U\}$, given by
equations~\eqref{DD} and~\eqref{DD-}.
\end{prop}

The second way of writing $h$ in terms of $u$ near the diagonal or
anti-diagonal is based on~\eqref{reprf}. This equation was used to
lift a global section $u\in\E_s$ to a section $(\ipoiss su,u)$
of~$\D_s$ over all of $\bigl(\proj\CC\sm\proj\RR\bigr)\times\uhp$,
but its right-hand side can also be used for functions $u\in\E_s(U)$
for open subsets $U\subset\uhp$ to define $h$ near points $(z,z)$ or
$(\bar z,z)$ with $z\in U$. This gives a new proof of the first
statement in Proposition~\ref{utoh1}, with the advantage that we now
also get some information off the diagonal and anti-diagonal:
\begin{prop}\label{utoh2} If $U$ is connected and simply connected,
then the section $(h,u)$ given in Proposition~\ref{utoh1} extends
analytically to $\bigl(U\cup\bar U\bigr)\times U$.
\end{prop}

\sssect{Formulation with sheaves} Proposition~\ref{prop-Dp} shows that
the component $h$ of a local section $(h,u)$ of $\D_s$ determines the
component $u$, which is locally independent of the second variable
and satisfies the Laplace equation. So there is a map from sections
of $\D_s$ to sections of $\E_s$. To formulate this as a sheaf
morphism we need to have sheaves on the same space. We denote the
projections from $\proj\CC\times \uhp$ on $\proj\CC$, respectively
$\uhp$, by $p_1$, We use the inverse image sheaf $p_2^{-1}\E_s$ on
$\proj\CC\times\uhp$, associated to the presheaf
$U \mapsto \E_s(p_2 U)$. (See, e.g., \S1, Chap.~II, in~\cite{Har-p}.)
The map $p_2$ is open, so we do not need a limit over open
$V \supset p_2 U$ in the description of the presheaf. Note that the
functions in $\E_s(p_2 U)$ depend only on~$z$, but that the
sheafification of the presheaf adds sections to $p_2^{-1}\E_s$ that
may depend on the first variable. In this way, $(h,u) \mapsto u$
corresponds to a sheaf morphism $C:\D_s \to p_2^{-1}\E_s$. We call
the kernel~$\K_s$.

We denote the sheaf of holomorphic functions on $\proj\CC$ by $\hol$.
Then $p_1^{-1}\hol$ is also a sheaf on $\proj\CC\times \uhp$. The
following theorem describes $\K_s$ in terms of $p_1^{-1}\hol$ and
show that the morphism $C$ is surjective.

\begin{thm}\label{thm-DKEexs}
The sequence of sheaves on $\proj\CC\times\uhp$
\be 0\lra \K_s \lra \D_s\stackrel C\lra p_2^{-1}\E_s\lra 0\ee
is exact. If a connected open set $U\subset\proj\CC\times\uhp$
satisfies $U\cap (\Dt^+\cup\Dt^-)\neq\emptyset$, then
$\K_s(U)=\{0\}$. The restriction of $\K_s$ to $\left( \proj\CC
\times\uhp\right) \sm\left( \Dt^+\cup\Dt^-\right)$ is locally
isomorphic to $p_1^{-1}\hol$ where holomorphic functions $\ph$
correspond to $(\z,z) \mapsto \bigl( \ph(\z)\,R(\z;z)^{-s} ,0\bigr)$.
The inductive limit of $ \K_s(U)$ over all neighborhoods~$U$ of
$\proj\RR\times\uhp$ in $\proj\CC\times\uhp$ is canonically
isomorphic to the space $\vomrig s$.
\end{thm}

\begin{proof}For the exactness, we only have to check the surjectivity
of $C:\D_s \lra p_2^{-1}\E_s$. For this we have to verify that for
any point $P_0=(\z_0,z_0)\in \proj\CC\times\uhp$, any solution of
$\Delta u=\lambda_su$ lifts to a section $(h,u)\in\D_s(U)$ for some
sufficiently small neighborhood $U$ of $P_0$. If
$P_0\in \Dt^+\cup\Dt^-$, then this is precisely the content of the
first statement of Proposition~\ref{utoh1}. If
$P_0\not\in \Dt^+\cup\Dt^-$, then we define $h$ near $P_0$ by the
formula
\be\label{h-int-c}
h(\z,z)\= \int_{z_0}^z \bigl[ \bigl( R_\z(\dv)/R_\z(z)
\bigr)^s,u\bigr]\ee
\eeqabc
instead, again with $\bigl( R_\z(z_1)/R_\z(z) \bigr)^s=1$ at $z_1=z$.
The next two assertions of the theorem follow from
Proposition~\ref{prop-Dp}. The relation with the rigid hybrid model
is based on~\eqref{pairhpf}.
\end{proof}

We end this section by making several remarks about the
equations~\eqref{A-de-glob} and their solution spaces $\can_s$ and
$D_s(U)$.

The first is that there are apparently very few solutions of these
equations that can be given in ``closed form." One example is given
by the pair $h(\z,z)= \frac{\z-\bar z}{2i}y^{-s}$, $u(z) = y^{1-s}$
(cf.~\eqref{canrep-y-s}). Of course one also has the translations of
this by the action of~$G$, and in Example~\ref{exPol} after
Theorem~\ref{ansatz1} we will give further generalizations where $h$
 is still a polynomial times $y^{-s}$. One also has the local
 solutions of the form $(\ph(\z)R(\z;z)^{-s},0)$ for arbitrary
holomorphic functions $\ph(\z)$, as described in
Theorem~\ref{thm-DKEexs}.

The second observation is that the description of $\can_s$ in terms of
differential equations can be generalized in a very simple way to the
space $\can_s^+$ of semicanonical hyperfunction representatives
introduced in the Remark in \S\ref{sect-pm}: these are simply the
functions $h$ on $\bigl(\proj\CC\sm\nobreak\proj\RR\bigr)\times\uhp$
that satisfy the system of differential equations
\be
\begin{aligned}
 \, \p_z \bigl(h(\z,z) -u_-(z) \bigr) &\= - s \, \frac{\z-\bar
z}{(z-\bar z)(\z-z)}\, \bigl(h(\z,z) - u_+(z) \bigr)\,,\\
 \p_{\bar z} \bigl(h(\z,z)-u_+(z) \bigr) &\= \phantom{-} s\,
\frac{\z-z}{(z-\bar z)(\z-\bar z)} \, \bigl(h(\z,z)-u_-(z) \bigr)
\end{aligned}
\ee
for some function $u_+$ and $u_-$ of $z$ alone. This defines a sheaf
$\D_s^+$ which projects to $\D_s$ by $(h,u_+,u_-)\mapsto(h,u_+-u_-)$,
and we have a map from $\can_s^+$ to the space of global sections of
$\D_s^+$ defined by $h\mapsto(h,\Rd s^+h,\Rd s^-h)$ with $\Rd s^\pm$
defined as in~\eqref{Rdpm}. In some ways $\can_s^+$ is a more natural
space than $\can_s$, but we have chosen to normalize once and for all
by $u_-(z)=0$ in order to have something canonical.

The third remark concerns the surjectivity
of~$C:\D_s\to p_2^{-1}\E_s$. We know from Theorem~\ref{thm-DKEexs}
that any solution~$u$ of the Laplace equation can be completed
locally to a solution $(h,u)$ of the differential
equations~\eqref{A-de-glob}. We now show that such a lift does not
necessarily exist for a~$u$ defined on a non-simply connected subset
of~$\uhp$. Specifically, we will show that there is no section of
$\D_s$ of the form $(h,q_{1-s}(z,i))$ on any open set
$U\subset\proj\CC\times\uhp$ whose image under~$p_2$ contains a
hyperbolic annulus with center~$i$.

Now the disk model is more appropriate. We work with coordinates
$\x=i\frac{1+\z}{1-\z} \in \CC^\ast$ and
$w=i\frac{z-i}{z+i} \in \disk$. The differential equations
\eqref{A-de1-glob} and \eqref{A-de2-glob} take the form
\beqabc
\begin{align}
\label{De1p}
(1-r^2)\,\p_w h + s \,\frac{1-\bar w \x}{\x-w}\,(h-u)&\=0\,,\\
\label{De2p}
(1-r^2)\,\p_{\bar w}(h-u) + s \,\frac{\x-w}{1-\bar w \x}\, h&\=0\,,
\end{align}
with $r=|w|$, and \eqref{br-pot} becomes
\be \label{br-potp}
\bigl[ R^\ci(\x;\dv)^s , u(\x,\dv)\bigr] \= d \bigl( R^\ci(\x;\dv)\,
h(\x,\dv)\bigr)\,, \ee
\eeqabc
with the Poisson kernel $R^\ci$ in the circle model, as
in~\eqref{Rcirc-p}.

\begin{prop}\label{prop-noq} Let $A\subset\disk$ be an annulus of the
form $r_1<|w|<r_2$ with $0\leq r_1<r_2\leq 1$, and let
$V \subset\CC^\ast$ be a connected open set that intersects the
region $r_1<|\x|<r_1^{-1}$in~$\CC^\ast$. Then $\D_s(V\times A)$ does
not contain sections of the form $(h,Q_{1-s,n})$ for any $n\in \ZZ$.
\end{prop}
\begin{proof}
Suppose that such a section $(h,Q_{1-s,n})$ exists. Take
$\rho \in (r_1,r_2)$ such that~$V$ intersects the annulus
$A_\rho=\{\rho<|\x|<\rho^{-1}\}$. Let $C$ be the contour $|w|=\rho$.
Then the function~$f$ given by
\[ f(\xi) \= \int_C \bigl[ R^\ci(\x;\dv)^s,Q_{1-s,n}\bigr] \]
is defined and holomorphic on $A_\rho$. For $\x\in V\cap A_\rho$ we
know from~\eqref{br-potp} that the closed differential form
$\bigl[R^\ci(\x;\dv)^s,Q_{1-s,n}]$ on~$A$ has a potential. Hence
$f(\x)=0$ for $\x\in V\cap A_\rho$, and then $f=0$ in $A_\rho$. In
particular, $f(\x)=0$ for $\x\in\uc$. In view of \eqref{p-exp} this
implies that the expansion
$R^\ci(\x;\dv)^s = \sum_{m\in\ZZ} a_m(\x)\,P_{\!s,m}$
(with $\x\in \uc$) satisfies $a_{-n}(\x)=0$. The function
$R^\ci(\x;\dv)^s$ is the Poisson transform
$\poiss{1-s}\,\dt_{1-s,\x}$ of the distribution
$\dt_{1-s,\x}: \ph^\ci \mapsto \nobreak \ph^\ci(\x)$ on $\V\om{1-s}$.
This delta distribution has the expansion
$\dt_{1-s,\x}= \sum_{m\in \ZZ} \x^{-m}\eph_{1-s,m}$. Hence
$R^\ci(\x;\dv)^s 
= \sum_{m\in \ZZ} \x^{-m} \, \frac{(-1)^m\Gf(1-s)}{\Gf(1-s+m)}
P_{\!1-s,m}$, in which all coefficients are non-zero. Since
$P_{\!1-s,m}=P_{\!s,m}$, this contradicts the earlier
conclusion.\end{proof}

This non-existence result is a monodromy effect. In a small
neighborhood of a point $(\x_0,w_0)\in\uc\times A$ we can construct a
section $(h,Q_{1-s,n})$ of $\D_s$ as in~\eqref{h-int-c}:
\be \label{hwi} h(\x,w) \= \int_{w_0}^w \bigl[ \bigl(
R^\ci(\x;w')/R^\ci(\x;w)\bigr)^s, Q_{1-s,n}(w')\bigr]_{w'} \,.\ee
If we now let the second variable go around the annulus~$A$, then
$h(\x,w)$ is changed to $h(\x,w)+h_0(\x,w)$, where $h_0$ is defined
by the same integral as~$h$ but with the path of integration being
the circle $|w'|=|w_0|$. Using
 \[ \bigl[ \bigl( R^\ci(\x;w')/R^\ci(\x;w)\bigr)^s,
 Q_{1-s,n}(w')\bigr]_{w'} \= R^\ci(\x;w)^{-s} \;\bigl[
 R^\ci(\x;w')^s,Q_{1-s,n}(w')\bigr]_{w'}\]
and the absolutely convergent expansion $R^\ci(\x;w')^s = 
\sum_{m\in \ZZ}\xi^{-m}\,\frac{(-1)^m \,\Gf(1-s)}{\Gf(1-s+m)} 
\allowbreak P_{s,m}(w')$ from the proof above, we find from the
explicit potentials in Table~\ref{tab-Grpot} in~\S\ref{sect-pot} that
only the term $m=-n$ contributes and that $h_0$ is given by
\be h_0(\x,w)\= \pi i\,\frac{(-1)^n\, \Gf(1-s)}{\Gf(1-s-n)}\,
R^\ci(\x;w)^{-s}\,. \ee
(Here we have also used~\eqref{Grf} to replace $[\ ,\ ]$ by
$\{\ ,\ \}$.)
\bigskip




\section{Eigenfunctions near $\p\H$ and the transverse Poisson
transform}\label{sect-gb}
The space $\E_s$ of $\lambda_s$-eigenfunctions of the Laplace operator
embeds canonically into the larger space $\F s$ of germs of
eigenfunctions near the boundary of~$\H$. In~\S\ref{sect-speg} we
introduce the subspace $\W\om s$ of $\F s$ consisting of
eigenfunction germs that have the behavior
$y^s \times(\text{analytic across $\RR$})$ near $\RR$, together with
the corresponding property near~$\infty\in \proj\RR$, and show that
$\F s$ splits canonically as the direct sum of $\E_s$ and $\W\om s$.
In \S\ref{sectbg} the space $\W\om s$ is shown to be isomorphic to
$\V\om s$ by integral transformations, one of which is called the
{\it transverse Poisson transformation} because it is given by the
same integral as the usual Poisson transformation
$\V\om s\rightarrow\E_s$, but with the integral taken across rather
than along $\proj\RR$. \alsochanged{}This transformation gives
another model $\W\om s$ of the principal series representation
$\V \om s$, which has proved to be extremely useful in the
cohomological study of Maass forms in~\cite{BLZ-p}. In
\S\ref{sect-dual-p} we describe the duality of $\V\om s$ and
$\V{-\om}{1-s}$ in \eqref{dual} in terms of a pairing of the
isomorphic spaces $\W\om s$ and $\E_{1-s}$. In \S\ref{sect-winf} we
construct a smooth version $\W\infty s$ of $\W\om s$ isomorphic to
$\V\infty s$ by using jets of $\ld_s$-eigenfunctions of the Laplace
operator. This space is also used in~\cite{BLZ-p}.

\subsection{Spaces of eigenfunction germs}\label{sect-speg}
Let $\F s$ be the space of germs of eigenfunctions of $\Dt$, with
eigenvalue $\ld_s=s(1-s)$, near the boundary of $\H$, i.e.,
\be\label{F-def} \F s\ = \limind U \E_s(U\cap\uhp)\,, \ee
where the direct limit is taken over open neighborhoods $U$ of $\H$ in
$\proj\CC$ (for either of the realizations $\uhp\subset\proj\CC$ or
$\disk\subset\proj\CC$). This space canonically contains $\E_s$
because an eigenfunction in $\uhp$ is determined by its values near
the boundary (principle of analytic continuation). The action of $G$
in $\F s$ is by $f\,|\,g(z) = f(gz)$. The functions $Q_{s,n}$ and
$Q_{1-s,n}$ in~\eqref{LegQ} represent elements of~$\F s$ not lying
in~$\E_s$. Clearly we have $\F {1-s}=\F s$.

Consider $u,v\in \F s$, represented by elements of $\E_s(U\cap\uhp)$
for some neighborhood $U$ of $\p\H$ in~$\proj\CC$. Then the Green's
form $[u,v]$ is defined and closed \change{and closed inserted}in
 $U$, and for a positively oriented closed path $C$ in $U$ which is
homotopic to $\p\H$ in $U\cap(\H\cup\p\H)$, the integral
\be\label{bt-def} \bt(u,v) \= \frac1{\pi i}\int_C[u,v] \= \frac 2\pi
\int_C\{u,v\}\ee
is independent of  the choice of~$C$ or of the set $U$ on
which the representatives of $u$ and $v$ are defined. This defines a
$G$-equivariant antisymmetric bilinear pairing
\be \bt:\F s\times\F s \lra \CC\,.\ee
If both $u$ and $v$ are elements of $\E_s$ we can contract $C$ to a
point, thus arriving at $\bt(u,v)=0$. Hence $\bt$ also induces a
bilinear pairing $\E_s \times
(\F s/\E_s) \lra\CC$.

For each $z\in \H$, the element $q_s(\dv,z)$ of $\F s$ is not in
$\E_s$. By $(\Pi_s u) (z) = \bt\bigl(u, q_s(\dv,z) \bigr)$ we define
a $G$-equivariant linear map $\Pi_s:\F s \lra \E_s$. Explicitly,
$u_{\mathrm{in}}(z):=\Pi_s u(z)$ is given by an integral
$\frac1{\pi  i} \int_C[u(z'),q_s(z',z)]_{z'}$, where $z$ is inside
the path of integration~$C$. By deforming $C$ we thus obtain
$u_{\mathrm{in}}(z)$ for all $z\in \uhp$, so
$u_{\mathrm{in}} \in \E_s$. We can also define
$u_{\mathrm{out}}(z) := \frac{-1}{\pi  i} \int_C[u(z'),q_s(z',z)]_{z'}$
\change{Minus sign in definition of $u_{\mathrm{out}}$.}%
where now $z$ is between the boundary of~$\H$ and the path of
integration. For $u\in \E_s$ we see that $u_{\mathrm{out}}=0$. 
More generally, 
Theorem~\ref{thm-psC-p} shows that
\begin{equation}\label{u-decomp}
 u\= u_{\mathrm{out}} +
u_{\mathrm{in}} \qquad\qquad(\forall\,u\in \F s)\,.\end{equation}
The $G$-equivariance of $[\cdot,\cdot]$ implies that the maps 
$\Pi_s$ and $1-\Pi_s$ 
are $G$-equivariant. This gives the following result.\change{The present
propositions \eqref{prop-EFexact} and \ref{propsplit} formed one proposition.}
\begin{prop}\label{prop-EFexact} The $G$-equivariant maps
$\Pi_s:u\mapsto u_{\mathrm{in}}$ and $1-\Pi_s:
u\mapsto u_{\mathrm{out}}$ split the exact sequence of $G$-modules
\be\label{EF-ex} \xymatrix@C=1.4cm{ 0 \ar[r]
& \E_s \ar[r]
& \F s \ar[r] \ar@/_/[l]_{\Pi_s}
& \F s/\E_s \ar[r] \ar@/_/[l]_{1-\Pi_s}
& 0 }\ee
\end{prop}\smallskip

We now define the subspace $\W\om s$ of~$\F s$. It is somewhat easier
in the disk model:
\begin{defn}\label{Wom-def-p}The space $\W\om s$ consists of those
boundary germs $u\in \F s$ that are of the form
\[ u(w) \= 2^{-2s} (1-|w|^2)^s A^\ci(w)\]
where $A^\ci$ is a real-analytic function on a two-sided neighborhood
of $\uc$ in~$\proj\CC$.\end{defn}

In other words, representatives of elements of $\W\om s$, divided by
the factor $(1-\nobreak|w|)^s$, extend analytically across the
boundary~$\uc$.
(The factor $2^{-2s}$ is included for compatibility with other
models.)

\smallskip
The next proposition shows that $\W\om s$ is the canonical direct
complement of $\E_s$ in~$\F s$.

\begin{prop}\label{propsplit}
The \change{simpler formulation}kernel of
$\Pi_s: \F s \rightarrow\E s$ is equal to the space $\W\om s$, and we
have the direct sum decomposition  of $G$-modules 
\be\label{FEW} \F s \= \E_s \oplus \W\om s\,, \ee
given by $u\leftrightarrow (u_{\mathrm{in}},u_{\mathrm{out}})$.
\end{prop}
Notice that all the spaces in the exact sequence~\eqref{EF-ex} are the
same for $s$ and $1-s$, but that $\Pi_s$ and $\Pi_{1-s}$ give
different splittings and that $\W\om{1-s}\neq \W\om s$ (for
$s \neq\frac12$).
\begin{proof}
In view of Proposition~\ref{prop-EFexact} it  remains to
show that $\W\om s$ is equal to the image of
$u\mapsto u_{\mathrm{out}}$.

The asymptotic behavior of $Q_{s-1}$ in~\eqref{qasb} gives for $w'$ on
the path of integration~$C$ and $w$ outside~$C$ in the definition of
$u_{\mathrm{out}}(w)$
\[ q_s(w,w') \= \Bigl(\frac 2{\rho^\disk(w',w)+1}\Bigr)^s
\;  f_s\Bigl(
\frac2{\rho^\disk(w',w)+1}\Bigr)\,,\]
where $f_s$ is analytic at~$0$. With \eqref{Ddist}:
\[ \frac2{\rho(w',w)+1} \= \frac{
(1-|w'|)^2}{|w-w'|^2+(1-|w'|^2)(1-|w|^2)}\;
(1-|w|^2)\,.\]
We conclude that if $w'$ stays in the compact set $C$, and $w$ tends
to $\uc$, we have
\[ u_{\mathrm{out}}(w) \= (1-|w|^2)^s \;\bigl( \text{analytic function
of $1-|w|^2$})\,.\]
So $u_{\mathrm{out}}\in \W\om s$.

For the converse inclusion it suffices to show that
$\E_s\cap\W\om s=\{0\}$. This follows from the next lemma, which is
slightly stronger than needed here.
\end{proof}
\begin{lem}\label{lem-Ebd0}\change{Formulation improved}Let $u$ be a solution of
$\Dt u=\ld_s u$  on some annulus $1-\dt\leq|w|^2<1$ with
$\dt>0$. Suppose that $u$ is of the form 
\be\label{est} u(w) \= (1-|w|^2)^s A(w) + \oh \bigl( (1-|w|^2) ^{s+1}
\bigr)\,,\ee
with a continuous function $A$ on  the closed annulus
$1-\dt\leq|w|^2\leq1$.  Then $u=0$.
\end{lem}
\begin{proof}On the annulus the function $u$ is given by its polar
Fourier series, with terms
\[ u_n(w) \= \int_0^{2\pi} e^{-2in\th} f(e^{i\th}w)\,
\frac{d\th}{2\pi}\,.\]
Each $u_n$ satisfies the estimate \eqref{est}, with $A$ replaced by
its Fourier term $A_n$. Moreover, the $G$-equivariance of $\Dt$
implies that $u_n $ is a $\ld_s$-eigenfunction of~$\Dt$. It is the
term of order $n$ in the expansion~\eqref{p-exp}. In particular,
$u_n$ is a multiple of $P_{\!s,n}$. In \S\ref{sect-Leg} we see that
$P_{\!s,n}$ has a term $(1-\nobreak |w|^2)^{1-s}$ in its asymptotic
behavior near the boundary, or a term
$(1-\nobreak |w|^2)^{1/2}\log(1-\nobreak |w|^2)$ if $s=\frac12$. So
$u_n$ can satisfy \eqref{est} only if it is zero.
\end{proof}
\noindent
\emph{Remark. }The proof of the lemma  gives the stronger assertion: If
$u\in \F s$ satisfies \eqref{est}, then $u\in \W\om s$ and
$\Pi_s u=0$.\medskip


Returning to the definition of~$\W\om s$, we note that the action $g:
w\mapsto \frac{A w+B}{\bar B w+\bar A}$ in~$\disk$ gives for the
function~$A^\ci$
\be\label{trfcimoanal} A^\ci | g (w) \= \bigl|\bar B w + \bar A|^{-2s}
A^\ci\Bigl( \frac{Aw+B}{\bar B w+\bar A}\Bigr)\,, \ee
first for $w\in \disk$ near the boundary, and by real-analytic
continuation on a neighborhood of $\uc$ in~$\proj\CC$. On the
boundary, where $|w|=1$, this coincides with the action of~$G$ in the
circle model, as given in~\eqref{trfcimo-p}. In~\eqref{trfcihmo} the
action in the circle model of~$\V\om s$ is extended to holomorphic
functions on sets in~$\proj\CC$. That action and the action
in~\eqref{trfcimoanal} coincide only on~$\uc$, but are different
elsewhere. This reflects that $A^\ci$ is real-analytic, but not
holomorphic.

The restriction of $A^\ci$ to~$\uc$ induces the \emph{restriction map}
\be \label{rhodef} \rho_s:\W\om s\lra \V\om s\,,\ee
which is $G$-equivariant.


Examples of elements of $\W\om s$ are the functions $Q_{s,n}$,
represented by elements of $\E_s(\disk\sm\{0\})$, whereas the
functions $Q_{1-s,n}$ belong to $\F s$ but not to $\W\om s$.

We note that the factor $2^{-2s}(1-\nobreak|w|^2)^s$ corresponds to
$\bigl( \frac y{|z+i|^2}\bigr)^s$ on the upper half plane. So in the
upper half plane model the elements of $\W\om s$ are represented by
functions of the form $u(z) = \bigl( \frac
y{|z+i|^2}\bigr)^s A^\prj(z)$ with $A^\prj$ real-analytic on a
neighborhood of $\proj\RR$ in~$\proj\CC$. The transformation behavior
for $A^\prj$ turns out to coincide on~$\proj\RR$ with the action
of~$G$ in the projective model of~$\V\om s$ in~\eqref{projac-p}.
Outside $\proj\RR$ it differs from the action in~\eqref{grotesque-p}
on holomorphic functions. The restriction map~$\rho_s:
\W\om s\rightarrow\V\om s$ is obtained by
$u\mapsto A^\prj|_{\proj\RR}$.

In the line model we have $u(z) = y^s A(z)$ near~$\RR$ and
$u(z) = (y/|z|^2)^s A^\infty(-1/z)$ near~$\infty$, with $A$ and
$A^\infty$ real-analytic on a neighborhood of~$\RR$ in~$\CC$. The
action on $A$ is given by
\be \label{A-action} A\,\Big|\,\matc abcd(z) \= |cz+d|^{-2s}\, A\Bigl(
\frac{az+b}{cz+d}\Bigr)\,, \ee
coinciding on~~$\RR$ with the action in the line model. Restriction of
$A$ to~$\RR$ induces the description of~$\rho_s$ in the line model.
The factors $2^{-2s}(1-\nobreak|w|^2)^s$,
$\bigl( y/|z+\nobreak i|^2\bigr)^s$, $y^s$ and $\bigl(y/|z|\bigr)^s$
have been chosen in such a way that $A^\ci$, $A^\prj$, $A$ and
$A^\infty$ restrict to elements of the circle, projective and line
models, respectively, of $\V\om s$, related by \eqref{trfcimo-p}
and~\eqref{line-proj}.
\smallskip

The space $\W\om s$ is the space of global sections of a sheaf, also
denoted $\W\om s$, on~$\p\H$, where in the disk model $\W\om s(I) $
for an open set $I$ in $\uc$ corresponds to the real-analytic
functions~$A^\ci$ on a neighborhood of $I$ in $\proj\CC$ such that
$(1-\nobreak|w|^2)^s A^\ci(w)$ is annihilated by $\Dt-\ld_s$.
Restriction gives $\rho_s : \W\om s(I) \lra \V\om s(I)$ for each
$I\subset\uc$. In the line model, $\W\om s(I)$ for $I\subset \RR$ can
be identified with the space of real-analytic functions $A$ on a
neighborhood $U$ of $I$ with
$y^s A(z)\in \ker\left( \Dt-\ld_s\right)$ on $U\cap\uhp$; for
$I\sm \proj\RR\sm\{0\}$ we use $(y/|z|^2)^s A^\infty(-1/z)$. The
function $z\mapsto y^s$ is an element of $\W\om s(\RR)$, but not of
$\W\om s=\W\om s(\proj\RR)$.

Another example is the function $z\mapsto y^{1-s}$, which represents
an element of $\W\om {1-s}(\RR)$, but not of $\W\om {1-s} = \W\om
{1-s}
(\proj\RR)$. It is the Poisson transform of the distribution
$\dt_{s,\infty}$, which has support $\{\infty\}$.

The \emph{support} $\supp(\al)$ of a hyperfunction $\al\in \V{-\om}s$
is the smallest closed subset $X$ of $\p\H$ such that each
$g\in \HHH_s$ representing~$\al$ extends holomorphically to a
neighborhood of $\p\H \sm X$.
\begin{prop}\label{prop-W-supp}The Poisson transform of a
hyperfunction $\al \in \V{-\om}s$ represents an element of
$\W\om{1-s} \bigl( \p\H \sm \supp(\al)\bigr)$.
\end{prop}
This statement is meaningful only if $\supp(\al)$ is not the whole
of~$\p\H$. In Theorem~\ref{thm-W-supp} we will continue the
discussion of the relation between support of a hyperfunction and the
boundary behavior of its Poisson transform.
\begin{proof}Let $g\in \HHH_s$ be a representative of
$\al\in \V{-\om}s$. In the Poisson integral in \eqref{poiss-ci} we
can replace the integral over $C_+$ and $C_-$ by the integral
\be \poiss s\al(w) \= \frac{(1-|w|^2)^{1-s}}{2\pi i} \int_C g(w)
\bigl(
(1-w/\x)(1-\bar w \x)\bigr)^{s-1}\, \frac{d\x}\x\,, \ee
where $C$ is a path inside the domain of $g$ encircling $\supp(\al)$.
For $w$ outside $C$ the integral defines a real analytic function on
a neighborhood of $\p\disk$, so there the boundary behavior is
$(1-\nobreak |w|^2)^{1-s}\times\,(\text{analytic})$. Adapting $C$, we
can arrange that any point of $\p\disk \sm \supp(\al)$ is inside this
neighborhood.
\end{proof}


\sssect{Decomposition of eigenfunctions} \alsochanged{Notation $A$ for
the annulus changed to $R$.}We close this subsection by generalizing
the decomposition~\eqref{u-decomp} from $\F s$ to $\E_s(R)$, where
$R$ is any annulus $0\leq r_1<|w|<r_2\leq 1$ in~$\disk$. For
$u\in \E_s(R)$ we define
\[ u_{\mathrm{in}} \in \E_s\bigl( \{ |w|<r_2\}\bigr)\quad\text{ and }
u_{\mathrm{out}}\in \E_s\bigl(\{ |w|>r_1\}\bigr)\,,\]
by $u_{\mathrm{in}}(z)=\frac1{\pi i}\int_{C}[u,q_s(\dv,z)]$ and
$u_{\mathrm{out}}(z)=\frac{-1}{\pi i}\int_{C}[u,q_s(\dv,z)]$, 
where
$C\subset R$ is a circle containing the argument of $u_{\mathrm{in}}$
in its interior, respectively the argument of $u_{\mathrm{out}}$ in
its exterior. Then \eqref{u-decomp} holds in the annulus~$R$.
Explicitly, any $u\in \E_s(R)$ has an expansion of the form
\be u\= \sum_{n\in \ZZ}\left( a_n Q_{s,n}+b_n
P_{s,n}\right)\qquad\text{on }r_1<|w|<r_2 \,,\ee
and $u_{\mathrm{in}}$ and $u_{\mathrm{out}}$ are then given by
\be u_{\mathrm{in}} \= \sum_{n\in \ZZ} b_n P_{s,n}\,,\qquad
u_{\mathrm{out}} \=\sum_{n\in \ZZ} a_n Q_{n,s}\,. \ee

\subsection{The transverse Poisson map}\label{sectbg}
In the last subsection we defined restriction maps
$\rho_s: \W\om s\to\V\om s$, and more generally
$\W\om s(I)\to \V\om s(I)$. We now show that these restriction maps
are isomorphisms and construct the explicit inverse maps. We in fact
give two descriptions of $\rho_s^{-1}$, one in terms of power series
and one defined by an integral transform (transverse Poisson map);
the former is simpler, and also applies in the $C^\infty$ setting
(treated in \S\ref{sect-winf}), while the latter
(which is motivated by the power series formula) gives a much stronger
statement in the context of analytic functions. \change{Reformulated.
DON do you agree? Don was right, and did not.}



\sssect{Power series version} Let $u\in\W\om s(I)$, where we work in
the line model and can assume that $I\subset\RR$ by locality. Write
$z$ as $x+iy$ and for $x\in I$ expand the real-analytic function $A$
such that $u(z)=y^s A(z)$ as a power series
$\sum_{n=0}^\infty a_n(x)y^n$ in $y$, convergent in some neighborhood
of $I$ in~$\CC$. By definition, the constant term $a_0(x)$ in this
expansion is the image $\ph=\rho_s(u)$ of $u$ under the restriction
map. The differential equation $\Dt u=\ld_su$ of~$u$ translates into
the differential equation
\be\label{deq-b}
 y\,\bigl( A_{xx}\,+\,A_{yy}\bigr)\,+\,2s\,A_y\=0\,.\ee
Applying this to the power-series expansion of~$A$ we find that
\[ a_{n-2}''(x)\;+\;n(n+2s-1)\,a_n(x)\=0 \]
for $n\ge2$, and that $a_1\equiv0$. Together with the initial
condition $a_0=\ph$ this gives
\be a_n(x)\=\begin{cases}
 \dfrac{(-1/4)^k\,\Gf(s+\frac12)}{k!\,\Gf(k+s+\frac12)}\,\ph^{(2k)}(x)
  &\text{ if $n=2k$\,,} \\ \qquad 0 &\text{ if $2\nmid n$\,,}
  \end{cases}\label{Ansatz}\ee
and hence a complete description of $A$ in terms of~$\ph$. Conversely,
if $\ph$ is any analytic function in a neighborhood of $x\in\RR$,
then its Taylor expansion at~$x$ has a positive radius of convergence
$r_x$ and we have $\ph^{(n)}(x)= \oh(n!\,c^n)$ for any $c>r_x^{-1}$.
{}From Stirling's formula or the Legendre duplication formula we see
that $4^{-k}/k!\,\Gf(k+s+\frac12)= \oh\bigl(k^{-\re(s)}/(2k)!\bigr)$,
so the power series $\sum_{n\ge0}a_n(x)y^n$ with $a_n(x)$ defined
by~\eqref{Ansatz} converges for $|y|<r_x\,$. By a straightforward
uniform convergence argument, the function $A(x+iy)$ defined by this
power series is real-analytic in a neighborhood of~$I$, and of course
it satisfies the differential equation $A_{xx}+A_{yy}+2sy^{-1}A_y=0$,
so the function $u(z) = y^s A(z)$ is an eigenfunction of~$\Dt$ with
eigenvalue~$\ld_s$. This proves:
\begin{thm} \label{ansatz1} Let $I$ be an open subset of $\RR$. Define
a map from analytic functions on~$I$ to the germs of functions on a
neighborhood of $I$ in~$\CC$ by
\be\label{tpoiss-series} (\tpoiss s \ph)(x+iy) \= y^s
\sum_{k=0}^\infty \frac{\ph^{(2k)}(x)}{k!\;
\bigl(s+\frac12\bigr)_k}\, (-y^2/4)^k \,, \ee
with the Pochhammer symbol $\bigl(\frac12+s\bigr)_k =
\prod_{j=0}^{k-1}\bigl(\frac12+s+j\bigr)$. Then $\tpoiss s$ is an
isomorphism from $\V\om s(I)$ to $\W\om s(I)$ with inverse~$\rho_s$.
\end{thm}
\noindent Of course, we can now use the $G$-equivariance to deduce
that the local restriction map $\rho_s:\W\om s(I)\to\V\om s(I)$ is an
isomorphism for every open subset $I\subset\proj\RR$ and that the
global restriction map $\rho_s:\W\om s\to\V\om s$ is an equivariant
isomorphism. The inverse maps, which we still denote $\tpoiss s$, can
be given explicitly in a neighborhood of infinity using the functions
$\ph_\infty$ and $A^\infty$ as usual for the line model or by the
corresponding formulas in the circle model. The details are left to
the reader.

\begin{examples}
\exampl\label{exConst} Take $\ph(x)=1$. Then \eqref{tpoiss-series}
gives $\tpoiss s \ph(z)=y^s$ in $\W\om s(\RR)$. More generally, if
$\ph(x)=e^{i\al x}$ with $\al\in\RR$, then $\tpoiss s\ph$ is the
function $i_{s,\al}$ defined in \eqref{i-def}.

\smallskip
\exampl\label{exPol} We can generalize~\ref{exConst} from $\ph=1$ to
arbitrary polynomials:
\be\label{ExPol} \ph(x)\=\binom{-2s}m \,x^m\quad \Rightarrow\quad
\tpoiss s\ph(z)\=y^s\,\sum_{k+\ell=m}\binom{-s}{
  k}\binom{-s}{\ell}\,z^k\,\bar z^\ell\;. \ee
This can be checked either from formula~\eqref{tpoiss-series} or,
using the final statement of Theorem~\ref{ansatz1}, by verifying that
the expression on the right belongs to~$\E_s$ and that its quotient
by~$y^s$ is analytic near~$\RR$ and restricts to
$\binom{-2s}{ m} x^m\,$ when $y=0$.

\smallskip
\exampl\label{exRa} Let $a\in \CC\sm I$ . Then \eqref{tpoiss-series}
and the binomial theorem give
\be\label{exRaEq}
\ph(x)=(x-a)^{-2s}\quad\Rightarrow\quad \tpoiss s\ph(z)=
y^s\,\sum_{k=0}^\infty\;\binom{-s}{ k}
\,\frac{y^{2k}}{(x-a)^{2s+2k}}\= R(a;z)^s\,, \ee
(Here the branches in $(x-a)^{-2s}$ and $R(a;z)^s$ have to be taken
consistently.)
Again, we could skip this calculation and simply observe that
$R(a;\dv)^s\in\W\om s(I)$ and that $\ph(x)$ is the restriction
$y^{-s}R(a;x+\nobreak iy)^s\bigr|_{y=0}$. If $|a|>|x|$, then
expanding the two sides of~\eqref{exRaEq} by the binomial theorem
gives another proof of~\eqref{ExPol} and makes clear where the
binomial coefficients in that formula come from.

\smallskip
\exampl\label{exRzs} Our fourth example is
\be\label{tP-R}
\ph(x)=R(x;z_0)^s\quad\Rightarrow\quad \tpoiss s\ph(z)\= \bb(s)^{-1}\,
q_s(z,z_0)\qquad(z_0\in\uhp)\,, \ee
where the constant $\bb(s)$ is given in terms of beta or gamma
functions by
\be\label{btdef-p}\bb(s)\=\Beta\bigl(s,\frac12\bigr)\=
\frac{\Gf(s)\,\Gf(\frac12)}{\Gf(s+\frac12)}\;.\ee
Formula~\eqref{tP-R} is proved by remarking that the function on the
right belongs to $\W\om s$ and that its image under $\rho_s$ is the
function on the left (as one sees easily from the asymptotic behavior
of the Legendre function $Q_{s-1}(t)$ as $t\to\infty$). Obtaining it
from the power series in~\eqref{tpoiss-series} would probably be
difficult, but we will see at the end of the section how to get it
from the integral formula for $\tpoiss s$ given below.

\smallskip
\par\noindent\emph{Remark.} Equation~\eqref{Ansatz} shows that the
function $y^{-s}\cdot \tpoiss s\ph(x+\nobreak iy)$ is even in~$y$ (as
is visible in Examples~1 and~2 above). In the projective model,
$A^\prj(z) = \bigl( \frac y{|z+i|^2}\bigr)^{-s} \, u(z)=|z+i|^{2s}A(z)$
is not even in~$y$. In the circle model, related to the projective
model by $w=\frac{z-i}{z+i}$, the reflection $z\mapsto \bar z$
corresponds to $w\mapsto 1/\bar w$
(or $r\mapsto r^{-1}$ in polar coordinates $w=re^{i\th}$), and the
function $A^\ci(w) = 2^{2s}(1-\nobreak |w|^2)^{-s} u(w)$ satisfies
$A^\ci(1/\bar w)=|w|^{2s}A^\ci(w)$. For example, equations
\eqref{Leg-d} and~\eqref{Leg} say that the function in
$\W\om s\bigl(\disk\sm\{0\}\bigr)$ whose image under $\rho_s$ is
$w^n$ ($n\in\ZZ$)
corresponds to $A^\ci(w)\=\bar w^{-n}F\bigl(s-n,s;2s;1-|w|^2\bigr)$,
and this equals $w^n|w|^{-2s}F\bigl(s-n,s;2s;1-|w|^{-2}\bigr)$ by a
Kummer relation. Note that if we had used the factor
$\bigl( \frac{1-|w|}{1+|w|}\bigr)^s$ instead of $2^{-2s}(1-|w|^2)^s$
in Definition~\ref{Wom-def-p}, we would have obtained functions
$A^\ci$ that are invariant under $w\mapsto 1/\bar w$.
\end{examples}

\sssect{Integral version} If $\ph$ is a real-analytic function on an
interval $I\subset\RR$, then we can associate to it two extensions,
both real-analytic on a sufficiently small complex neighborhood
of~$I$: the holomorphic extension, which we will denote by the same
letter, and the solution $A$ of the differential equation
\eqref{deq-b} given in Theorem~\ref{ansatz1}.
The following result shows how to pass explicitly from $\ph$ to $A$, 
and from $A$ to~$\ph$, and show that their domains coincide.
\change{Theorem\ref{thm-phi-A} is the combination of two theorems in previous
versions. Also formulation improved.}

\begin{thm}\label{thm-phi-A}
 Let $\ph \in \V\om s(I)$ for some open interval $I\subset\RR$,
and write $\tpoiss s \ph(z) =y^s\, A(z)$ with a real-analytic
function~$A$ defined in some neighborhood of~$I$.
Let $U= \bar U\subset\CC$ be a connected and simply connected
subset of~$\CC$, with $I= U\cap \RR$.  Then the
following two statements are equivalent:
\begin{enumerate}
\item[i)]  $\ph$ extends holomorphically to all of~$U$.
\item[ii)]  $A$ extends real-analytically to all of~$U$.
\end{enumerate}
Moreover, the two functions define one another in the following
way.
\begin{itemize}
\item[a)]  Suppose that $\ph$ is holomorphic in~$U$. Then the function
$u=\tpoiss s \ph$ is given for $z\in U\cap\uhp$ by
\be\label{Wdef-p} u(z) \= \frac1{i\, \bb(s)} \,
 \int_{\bar z}^z R\bigl(\z;z\bigr)^{1-s} \,\ph(\z)\, d\z\,, \ee
where $\bb(s)$ is given by~\eqref{btdef-p} and the integral is taken
along any piecewise $C^1$-path in~$U$ from $\bar z$ to $z$
intersecting $I$ only once, with the branch of $R(\z;z)^{1-s}$
continuous on the path and equal to its standard value at the
intersection point with~$I$.
\item[b)] Suppose that $u(z) = \tpoiss s \ph(z) = y^s A(z)$ with $A$
real-analytic in~$U$. Then the holomorphic extension of $\ph$ to $U$
is given by
\be \label{rint-p}\ph(\z) \= \begin{cases}
    \displaystyle\frac {2\, \bb(s)\sin\pi s}\pi\,\int_{\x_0}^\z
    \bigl[u(\dv),\bigl(-R(\z;\dv)\bigr)^s\bigr]
     &\text{ if $\z\in U\cap\uhp$}\,, \smallskip\\
     A(\z) &\text{ if $\z\in I\,$,} \smallskip\\
  \displaystyle\frac {2\, \bb(s)\sin\pi s}\pi
  \,\int^{\bar\z}_{\x_0}\bigl[\bigl(-R(\z;\dv)\bigr)^s,\,
  u(\,\cdot\,)\bigr]
          &\text{ if $\z\in U\cap\uhp^-$,} \end{cases}
\ee
where the integrals are along piecewise $C^1$-paths in $U\cap\uhp$
from any $\x_0\in I$ to~$\z$, respectively~$\bar \z\,$, with the
branch of $\bigl(-R(\z;z) \bigr)^s$ fixed by
$\,|\arg\bigl(-R(\z;z) \bigr)|<\pi\,$ for $z$ near~$\x_0$.
\end{itemize}
\end{thm}


We  note the formal similarity between the formula \eqref{Wdef-p} for
$\tpoiss s\ph$ and the formula~\eqref{poiss-l} for the Poisson map:
 the integrand is exactly the same, but in the case of $\poiss s$ the
integration is over $\proj\RR$ (or $\uc$), while in the formula
for~$\tpoiss s$ it is over a path which crosses $\proj\RR\,$. We
therefore call $\tpoiss s$ the \emph{transverse Poisson
map}.


We have stated the  theorem  only for neighborhoods of intervals
in $\RR$, but because everything is $G$-equivariant they can easily
be transferred to any interval in $\proj\RR$. (Details are left to
the reader.) Alternatively, one can work in the projective or the
circle model. This will be discussed after we have given the  proof.




\begin{proof}[Proof of Theorem~\ref{thm-phi-A}] First we show that
\eqref{Wdef-p} gives $A$ on~$U\cap \uhp$ starting from a holomorphic
$\ph$ on~$U$.  Define $\tpoiss s \ph$ locally by
\eqref{tpoiss-series}. For $x\in I$ we denote by $r_x$ the radius of
the largest open disk with center~$x$ contained in~$U$. Using the
identity
 \[\frac{(2k)!}{4^k\,k!\,\Gf(k+s+\frac12)}
 \=\frac{\Gf(k+\frac12)}{\Gf(\frac12)\,\Gf(k+s+\frac12)}
 \=\frac1{\Gf(s)\,\Gf(\frac12)}\,\int_0^1(1-t)^{s-1}\,t^{k-\frac12}\,dt\]
(duplication formula and beta function), we find for $x\in I$ and
$0<y<r_x$ the formula
 \begin{align}\nonumber
 \bb(s)\,&(\tpoiss s\ph)(x+iy)\= y^s\int_0^1 (1-t)^{s-1}\,t^{-1/2} \,
  \biggl(\sum_{k=0}^\infty \frac{\ph^{(2k)}(x)}{(2k)!}
  (-ty^2)^k\biggr)\,dt \\
\nonumber&\= \frac12\,y^s\int_0^1 (1-t)^{s-1}\,t^{-1/2}\,
   \left(\ph(x+iy\sqrt t)\,+\,\ph(x-iy\sqrt t)\right)\,dt
   \displaybreak[0] \\
\label{intermediate}
&\= y^s \int_{-1}^1 (1-t^2)^{s-1} \ph(x+iyt)\, dt
\qquad\bigl(t=t^2\bigr) \displaybreak[0] \\
\nonumber&\= y^s\,\int_{-y}^y\,\biggl(\frac
{y^2}{y^2-\eta^2}\biggr)^{1-s}\,\ph(x+i\eta)\,y^{-1}\,d\eta
    \qquad\bigl(t=\eta/y\bigr) \displaybreak[0] \\
\nonumber&\= -i\,\int_{x-iy}^{x+iy} \biggl(\frac{y}{(\z-z)(\z-\bar z)
} \biggr)^{1-s} \,\ph(\z)\,d\z\,,
\end{align}
where the path of integration is the vertical line from $x-iy$ to
$x+iy$. The integral converges at the end points. The value of the
factor $\bigl(y/(\z-\nobreak z)(\z-\nobreak\bar z) \bigr)^{1-s}$ is
based on the positive value $y^{-1}$ of
$y/(\z-\nobreak z)(\z-\nobreak\bar z)$ at $\z=x$. Continuous
deformation of the path does not change the integral, as long as we
anchor the branch of the factor
$\bigl(y/(\z-\nobreak z)(\z-\nobreak\bar z) \bigr)^{1-s}$ at the
intersection point with~$I$. (This holds even though that factor is
multivalued on $U\sm\{z,\bar z\}$. We could also allow multiple
crossings of~$I$, but then would have to prescribe the crossing point
at which the choice of the branch of the Poisson kernel is anchored.)
This proves~\eqref{Wdef-p} for points $z\in U\cap\uhp$ sufficiently
near to~$I$, and the extension to all of $U\cap\uhp$ is then
automatic since the integral makes sense in the whole of that domain
and is real-analytic in~$z$.\medskip

To show that \eqref{rint-p} gives $\ph$ on $U$ if we start from a
given~$A$ we also consider first the case that $\z=X+iY\in U\cap\uhp$
 and that the vertical segment from $X$ to $\z$ is contained in~$U$.
Since we want to integrate up to $z=\z$, we will use the Green's form
$\om_\z(z)=\bigl[u(z),\bigl(-R(\z;z)\bigr)^s\bigr]$ rather than
$\bigl[\bigl(-R(\z;z)\bigr)^s,u(z)\bigr]$ or
$\bigl\{u(z),\bigl(-R(\z;z)\bigr)^s\bigr\}$, which would have
non-integrable singularities at this end-point. (The minus sign is
included because $R(\z;z)$ is negative on the segment.) Explicitly,
this Green's form is given for $z=x+iy\in U\cap\uhp$ by
\begin{align}
\nonumber
\om_\z(z)&\=\bigl(-R(\z;z)\bigr)^s\, \biggl(\frac{\p u}{\p z}\,dz
\,+\,\frac{is}{2y}\,\frac{z-\z}{\bar z-\z}\,u\,d\bar z\biggr)
\displaybreak[0]
\\
\nonumber
&\= \bigl(-y\, R(\z;z)^s\bigr) \,\biggl(\frac{\p A}{\p z}\,dz
\,-\,\frac{is}{2y}\,A\,dz \,+\, \frac{is}{2y}\,\frac{z-\z}{\bar
z-\z}\,A\,d\bar z\biggr)
\displaybreak[0]\\
\label{omZ}
&\= \bigl(-y \,R(\z;z)\bigr)^s\,\biggl[\biggl(\frac{\p A}{\p z}
\,-\,\frac s{\bar z-\z}\,A\biggr)\,dx \,+\, \biggl(i\frac{\p A}{\p
  z}\,+\, \frac sy\, \frac{x-\z}{\bar z-\z}\, A\biggr)\,dy\biggr]\;.
\end{align}
If we restrict this to the vertical line $z=X+itY$ ($0<t<1$) joining
$X$ and $\z$, it becomes
\begin{align*} \om_\z(X+itY)&\= \,\biggl(\frac{iY}2\,A_x(X+itY)
    \,+\, \frac Y2\,A_y(X+itY)\,+\,\frac
    s{t(1+t)}\,A(X+itY)\biggr)\,\frac{t^{2s}\, dt}{(1-t^2)^s}
      \displaybreak[0]
\\
&\= \sum_{k=0}^\infty\frac{\Gf(s+\h)}{k!\,\Gf(k+s+\h)}
\biggl[\ph^{(2k+1)}(X)\,\bigl(iY/2\bigr)^{2k+1}\\
      &\qquad\qquad\qquad \,+\,\biggl(\frac kt+\frac
      s{t(1+t)}\biggr)\,\ph^{(2k)}(X)\, \bigl(iY/2\bigr)^{2k}\biggl]
       \,t^{2k}\,\frac{t^{2s}}{(1-t^2)^s}\,dt \;,
\end{align*}
where $\ph$ is the holomorphic function near~$I$ with $\ph=A$ on $I$.

Now we use the beta integrals
\begin{align*} \int_0^1 t^{2k}\,&\frac{t^{2s}}{(1-t^2)^s}\,dt
   \=\frac12\int_0^1t^{k+s-\h}(1-t)^{-s}dt
   \=\frac{\Gf(k+s+\h)\,\Gf(1-s)}{2\,\Gf(k+\frac32)} \\
    &\=\frac{\Gf(s+\frac12)\,\Gf(1-s)}{2\,\Gf(\frac12)}
    \cdot\frac{k!\,\Gf(k+s+\frac12)}{\Gf(s+\frac12)}
    \cdot\frac{2^{2k+1}}{(2k+1)!}\;,\displaybreak[0] \\
   \int_0^1 \biggl(\frac kt \,&+\,\frac s{t(1+t)}\biggr)\,t^{2k}\,
   \frac{t^{2s}}{(1-t^2)^s}\,dt \=
     \int_0^1\biggl(k\,\frac{t^{2k+2s-1}}{(1-t^2)^s}
     \,+\,s\,\frac{t^{2k+2s-1}-t^{2k+2s}}{(1-t^2)^{s+1}}\biggr)\,dt
     \displaybreak[0]\\
   &\= \frac k2\,\int_0^1t^{k+s-1}(1-t)^{-s}dt \,+\, \frac
   s2\,\int_0^1\bigl(t^{k+s-1}-t^{k+s-\h}\bigr)\,(1-t)^{-s-1}\,dt
   \displaybreak[0]\\
   &\=\frac k2\,\frac{\Gf(k+s)\,\Gf(1-s)}{\Gf(k+1)} \,+\,\frac s2\,
	 \biggl(\frac{\Gf(k+s)\,\Gf(-s)}{\Gf(k)}
	 \,-\,\frac{\Gf(k+s+\frac12)\,\Gf(-s)}{\Gf(k+\frac12)}\biggr)
	 \displaybreak[0]\\
    &\=\frac{\Gf(s+\frac12)\,\Gf(1-s)}{2\,\Gf(\frac12)}\cdot
    \frac{k!\,\Gf(k+s+\frac12)}{\Gf(s+\frac12)}\cdot\frac{2^{2k}}{(2k)!}
\end{align*}
(the second calculation is valid initially for $\re(s)<0$,
$\re(k+s)>0$, but then by analytic continuation for $\re(s)<1$,
$\re(k+s)>0$, where the left-hand side converges) to get
 \be \label{uhpint}
 \int_X^\z \om_\z\=
 \frac{\Gf(s+\frac12)\,\Gf(1-s)}{2\,\Gf(\frac12)}\sum_{n=0}^\infty
 \ph^{(n)}(X) \frac{(iY)^n}{n!} \= \frac{\pi}{2\,\bb(s)\,\sin\pi s}\,
 \ph(\z)\,. \ee
Furthermore, we see from~\eqref{omZ} that the $dx$-component of the
1-form $\om_\z(x+iy)$ extends continuously to $U\cap\overline\uhp$
and vanishes on $I$, so $\int_{x_0}^X\om_\z$ vanishes for any
$\x_0\in I$ and we can replace the right-hand side of~\eqref{uhpint}
by $\int_{\x_0}^\z\om_\z\,$. On the other hand, the fact that the
$1$-form is closed means that we can integrate along any path from
$\x_0$ to $\z$ inside $U\cap\uhp$, not just along the piecewise
linear path just described, and hence also that we can move $\z$
anywhere within $U\cap\uhp$, thus obtaining the analytic continuation
of~$\ph$ to this domain as stated in~\eqref{rint-p}.

If $\z=X-iY$ ($Y>0$) belongs to $\lhp\cap U$, then the calculation is
similar. We suppose that the segment from $X$ to $\bar \z$ is in $U$,
and parametrize it by $z=X+itY$. The differential form is
\begin{align*}\bigl [ \bigl( -&R( \z;z)\bigr)^s, u\bigr]
\\
&\= \bigl(-y\, R( \z;z) \bigr)^s \biggl( \biggl( \frac{\partial
A}{\partial \bar z}( z) +\frac s{ \z-z}\, A( z)
\biggr) \,dx
+ \biggl( -i\frac{\partial A}{\partial \bar z}( z) + \frac s y\;
\frac{ \z-x}{ \z-z}\, A( z)
\biggr) \, dy\biggr)\,,\end{align*}
which leads to the integral
\begin{align*}
\int_X^{\bar \z} \om_\z &\= \int_0^1 \frac{t^{2s}}{(1-t^2)^s} \bigl(
-\frac {iY}2 \, A_x(X+itY) +\frac Y2 A_y(X+itY) + \frac st \,
\frac1{1+t} A(X+itY)\bigr)\, dt
\\
&\=\sum_{k\geq 0} \frac{\Gf(s+\frac12)}{k!\; \Gf(s+\frac12+k)} \biggl(
\ph^{(2k+1)}(X) \, (-iY/2)^{2k+1} \\
&\qquad\hbox{}
+\biggl( \frac kt +\frac s{t(1+t)} \biggr) \,\ph^{(2k)}(X)\,
(-iY/2)^{2k} \biggr) \frac{t^{2s+2k}}{(1-t^2)^s}\, dt\,,
\end{align*}
which is the expression that we obtained in the previous case with $Y$
replaced by $-Y$. We replace $Y$ by $-Y$ in \eqref{uhpint}, and
obtain the statement in~\eqref{rint-p} on $U\cap\lhp$ as well.
\end{proof}

It is not easy to find examples that illustrate the integral
transformation~\eqref{rint-p} explicitly, i.e., examples of functions
in $\W\om s$ for which the Green's form $[u(\dv),R(\z;\dv)^s]$ can be
written explicitly as $dF$ for some potential function $F(\dv)$. One
case which works, though not without some effort, is
$u(z)=y^s=\tpoiss s(1)$
(Example~1). Here the needed potential function is given by Entry~6 in
Table~\ref{tab-Grpot} in~\S\ref{sect-pot}, and a somewhat lengthy
calculation, requiring careful consideration of the branches and of
the behavior at the end points of the integral, lets us deduce
from~\eqref{rint-p} that the inverse transverse Poisson transform of
the function $y^s\in\W\om s(\RR)$ is indeed the constant
function~$1$.

\sssect{Other models} The two integral formulas above were formulated
in the line model. To go to the projective model, we consider first
$U\subset\CC$ as in the theorems not intersecting the half-line
$\,i\,[1,\infty)$. In that case we find by \eqref{line-proj}
and~\eqref{Rproj-p}
\beqabclbl{tPprj}
\begin{align}\label{tP-prj}
\tpoiss s \ph^\prj(z) &\= \frac1{i\,\bb(s)} \int_{\bar z}^z
R^\prj(\z;z)^{1-s} \,\ph^\prj(\z)\,\frac{d\z}{1+\z^2}\qquad(z\in
U\cap\uhp)\,,
\displaybreak[0]
\\
\label{itP-prj}
\ph^\prj(\z)&\=
\begin{cases}\displaystyle\frac {2\, \bb(s)\sin\pi
s}\pi\,\int_{\x_0}^\z
\bigl[u(\dv),\bigl(-R^\prj(\z;\,\dv\,)\bigr)^s\bigr]
     &\text{ if $\z\in U\cap\uhp$}\,, \\
     A^\prj(\z)
     &\text{ if $\z\in U\cap\RR=I\,$,} \\
       \displaystyle\frac {2\, \bb(s)\sin\pi s}\pi \,\int^{\bar
       \z}_{\x_0}\bigl[\bigl(-R^\prj(\z;\,\dv\,)\bigr)^s,\,
       u(\,\cdot\,)\bigr]
          &\text{ if $\z\in U\cap\uhp^-$,} \end{cases}
\end{align}
\eeqabc
with $u(z) = \bigl( \frac y{|z+i|^2}\bigr)^s\, A^\prj(z)$, where the
paths of integration and the choices of branches in the Poisson
kernels are as in the theorems, suitably adapted. These formulas then
extend by $G$-equivariance to any connected and simply connected open
set $U=\nobreak \bar U\subset\proj\CC\!\sm\!\{i,-i\}$ and any
$\x_0\in U\cap\proj\RR$, giving a local description of the
isomorphism $\V\om s \cong \W\om s$ on all of~$\proj\RR$. Note that
the integrals in~\eqref{tPprj} make sense if we take for $U$ an
annulus $1-\e < \bigl|\frac{z-i}{z+i}\bigr|<1+\e$ in~$\proj\CC$,
which is not simply connected, but the theorem then has to modified.
We will explain this in a moment.

In the circle model, we have
\beqabclbl{tPci}
\begin{align}\label{tP-ci}
\tpoiss s \ph^\ci(w) &\= \frac1{2\,\bb(s)} \int_w^{1/\bar w}
R^\ci(\eta;w)^{1-s} \,\ph^\ci(\eta)\,\frac{d\eta}\eta
\qquad(w\in U\cap\disk)\,,
\displaybreak[0]\\
\label{itP-ci}
\ph^\ci(\eta)&\=
\begin{cases}\displaystyle\frac {2\, \bb(s)\sin\pi
s}\pi\,\int_{\x_0}^\eta
\bigl[u(\dv),\bigl(-R^\ci(\eta;\,\cdot\,)\bigr)^s\bigr]
     &\text{ if $\eta\in U$\,, $|\eta|<1$}\,, \\
     A^\ci(\z)
     &\text{ if $\eta\in U\cap\uc\,$,} \\
       \displaystyle\frac {2\, \bb(s)\sin\pi s}\pi \,\int^{1/\bar
       \eta}_{\x_0}\bigl[\bigl(-R^\ci(\eta;\,\cdot\,)\bigr)^s,\,
       u(\,\cdot\,)\bigr]
          &\text{ if $\eta\in U$\,, $|\eta|>1\,$, } \end{cases}
\end{align}
\eeqabc
with $u(w) = 2^{-2s}(1-\nobreak |w|\bigr)^s\, A^\ci(w)$, for
$w\in U\cap\disk$, with $U$ open in $\CC\sm\{0\}$, connected, simply
connected and invariant under $w\mapsto1/\bar w$, and with
$\x_0\in U\cap\uc$, with the paths of integration and the choice of
branches of the Poisson kernel again suitably adapted from the
versions in the line model.

If $U$ is an annulus of the form $\e<|w|<\e^{-1}$ with $\e\in (0,1)$,
we still can apply the relations in~\eqref{tPci}, provided we take in
\eqref{tP-ci} the path from $w $ to $1/\bar w$ homotopic to the
shortest path. If we change to a path that goes around a number of
times, the result differs from $\tpoiss s \ph
(w)$ by an integral multiple of
$\frac{\pi i}{\bb(s)}\, \poiss s \ph^\ci(w)$. In \eqref{itP-ci} we
can freely move the point $\x_0$ in $\p\disk$, without changing the
outcome of the integral.

Let us use~\eqref{tP-ci} to verify the formula for
$\tpoiss s\bigl(R(\dv;z_0)^s\bigr)$ given in Example~3. By
$G$-equivariance, we can suppose that $z_0=i$. Now changing to circle
model coordinates, we find with the help of~\eqref{Rcirc-p} that the
function $\ph(x)=R(x;i)^s$ corresponds to $\ph^\ci(\x)=\nobreak1$ and
that the content of formula~\eqref{tP-R} is equivalent to the formula
$$ \begin{aligned}
&\int_r^{1/r}\,\biggl(\frac{\bigl(1-r/\eta\bigr)\bigl(1-r\eta\bigr)}
{1-r^2}\biggr)^{s-1} \,\frac{d\eta}\eta  \=\bigl(1-r^2\bigr)^s\,
\int_0^1\frac{(1-t)^{s-1}\,t^{s-1}\,dt}{\bigl(1-t(1-r^2)\bigr)^s}\\
&\qquad\qquad\=(1-r^2)^s\,\frac{\Gf(s)^2}{\Gf(2s)}\,F(s,s;2s;1-r^2)
\=2\,Q_{s-1}\biggl(\frac{1+r^2}{1-r^2}\biggr)\,,
\end{aligned} $$
where in the first line we have made the substitution
$\eta=(1-t)r^{-1}+tr\,$.


\subsection{Duality}\label{sect-dual-p} We return to the bilinear form
$\bt$ on $\F s$ defined in~\eqref{bt-def}. We have seen that $\bt$ is
zero on $\E_s\times\E_s$. The next result describes $\bt$ on other
combinations of elements of $\F s$ in terms of the duality map
$\;\langle\ \,,\ \rangle:\,\V\om s\times\V{-\om}{1-s}\to\CC\,$
defined in~\eqref{dual}.
\begin{prop}\label{prop-WEdual} Let $u,\,v\in\F s$.
\begin{enumerate}
\item[a)] If $\,u\in \E_s$ and $v\in \W\om s$, then
  \be\label{bt1} \bt(u,v) \= \bb(s)^{-1} \, \langle \ph,\al \rangle
  \ee
with $\bb(s)$ as in~\eqref{btdef-p}, where $u=\poiss {1-s} \al$ with
$\al\in \V{-\om}{1-s}$ and $v=\tpoiss s \ph$ with $\ph\in\V\om s$.
\item[b)] If $\;u,\,v\in \W\om s$, then $\bt(u,v)=0$.
\item[c)] If $\,u\in \W\om{1-s}$ and $v\in \W\om s$, then
\be \bt(u,v) \= \bigl(s-\frac12\bigr)\,\langle \ph,\ps\rangle\,, \ee
with $\ph\in \V\om{1-s}$, $\ps\in \V\om s$ such that
$u=\tpoiss{1-s}\ph$ and $v=\tpoiss s \ps$.
\end{enumerate}
\end{prop}
\begin{proof}The bijectivity of the maps
$\poiss {1-s} :\V{-\om}s\lra \E_{1-s}=\E_s$ and
$\tpoiss s :\V\om s \lra \W\om s$ implies that we always have $\ph$
and $\al$ as indicated in~a). All transformations involved are
continuous for the topologies of $\V{-\om}{1-s}$ and $\V\om s$, so it
suffices to check the relation for $\ph=\eph_{s,m}$ and
$\al=\eph_{1-s,m}$. Now we use \eqref{Poisofe}, the result for
$\tpoiss s e_{s,n}$ in \S\ref{sect-bg}, and equations \eqref{pq-br}
and \eqref{de} to get the factor in~\eqref{bt1}.

For part~b) we write $u=(1-\nobreak |w|^2)^s A(w)$ and
$v=(1-\nobreak |w|^2)^s B(w)$ with $A$ and $B$ extending in a
real-analytic way across $\p\disk$. If we take for $C$ a circle
$|w|=r$ with $r$ close to~$1$, then
\[ [u,v] \= \frac1{2i}\, r(1-r^2)^{2s} \, (A B_r -B A_r)\, d\th\,.\]
It follows that the integral is $\oh\bigl((1-\nobreak r^2)^s\bigr)$ as
$r\uparrow1$, and hence vanishes.

In view of~b) we can restrict ourselves for c) to the case
$s\neq \frac12$. As in part a) it suffices to consider the relation
for basis vectors. We derive the relation from~\eqref{pqrel}:
\[\bt(Q_{1-s,m},Q_{s,n}) = \bt\left( \pi\cot\pi s\;
P_{\!s,m}+Q_{s,m},\,Q_{s,n}\right) = \pi\,\cot\pi s\; (-1)^n
\dt_{n,-m}\,.\qedhere\]
\end{proof}


\subsection{Transverse Poisson map in the differentiable
case}\label{sect-winf}
The $G$-module $\W\om s$, which is isomorphic to~$\V\om s$, turns out
to be very useful for the study of cohomology with coefficients in
$\V\om s$, discussed in detail in~\cite{BLZ-p}. There we also study
cohomology with coefficients in $\V p s$, with $p=2,3,\ldots,\infty$,
and for this we need an analogue $\W p s$ of $\W\om s$ related to $p$
times differentiable functions. In this subsection we define such a
space and show that there is an equivariant isomorphism
$\tpoiss s:\V p s\lra\W p s$. To generalize the restriction
$\rho_s:\W\om s\lra \V\om s$ we will define $\W p s$ not as a space
of boundary germs, but as a quotient of $G$-modules. In fact, we give
a uniform discussion, covering also the case~$p=\om$ treated in the
previous subsections.
\begin{defn}\label{GN-def}
For $p=2,3,\ldots,\infty,\om$ we define $\G^p_s$ and $\N^p_s$ as
spaces of functions $f\in C^2(\disk)$ for which there is a
neighborhood $U$ of $\p \disk=\ci$ in~$\CC$ such that the function
$\tilde f(w) = (1-\nobreak |w|)^{-s} \, f(w)$ extends as an element
of $C^p(U)$ and satisfies on $U$ the conditions
\[ \renewcommand\arraystretch{1.2}
\begin{array}{||c||c|c||}
\hline\hline
p& \text{for }\G^p_s & \text{for }\N^p_s\\ \hline \hline
\in \ZZ_{\geq 2}& \tilde\Dt_s \tilde f(w)=\o\bigl( (1-|w|^2)^{p}\bigr)
& \tilde f(w)=\o\bigl( (1-|w|^2)^{p}\bigr)
\\ \hline
\infty & \text{ the above condition for all $p\in \NN$ }&\text{ the
above condition for all $p\in \NN$ }\\ \hline
\om& \tilde\Dt_s \tilde f(w)=0 & \tilde f(w)=0 \\ \hline \hline
\end{array}
\]
where $\tilde\Dt_s $ is the differential operator corresponding to
$\Dt-\ld_s$ under the transformation $f\mapsto \tilde f$.
\end{defn}

In the analytic case $p=\om$, the space $\G^\om_s$ consists of
$C^2$-representatives of germs in $\W\om s$, and $\N^\om_s$ consists
of $C^2$-representatives of the zero germ in $\W\om s$, i.e.,
$\N^\om_s = C_c^2(\disk)$. Any representative of a germ can be made
into a $C^2$-germ by multiplying it by a suitable cut-off function.
Thus $\W\om s $ as as in Definition~\ref{Wom-def-p} is isomorphic to
$\G^\om_s/\N^\om_s$. We take $C^2$-representatives to be able to
apply $\Dt$ without the need to use a distribution interpretation.

In the upper half plane model, there is an equivalent statement with
$f^\ci$ replaced by $f^\prj$, and $2^{-2}\allowbreak
(1-\nobreak|w|^2)$ by $\frac y{|z+i|^2}$. The group $G$ acts on
$\G^p_s$ and $\N^p_s$ for $\allp$, by $f\,|\,g (z)=f(gz)$, and the
operator corresponding to $\Dt-\ld_s$ is
$\tilde\Dt_s=-y^2\bigl( \p_y^2+\p_x^2\bigr(-2sy\,\p_y$
(cf.~\eqref{deq-b}).

The definition works locally: $\G^p_s(I)$ and $\N^p_s(I)$, with
$I\subset \p\H$ open, are defined in the same way, with $\Om$ now a
neighborhood of $I$ in $\proj\CC$. In the case that $I\subset \RR$ in
the upper half plane model, we have $f(z) = y^s \tilde f(z)$ on
$\Om \cap\uhp$ with $\tilde f\in C^p(\Om)$. On can check that
$\G^p_s$ and $\N^p_s$ are sheaves on $\p\H$.

\sssect{Examples}The function $z\mapsto y^s$ is in $\G^\om_s(\RR)$.
The function $Q_{s,n}$ in~\eqref{LegQ} has the right boundary
behavior, but is not defined at $w=0\in \disk$. We can multiply it by
$re^{i\th} \mapsto \ch(r)$ with a smooth function $\ch$ that vanishes
near zero and is equal to one on a neighborhood on~$1$. In this way
we obtain an element of $\G^\om_s$.


\sssect{Restriction to the boundary} For $f\in \G^p_s$ the
corresponding function $f^\ci$ on $\Om$ has a restriction to $\uc$
that we denote by $\rho_s f$. It is an element of $\V p s$. In this
way, restriction to the boundary gives a linear map
\begin{equation}
\rho_s : \G^p_s \lra \V p s
\end{equation}
that turns out to intertwine the actions of $G$, and that behaves
compatibly with respect to the inclusions $\G^p_s \lra \G^q_s$ and
$\V p s\lra \V q s$ if $q<p$. This restriction map can be localized
to give $\rho_s:\G^p_s(I) \lra \V p s(I)$ for open intervals
$I\subset\p\H$.

\begin{lem}\label{lem-Nkr}Let $I\subset\p\H$ be open. For $\allp$ the
space $\N^p_{\,s}(I)$ is a subspace of $\G^p_s(I)$. It is equal to
the kernel of $\rho_s:\G^p_s(I)\lra \V p s(I)$.
\end{lem}
\begin{proof}The sheaf properties imply that we can work with
$I\neq\p\H$. The action of $G$ can be used to arrange $I\subset\RR$
in the upper half plane model.

Let first $p\in \NN$, $p\geq 2$. Suppose that $f(z) = y^s \tilde f(z)$
on $\Om \cap\uhp$ for some $\tilde f \in C^p(\Om)$, with $\Om$ a
neighborhood of $I$ in~$\CC$. The Taylor expansion at $x\in I$ gives
for $i,j\geq 0$, $i+j\leq p$
\begin{equation}
\label{Tay} \p_x^i \p_y^j \tilde f(x+iy) \= \sum_{n=i+j}^p
\frac{(n-i)!}{(n-i-j)!} a_{n-i}^{(i)}(x)
\, y^{n-i-j} + \o\bigl( y^{p-i-j}\bigr)\end{equation}
on $\Om$, with
\[ a_n(x) \= \frac 1{n!} \p_y^n \tilde f(x)\,.\]
 The differential operator $\Dt-\ld_s$ applied to $f$ corresponds to
 the operator $\tilde \Dt_s = -y^2 \p_x^2 -y^2\p_y^2-2sy\p_y$ applied
 to $\tilde f$ on the region $\Om \cap \uhp$. Thus we find
\begin{equation}\label{anrel}
\tilde \Dt_s \tilde f(x+iy) \= -2sa_1(x) - \sum_{n=2}^p \Bigl(
a_{n-2}''(x)+n(n+2s-1) a_n(x) \Bigr) y^n + \o(y^p)\,.\end{equation}

If $f\in \N^p_{\,s}(I)$, then $a_n=0$ for $0\leq n \leq p$, and
$\tilde\Dt_s \tilde f (z) = \o(y^p)$. So $f\in \G^p_s(I)$, and
$\rho_s f (x) = \tilde f(x) = a_0(x)$. Hence
$\N^p_s(I) \subset \ker \rho_s$.

Suppose that $f\in \G^p_s(I)$ is in the kernel of $\rho_s$. Then
$a_0=0$. {}From \eqref{anrel} we have $a_1=0$ and
$a_{n-2}'' = n(1-\nobreak 2s -\nobreak n)
a_n$ for $2\leq n\leq p$. Hence $a_n=0$ for all~$n\leq p$, and
$f\in \N^p_s(I)$.
\medskip

The case $p=\infty$ follows directly from the result for
$p\in \NN$.\medskip

In the \emph{analytic case}, $p=\om$, the inclusions
$\N^\om_s(I) \subset \G^\om_s(I)$ and $ \N^p_s(I) \subset \ker\rho_s$
are clear. If $f\in \G^\om_s(I)\cap \ker\rho_s$, then $\tilde f$ is
real analytic on $\Om$, and instead of the Taylor expansion
\eqref{Tay}, we have a power series expansion with the same
structure. Since
$(\ker \rho_s) \cap \G^\om_s(I) \subset \N^\infty_s(I)$, we have
$a_n=0$ for all $n$, hence the analytic function $\tilde f$ vanishes
on the connected component of $\Om$ containing~$I$. Thus,
$f\in \N^\om_s$.
\end{proof}

Relation \eqref{anrel} in this proof also shows that any
$f\in \G^p_s(I)$ with $I\subset\RR$ has the expansion
\be\label{anrel1}
f(x+iy) \= \sum_{0\leq k \leq p/2} \frac{(-1/4)^k\,
\Gf(s+\frac12)}{k!\, \Gf(s+k+\frac12)}\, \ph^{(2k)}(x)\,
y^{s+2k}+\o(y^{s+p})\quad(y\downarrow0\,,\; x\in I)\,, \ee
with $\ph=\rho_s f\in \V p s(I)$.

\sssect{Boundary jets} For $p=2,\ldots,\infty$ we define $\W p s$ as
the quotient in the exact sequence of sheaves on $\p\H$
\be \label{Wp-def}
 0 \lra \N^p_s\lra \G^p_s \lra \W p s\lra0\,. \ee
In the analytic case, $p=\om$, we have already seen that $\W\om s$ is
the quotient of $\G^\om_s/\N^\om_s$.


In the differentiable case $p=2,\ldots,\infty$, an element of
$\W p s(I)$ is given on a covering $I=\bigcup_j I_j$ by open
intervals $I_j$ by a collection of $f_j\in \G^p_s(I_j)$ such that
$f_j\equiv f_j' \bmod \N^p_s(I_j\cap I_{j'})$ in
$\G^p_s(I_j\cap I_{j'})$ if $I_j\cap I_{j'}\neq\emptyset$. To each
$j$ is associated a neighborhood $\Om_j$ of $I_j$ in $\proj\CC$ on
which $f_j^\ci$ is $p$ times differentiable. Add an open set
$\hat\Om\subset \H$ such that $\H \subset \hat \Om \cup
\bigcup_j \Om_j$. With a partition of unity subordinate to the
collection $\{\hat\Om\} \cup \{\Om_j\,:\, j\}$ we build one function
$f$ on $\H$ such that $f^\ci =(1-|w|^2)^{-s} f(w)$ differs from
$f_j^\ci$ on $\Om_j$ by an element of $\N^p_s(I_j)$. In this way we
obtain $\W p s(I) = \G^p_s(I)/\N^p_s(I)$ in the differentiable case
as well.

We have also
\[ 0 \lra \N^p_s(\p\H) \lra \G^p_s(\p\H) \lra \W p s(\p\H)\lra0\]
as an exact sequence of $G$ modules. We call elements of $\W p s$
\emph{boundary jets} if $p=2,\ldots,\infty$. The $G$-morphism
$\rho_s$ induces a $G$-morphism $\rho_s: \W p s(I) \lra \V p s(I)$
for $\allp$. The morphism is injective by Lemma~\ref{lem-Nkr}. In
fact it is also surjective:
\begin{thm}\label{thm-tpoiss}The restriction map
$\rho_s : \W p s(I)\lra \V p s(I)$ is an isomorphism for every open
set $I\subset\p\H$, for $\allp$.
\end{thm}
The case $p=\om$ was the subject of \S\ref{sectbg}.
Theorem~\ref{thm-phi-A}  described the inverse $\tpoiss s$ explicitly
with a transverse Poisson integral, and Theorem~\ref{ansatz1} works
with a power series expansion. It is the latter approach that
suggests how to proceed in the differentiable and smooth cases. We
denote the inverse by~$\tpoiss s$, or by $\tpoiss {s,p}$ if it is
desirable to specify~$p$.
\begin{proof}
In the \emph{differentiable case} $p\in \NN\cup\{\infty\}$, it
suffices to consider $\ph \in C_c^p(I)$ where $I$ is an interval
in~$\RR$. The obvious choice would be to define near~$I$
\be\label{temptation} f(z) \= \sum_{0\leq k \leq p/2}
\frac{(-1)^k}{4^k\; k!\; \bigl(s+\frac12\bigr)_k}\, \ph^{(2k)}(x)
\,y^{s+2k}\,.\ee
However, this is in general not in $C^p(\uhp)$, because each term
$\ph^{(2k)}(x) \, y^{s+2k}$ is only in $C^{p-2k}$. Instead we set
\be \label{f-ps-def}
f(z) \= y^s \int_{-\infty}^\infty \om(t) \, \ph(x+yt)\, dt \= y^{s-1}
\int_{-\infty}^\infty \om \bigl( \frac{t-x}y\bigr) \, \ph(t)\, dt
\,, \ee
where $\ph$ has been extended by zero outside its support, and where
$\om$ is an even real-analytic function on~$\RR$ with quick decay
that has prescribed moments
\be\label{moments} M_{2k} \;:=\; \int_{-\infty}^\infty t^{2k}\,
\om(t)\, dt \= \frac{(-1)^k (\frac12)_k}{(s+\frac12)_k}\qquad\text{
for even }k\geq 0\,.
\ee
(For instance we could take $\om$ to be the Fourier transform of the
product of the function
$ u \mapsto \Gf(s+\nobreak \frac12)\allowbreak\, \Bigl(\frac{|u|}2\Bigr)^{\frac12-s}
\allowbreak \, I_{s-1/2}(|u|)$ and an even function in
$C_c^\infty(\RR)$ that is equal to~$1$ on a neighborhood of $0$
in~$\RR$. This choice is even real-analytic.)
Replacing $\ph$ in~\eqref{f-ps-def} by its Taylor expansion up to
order~$p$, we see that this formally matches the
expansion~\eqref{temptation}, but it now makes sense and is
$C^\infty$ in all of~$\uhp$, as we see from the second integral. The
first integral shows that
\be \label{tf-int} \tilde f(z) \= y^{-s}f(z) = \int_{-\infty}^\infty
\om(t)
\ph(x+yt)\, dt\ee
extends as a function in $C^\om(\CC)$.

Inserting the power series expansion of order~$p$ of $\ph$ at $x\in I$
in~\eqref{tf-int} we arrive at $\tilde\Dt_s \tilde f(z) = \oh(y^p)$.
This finishes the proof in the differentiable and smooth cases.
\end{proof}


In the proof of Theorem~\ref{thm-tpoiss} we have chosen an
real-analytic Schwartz function $\om$ with prescribed moments. In the
case $p=2,3,\ldots$ we may use the explicit choice in the following
lemma, which will be used in the next chapter:
\begin{lem}\label{lem-smoothw} For any $s\not\in\frac12\ZZ$ and any
integer $N\ge0$ there is a unique decomposition
\be\label{smoothw} (t^2+1)^{s-1}\=\frac{d^N\al(t)}{dt^N}\,+\,\bt(t)
\ee
where $\al(t)=\al_{N,s}(t)$ is $(t^2+1)^{s-1}$ times a polynomial of
degree $N$ in $t$ and $\bt(t)=\bt_{N,s}(t)$ is $\oh(t^{2s-N-3})$ as
$|t|\to\infty$.
\end{lem}
\noindent We omit the easy proof. The first few examples are
\begin{align*}
(t^2+1)^{s-1}&\=\frac{d}{dt}\biggl[\frac{t(t^2+1)^{s-1}}{2s-1}\biggr]
   \,+\, \frac{2s-2}{2s-1}\,(t^2+1)^{s-2}\,,\\
 (t^2+1)^{s-1}&\=\frac{d^2}{dt^2}\biggl[\frac{(t^2+1)^{s-1}}{(2s-1)(2s-3)}
 \,+\, \frac{(t^2+1)^s}{2s(2s-1)}\biggr] \,+\,
 \frac{4(s-1)(s-2)}{(2s-1)(2s-3)}\,(t^2+1)^{s-3}\,,\\
 (t^2+1)^{s-1}&\=\frac{d^3}{dt^3}
 \biggl[\frac{2t(t^2+1)^{s-1}}{(2s+1)(2s-1)(2s-3)} \,+\,
   \frac{t(t^2+1)^s}{2s(2s+1)(2s-1)}\biggr] \\ &\qquad\quad\,+\,
\frac{4(s-1)(s-2)}{(2s+1)(2s-1)}\,
\biggl(\frac{2s+3}{2s-3}\,+\,3t^2\biggr)\,(t^2+1)^{s-4}\,.\end{align*}
In general we have
$$ \al_{N,s}(t)\=\frac1{2^N}\,\sum_{j=0}^{N/2}\binom{N-j-1}{N/2\,-\,1}\,
    \frac{(t^2+1)^{s-1+j}}{(s)_j\,(s-\frac{N+1}2+j)_{N-j}}$$
if $N\geq2$ is even, where $(s)_j=s(s+1)\cdots(s+j-1)$ is the
ascending Pochhammer symbol, and a similar formula if $N$ is odd, as
can be verified using the formula
$$ \frac1{n!}\frac{d^n}{dt^n}\,(t^2+1)^{s-1}\=
\sum_{0\le j\le n/2}\binom{n-j}j
   \binom{s-1}{n-j}\,(2t)^{n-2j}\,(t^2+1)^{s-n+j}\,.$$
\medskip

Let us compute the moments of $\bt=\bt_{N,s}$ as in \eqref{smoothw}.
 For $0\leq n < N$ we have
\[ \int_{-\infty}^\infty t^n\,\bt(t)\,dt \=
\int_{-\infty}^\infty\biggl((t^2+1)^{s-1}
  \,-\,\frac{d^N\al(t)}{dt^N}\biggr)\,t^n\,dt\,. \]
This is a holomorphic function of $s$ on $\re s<1$. We compute it by
considering $\re s<-\frac n2$:
\be\label{bti}
\begin{aligned}
\int_{-\infty}^\infty &t^n (t^2+1)^{s-1}\, dt \= \begin{cases}
\int_0^1 x^{\frac{n-1}2}(1-x)^{-s-\frac{n+1}2}\, dx&\text{ if $n$ is
even}\,,\\
0&\text{ if $n$ is odd}\,;
\end{cases}\\
&\= \sqrt\pi \tan\pi s\, \frac{\Gf(s)}{\Gf(s+\frac12)}
\cdot\begin{cases}
(-1)^k\, \frac{(\frac12)_k}{(s+\frac12)_k}
&\text{ if $n=2k$ is even}\,,\\
0&\text{ if $n$ is odd}\,.
\end{cases}
\end{aligned}\ee
So a multiple of $\bt_{N,s}$ has the moments that we need in the proof
of Theorem~\ref{thm-tpoiss}.
\bigskip

\section{Boundary behavior of mixed
eigenfunctions}\label{sect-conjectures}
In this section we combine ideas from \S\ref{sect-pms}
and~\S\ref{sect-gb}. Representatives $u$ of elements of $\W\om {1-s}$
have the special property that $\bigl(1-|w|^2\bigr)^{s-1}\, u(w)$
(in the circle model) or $y^{s-1}u(z)$ (in the line model) extends
analytically across the boundary $\p\H$. If such an eigenfunction
occurs in a section $(h,u)$ of the sheaf $\D_s$ of mixed
eigenfunctions we may ask whether a suitable multiple of~$h$ also
extends across the boundary. In \S\ref{sect-medb} we will show that
this is true locally (Theorem~\ref{thm-conj}), but not globally
(Proposition~\ref{prop-gle}).

In \S\ref{sect-interpolation} we use the differential equations
satisfied by $y^{-s}u$ for representatives $u$ of elements of
$\W\om s$ to define an extension $\A_s$ of the sheaf $\E_s$ from
$\uhp$ to~$\proj\CC$. We also extend the sheaf $\D_s$ on
$\proj\CC\times\uhp$ to a sheaf $\tD_s$ on $\proj\CC\times\proj\CC$
that has the same relation to $\A_{1-s}$ as the relation of $\D_s$ to
$\E_s=\E_{1-s}$. In \S\ref{sect-pse} we show that the power series
expansion of sections of $\A_s$ leads in a natural way to sections of
$\tD_{1-s}$, a result which is needed for the proofs
in~\S\ref{sect-medb}, and in \S\ref{sect-AD} we give the
generalization of Theorem~\ref{thm-DKEexs} to the sheaf~$\tD_s$.
Finally, in \S\ref{sect-gbg-D} we consider the sections of~$\D_s$
near $\proj\RR\times\uhp$.


\subsection{Interpolation between sheaves on $\uhp$ and its boundary}
\label{sect-interpolation}In this subsection we formulate results
from \S\ref{sectbg} in terms of a sheaf on $\proj\CC$ that is an
extension of the sheaf $\E_s$. This will be used in the rest of this
section to study the behavior of mixed eigenfunctions near the
boundary $\proj\CC\times \proj\RR$ of $\proj\CC\times \uhp$ and to
extend them across this boundary. We also define an extension $\tD_s$
of the sheaf $\D_s$ of mixed eigenfunctions.

For an open set $U\subset\CC$, let $\A_s(U)$ be the space of
real-analytic solutions $A(z)$ of~\eqref{deq-b} in~$U$. For
$U\subset\proj\CC$ containing~$\infty$ the definition is the same
except that the solutions have the form
$A(z)=|z|^{-2s}A^\infty(-1/z)$ for some real-analytic function
$A^\infty$ near~$0$ (which then automatically satisfies the same
equation). The action~\eqref{A-action} makes $\A_s$ into a
$G$-equivariant sheaf: $A\mapsto A|g$ defines an isomorphism
$\A_s(U)\cong\A_s(g^{-1}U)$ for any open $U\subset\proj\CC$ and
$g\in G$.


For any $U\subset\proj\CC$, the space $\A_s(U)$ can be identified via
$A(z)\mapsto u(z)=|y|^sA(z)$ with a subspace of the space
$\E_s\bigl(U\sm\nobreak\proj\RR\bigr)$ of $\lambda_s$-eigenfunctions
of the Laplace operator $\Delta=-y^2(\p_x^2+\p_y^2)$ in
$\proj\CC\sm\proj\RR$ (up to now we have considered the operator
$\Delta$ and the sheaf~$\E_s$ only on~$\uhp$), namely the subspace
consisting of functions which are locally of the form
$\,|y|^s\times(\text{analytic})\,$ near~$\RR$ and of the form
$\,|y/z^2|^s\times(\text{analytic})\,$ near~$\infty$.


If $U\subset\proj\CC\sm\proj\RR$, then the map $A\mapsto u$ is an
isomorphism between $\A_s(U)$ and $\E_s(U)$. (In this case the
condition ``real-analytic'' in the definition of~$\A_s(U)$ can be
dropped, since~$C^2$ or even distributional solutions of the
differential equation are automatically real-analytic.) At the
opposite extreme, if $U$ meets $\proj\RR$ in a non-empty set~$I$,
then any section of $\A_s$ over $U$ restricts to a section of
$\V\om s$ over $I$, and for any $I\subset\proj\RR$ we obtain from
Theorem~\ref{ansatz1} an identification between $\V\om s(I)$ and the
inductive limit of $\A_s(U)$ over all neighborhoods $U\supset I$. The
sheaf $\A_s$ thus ``interpolates'' between the sheaf $\E_s$ on
$\proj\CC\sm\proj\RR$ and the sheaf $\V\om s$ on $\proj\RR$. At
points outside $\proj\RR$ the stalks of $\A_s$ are the same as those
of~$\E_s$, while at points in $\proj\RR$ the stalks of the sheaves
$\A_s$, $\V\om s$ and $\W\om s$ are all canonically isomorphic. At
the level of open sets rather than stalks,  Theorem~\ref{thm-phi-A}
says  that the space $\A_s(U)$ for suitable $U$
intersecting $\proj\RR$ is isomorphic to $\hol(U)$ by a unique
isomorphism compatible with restriction to $U\cap\RR$, the
isomorphisms in both directions being given by explicit integral
transforms. Finally, from~\eqref{Ansatz} we see that if $U$ is
connected and invariant under conjugation, then any $A\in\A_s(U)$ is
invariant under $z\mapsto \bar z$. In the language of sheaves, this
says that, if we denote by $c:\proj\CC\to\proj\CC$ the complex
conjugation, the induced isomorphism $c:c^{-1}\A_s\to\A_s$ is the
identity when restricted to $\proj\RR$.

We now do the same construction for the sheaf $\D_s$ of mixed
eigenfunctions, defining a sheaf $\tD_s$ on $\proj\CC\times\proj\CC$
which bears the same relation to~$\D_s$ as $\A_s$ has to~$\E_s$. (We
could therefore have used the notation $\E^\ast_s$ instead of~$\A_s$,
but since $\A_s$ interpolates between two very different subsheaves
$\E_s$ and $\V\om s$, we preferred a neutral notation which does not
favor one of these aspects over the other. Also, $\E_s=\E_{1-s}$, but
$\A_s\ne\A_{1-s}\,$.)

Let $(h,u)$ be a section of the sheaf $\D_s$ in $U \cap \bigl(
\CC\times\nobreak\uhp\bigr)$, where $U$ is a neighborhood
in~$\CC\times\CC$ of a point $(x_0,x_0)$ with $x_0\in\RR$. The
function $u(z)$ is a $\ld_s$-eigenfunction of~$\Dt$, and we can ask
whether it ever has the form $y^s\, A(z)$ or $y^{1-s}\, A(z)$ with
$A(z)$ (real-\nobreak)\allowbreak analytic near~$x_0$. It turns out
that the former does not happen, but the latter does, and moreover
that in this case the function $h(\z,z)$ has the form
$y^{-s}\, B(\z,z)$ where $B(\z,z)$ is also analytic in a neighborhood
of $(x_0,x_0)\in \CC\times\CC$. To see this, we make the substitution
\be\label{hu-BA-sub}
 u(z)\= y^{1-s} \, A(z)\,,\qquad h(\z,z)\= y^{-s}\, B(\z,z)
\ee
in the differential equations \eqref{A-de-glob} to obtain that these
translate into the differential equations
\beqabclbl{AD-eq}
\begin{align}
(\z-z)\, \p_z B &\=-s\, B - \frac{is}2 \,(\z-\bar z)\, A\,,\\
(\z-\bar z)\,\p_{\bar z}\bigl( B - y A\bigr) &\=- s\, B-\frac{is}2\,
(\z-\bar z)\, A\,,
\end{align}
\eeqabc
for $A$ and~$B$, in which there is no singularity at $y=0$. (This
would not work if we had used $u=y^s A$, $h=y^\ast B$ instead.)

As long as $z$ is in the upper half plane, equations~\eqref{hu-BA-sub}
define a bijection between pairs $(h,u)$ and pairs $(B,A)$, and it
makes no difference whether we study the original differential
equations~\eqref{A-de-glob} or the new ones~\eqref{AD-eq}. The
advantage of the new system is that it makes sense for all $z\in \CC$
and defines a sheaf $\tD_s$ on $\CC\times\CC$ whose sections over
$U\subset\CC\times\CC$ are real-analytic solutions $(B,A)$
of~\eqref{AD-eq} in~$U$ with $B$ holomorphic in the first variable
and $A$ locally constant in the first variable. This sheaf is
$G$-equivariant with respect to the action $(B,A)|g = (B|g,A|g)$
given for $g=\matc abcd$ by
\be B|g(\z,z)\= |cz+d|^{2s} B(g\z,gz)\,,\qquad A|g(z)\=
|cz+d|^{2s-2}A(gz)\,, \ee
so it extends to a sheaf on all of $\proj\CC\times\proj\CC$ by setting
$\tD_s(U) = \tD_s(g^{-1}U)|g$ if $U$ is a small neighborhood of a
point $(\z_0,\infty)$ or $(\infty,z_0)$ and $g$ is chosen with
$g^{-1}U \subset \CC\times\CC$.
\smallskip

In \eqref{DD} and~\eqref{DD-} we give a formula for $h$ in terms of
$u$ near the diagonal and the antidiagonal where $(h,u)$ is a section
of~$\D_s$. In terms of~$A=y^{s-1}u $ and~$ B=y^s h$ this formula
becomes
\be\label{Bsum}
B(\z,z) \= \begin{cases}\displaystyle -\frac i2 (\z-\bar z)\sum_{n\geq
0} \frac{\p^n A}{\p z^n}(z)\, \frac{(\z-z)^n}{(1-s)_n}&\text{ for
$\z$ near $z$}\,,\\\displaystyle
-\frac i2 (\z-z) \sum_{n\geq 1} \frac{\p^n A}{\p\bar z^n}(z)\,
\frac{(\z-\bar z)^n}{(1-s)_n}
-\frac i2(\z-\bar z) A(z)
&\text{ for $\z$ near $\bar z$}\,.
\end{cases}
\ee
Now inspection shows that the right hand side of~\eqref{Bsum}
satisfies the differential equations~\eqref{AD-eq}, whether
$z\in \uhp$ or not, so $(B,A)$ with $B$ as in \eqref{Bsum} gives a
section of $\tD_s$ on neighborhoods of points $(z,z)$ and
$(\bar z,z)$ for all $z\in \CC$. From~\eqref{Bsum} it is not clear
that for $z\in\RR$ both expressions define the same function on a
neighborhood of~$z$. In the next subsection we will see that they do.



\subsection{Power series expansion}\label{sect-pse}
Sections of $\A_s$ are real-analytic functions of one complex
variable, and hence can be seen as power series in two variables. In
this subsection we show that the coefficients in these expansions
have interesting properties. They will be used in \S\ref{sect-medb}
to study the structure of sections of $\D_s$ and~$\tD_s$ near the
diagonal of $\proj\RR\times \proj\RR$.


Let $U\subset \CC$ be open, and let $z_0\in U$. We write the expansion
of a section $A$ of $\A_s$ at a point $z_0$ in the strange form (the
reason for which will become apparent in a moment)
\be\label{A-ps}
 A(z) \= \sum_{m,n\geq 0} \binom{m+s-1}{ m}\; \binom {n+s-1}{ n}
\;c_{m,n}(z_0) \, (z-z_0)^m\, (\overline{z-z_0})^n\,.\ee
Then we have the following result.
\begin{thm}\label{thm-ps}
Let $U\subset\CC$, $A\in \A_s(U)$, and for $z_0\in U$ define the
coefficients $c_{m,n}(z_0)$ for $m,n\geq0$ by~\eqref{A-ps}. Let
$r:U\rightarrow\RR_+$ be continuous. Then the series \eqref{A-ps}
converges in $|z-\nobreak z_0|<r(z_0)$ if and only if the series
\be \label{Aphi}
 \Ph_A(z_0;v,w) \;:=\; \sum_{m,n\geq 0} c_{m,n}(z_0) \, v^m w^n \ee
converges for $|v|, |w|<r(z_0)$. The function defined by~\eqref{Aphi}
has the form
\be\label{Phi-R-L} \Ph_A(z_0;v,w) \= \frac{B(z_0+v,z_0)-B(\bar
z_0+w,z_0)}{y_0 +(v-w)/2i} \ee
for a unique analytic function $B$ on
\[ U'\= \bigl\{ (\z,z)\in \CC\times U\;:\; |\z-z|<r(z)\bigr\} \;\cup\;
\bigl\{ (\z,z)\in \CC\times U\;:\; |\z-\bar z|<r(z)\bigr\}\]
satisfying $B(\z,z) = y\, A(z)$ and $B(\bar z,z)=0$ for $z\in U$, and
the pair $(B,A)$ is a section of $\tD_{1-s}$ over~$U'$.
\end{thm}
\begin{proof}
The fact that $\binom{m+s-1}m = m^{\oh(1)}$ as $m\rightarrow\infty$
implies the relation between the convergence of \eqref{A-ps}
and~\eqref{Aphi}. (We use here that a power series
$\sum c_{mn,n}v^nw^m$ in two variable converges for $|v|,|w|<r$ if
and only if its restriction to $w=\bar v$ converges for $|v|<r$.)
The differential equation \eqref{deq-b} is equivalent to the very
simple recursion
\be\label{c-recur}
2iy_0\, c_{m,n}(z_0) \= c_{m,n-1}(z_0)-c_{m-1,n}(z_0)\qquad(m,n\geq 1)
\ee
for the coefficients $c_{m,n}(z_0)$. (This was the reason for the
choice of the normalization in~\eqref{A-ps}.\,) This translates into
the fact that $\left(2iy_0+v-w\right)\allowbreak \Ph_A(z_0;v,w) $ is
the sum of a function of~$v$ alone and a function of~$w$ alone, i.e.,
we have
\be\label{Phq} \Ph_A(z_0;v,w) \= \frac{L_A(z_0;v) -
R_A(z_0;w)}{y_0+(v-w)/2i} \,,\ee
where, if we use the freedom of an additive constant to normalize
$R_A(z_0;0)=0$, the functions $L_A$ and $R_A$ are given explicitly in
terms of the boundary coefficients $\{c_{j,0}(z_0)\}_{j\geq 0}$ and
$\{c_{0,j}(z_0)\}_{j\geq 1}$ by
\be\label{La-Ra}
\begin{aligned}
2i\,L_A(z_0;v) &\= (v+2iy_0)\, \sum_{m\geq 0} c_{m,0}(z_0)\, v^m\,,\\
2i\,R_A(z_0;w)&\= c_{0,0}(z_0)\, w + (w-2iy_0)\sum_{n\geq 1}
c_{0,n}(z_0)\, w^n\,.
\end{aligned}
\ee
(Multiplied out, this says that coefficients $c_{m,n}(z_0)$
satisfying~\eqref{c-recur} are determined by their boundary values by
\be\label{c-sum}
\begin{aligned}
c_{m,n}(z_0) &\= \sum_{j=1}^m \frac{(-1)^n}{(2iy_0)^{m+n-j}}\,\binom
{m\!-\!j\!+\!n\!-\!1 }{ m\!-\!j } \, c_{j,0}(z_0) \\
&\qquad\hbox{} + \sum_{j=1}^n
\frac{(-1)^{n-j}}{(2iy_0)^{m+n-j}}\,\binom {n\!-\!j\!+\!m\!-\!1 }{
n\!-\!j } \, c_{0,j}(z_0)
\,,
\end{aligned}
\ee
which of course can be checked directly.)


We define $B$ on~$U'$ (now writing $z$ instead of~$z_0$) by
\be\label{B-def} B(\z,z) \= \begin{cases}
L_A(z,\z-z)&\text{ if } |\z-z|<r(z)\,,\\
R_A(z,\z-\bar z)&\text{ if }|\z-\bar z|<r(z)\,.
\end{cases}
\ee
These two definitions are compatible if the discs in question overlap
(which happens if $r(z)>|y_0|$), because the convergence
of~\eqref{Aphi} for $|v|$, $|w|<r(z)$ implies that the fraction
in~\eqref{Phq} is holomorphic in this region, and hence that its
numerator vanishes if $z_0+v=\bar z_0+w$.\smallskip

Surprisingly, the function $B$ thus defined constitutes, together with
the given section~$A$ of $\A_s$, a section $(B,A)$ of $\tD_{1-s}$ for
$\z$ near $z$ or~$\bar z$. To see this, we apply the formulas
\eqref{Bsum}, with $s$ replaced by $1-s$, and express the derivatives
of~$A$ in the coefficients $c_{m,n}(z)$ with help of~\eqref{A-ps}. We
find that the first expression in~\eqref{Bsum} is equal to
$L_A(z;\z-\nobreak z)$, and the second one
to~$R_A(z;\z-\nobreak \bar z)$.
\end{proof}

\begin{examples}
\exampl Let $A(z)\in \A_s\bigl(\CC\sm\{0\}\bigr)$ be the function
$ |z|^{-2s}$. For any $z_0\neq0$ the binomial theorem gives
$c_{m,n}(z) = (-1)^{m+n}\,|z_0|^{-2s} \,\allowbreak z_0^{-m}\, 
\allowbreak\bar z_0^{-n}$ and
\be \Ph_A(z_0;v,w)\= \frac{|z_0|^{2-2s}}{(z_0+v)(\bar z_0+w)} \=
\frac1{2iy_0+v-w} \Bigl( \frac{|z_0|^{2-2s}}{\bar z_0+w} -
\frac{|z_0|^{2-2s}}{z_0+v} \Bigr)
\,, \ee
in accordance with \eqref{Phi-R-L} with the solution
$B(\z,z)=\frac i2\, |z|^{2-2s} \, \left( \z^{-1}-\bar z^{-1}\right)$,
defined on $z\neq0$, $\z\neq 0$. (The regions $|\z-\nobreak z|<|z|$
and $|\z-\nobreak \bar z|<z$ do not overlap.)
\exampl If $z_0\in \RR$, then \eqref{c-recur} says that $c_{m,n}(z_0)$
depends only on $n+m$, so the generating function $\Ph_A$ has an
expansion of the form
\[ \Ph_A(z_0;v,w) \= \sum_{N=0}^\infty C_N(z_0) \,
\frac{v^{N+1}-w^{N+1}}{v-w}\,.\]
Hence in this case we have
$A(z) \= \sum_{N\geq 0} C_N(z_0)P_N(z-\nobreak z_0)$ where $P_N$ is
the section of~$\A_s$ defined by
\be P_N(z) \;:=\; (-1)^N \sum_{m,n\geq 0,\, m+n=N} \,\binom{-s}m\;
\binom{-s}n\, z^m\, \bar z^n\,, \ee
a polynomial that already occurred in~\eqref{ExPol}. \exampl Let
$A(z) = y^{-s} p_{s,k}(z,i)$, defined in~\eqref{psn-def-p}, with
$z_0=i$ and $k\geq 0$. We describe
$A(z)=\frac{\Gf(s+k)}{k!\; \Gf(s-k)}\, \tilde A(w)$ first in the
coordinate $w=\frac{z-i}{z+i}$ of the disk model. Taking into account
\eqref{Leg-d} and~\eqref{Leg} we obtain
\begin{align*}
\tilde A(w)&\= w^k\, \Bigl(\frac{1-w\,\bar w}{|1-w|^2} \Bigr)^{-s} \,
(1-w\,\bar w)^s\; \hypg21{s,s+k;1+k,w\,\bar w} \,.
\end{align*}
Set $p=(z-\nobreak i)/2i$, so that $w=p/(p+1)$. Then
\begin{align*}
\tilde A(w) &\= (1-w)^{s}\,(1-\bar w)^{s}\,\sum_{l\geq 0}
\frac{(s)_\ell\,
(s+k)_{\ell}}{(1+k)_\ell\, \ell!} \, w^{k+\ell} \bar
w^\ell\displaybreak[0]\\
&\= \sum_{\ell\geq 0} \binom{-s}\ell\; \binom{-s-k}\ell\;
\binom{\ell+k}\ell^{-1} \; p^{k+\ell} \,\bar
p^\ell\,(1+p)^{-s-k-\ell}\, (1+ \bar p)^{-s-\ell}\displaybreak[0]\\
&\= \sum_{\ell,i,j\geq 0} \binom{-s}\ell\; \binom{-s-k}\ell\;
\binom{\ell+k}\ell^{-1} \; \binom{-s-k-\ell}i\; \binom{-s-\ell}j\;
p^{k+\ell+i} \,\bar p^{\ell+j} \displaybreak[0]\\
&\=\sum_{m\geq k,\,n\geq 0} p^{m}\, \bar p^n\; \binom{-s-k}{m-k}\;
\binom{-s}n\; \binom{n+k}k^{-1}\, \sum_{l=0}^n \binom {m-k} \ell \;
\binom{n+k}{n-\ell}
\displaybreak[0]\\
&\= \sum_{m\geq k,\, n\geq 0} \Bigl(\frac{z-i}{2i}\Bigr)^m \, \Bigl(
\frac{\bar z+i}{-2i}\Bigr)^n \; \binom{-s-k}{m-k}\; \binom{-s}n \;
\binom{n+k}k^{-1}\binom{m+n}n\,.
\end{align*}
Hence $A$ has an expansion as in~\eqref{A-ps} with
\beqabclbl{Cmn[k]}
\be c^{[k]}_{m,n}(i) \;:=\; c_{m,n}(i) \=
 (-1)^m \, (2i)^{-m-n} \; (1-s)_k\;\binom{m+n}{n+k} \ee
($=0$ if $m<k$), which satisfies the recursion~\eqref{c-recur}.
\medskip

The analogous computation for $k< 0$ gives
\be c_{m,n}^{[k]}(i)\=
(-1)^m (2i)^{-m-n}\;(1-s)_k \;\binom{m+n}{m-k} \ee
\eeqabc
($=0$ if $n<-k$).\footnote{The Pochhammer symbol $(x)_k$ is defined
for $k<0$ as $(x-\nobreak1)^{-1}\cdots (x-\nobreak|k|)^{-1}$, so that
$(x)_k=\Gf(x+\nobreak k)/\Gf(x)$ in all cases.}

In this example we can describe the form of the function~$B$ up to a
factor without computation by equivariance: since $z\mapsto
p_{s,n}(z,i)$ transforms according to the character
$\matr{\cos\th}{\sin \th}{-\sin\th}{\cos\th} \mapsto e^{2ik\th}$, the
function $h=y^{s-1}B$ should do the same near points of the diagonal
or the antidiagonal. Thus, for $k\geq 0$ we know that $B(\z,i)$ is a
multiple of $\bigl(\frac{\z-i}{\z+i}\bigr)^k$ near $\z=i$, and
vanishes near $\z=-i$, while for $k<0$, we have $B(\z,i)=0$ for $\z$
near~$i$, and $B(\z,i)$ is a multiple of
$\bigl( \frac{\z-i}{\z+i}\bigr)^k$ for $\z$ near~$-i$. The explicit
computation using \eqref{B-def}, \eqref{La-Ra} and~\eqref{Cmn[k]}
confirms these predictions, giving
$B(\z,i) = (-1)^k\,(1- \nobreak s)_k\, 
\bigl(\frac{\z-i}{\z+i}\bigr)^k$ if $k\geq 0$ and $\z$ is near~$i$,
and $B(\z,i)=-(-1)^k \,
 \frac{\Gf(k+\nobreak1-\nobreak s)}{\Gf(1-\nobreak s)}\, 
\bigl(\frac{\z-i}{\z+i}\bigr)^k$ if $k<0$ and $\z$ is near~$-i$.
\smallskip

Note that, since any holomorphic function of~$\z$ near~$i$
(resp.~$-i$) can be written as a power series in $\frac{\z-i}{\z+i}$
(resp.~$\frac{\z+i}{\z-i}$) we see that this example is generic for
the expansions of $A$ and $B$ for any section $(B,A)$ of $\D_s$ near
$(\z,z)=(\pm i,i)$, and hence by $G$-equivariance for $z$ near any
$z_0\in\uhp$ and $\z$ near $z_0$ or~$\bar z_0$.
\end{examples}


\noindent\emph{Remark. } We wrote formula \eqref{A-ps} as the
expansion of a fixed section $A\in \A_s(U)$ around a variable
point~$z_0\in U$. If we simply define a function $A(z)$ by
\eqref{A-ps}, where $z_0$ (say in $\uhp$) is fixed, then we still
find that the differential equation
$(\Dt-\nobreak \ld_s) \bigl( y^s\, A(\dv,z_0)\bigr)=0$ is equivalent
to the recursion \eqref{c-recur} and to the splitting~\eqref{Phq} of
the generating function $\Ph_A$ defined by~\eqref{Aphi}. In this way
we have constructed a very large family of (locally defined)
$\ld_s$-eigenfunctions of~$\Dt$: for any $z_0\in \uhp$ and any
holomorphic functions $L(v)$ and $R(w)$ defined on discs of radius
$r\leq y_0$ around~$0$, we define coefficients $c_{m,n}$ either by
\eqref{Phq} and~\eqref{Aphi} or by \eqref{La-Ra} and~\eqref{c-sum};
then the function $u(z)=y^s\, A(z)$ with $A$ given by \eqref{A-ps} is
a $\ld_s$-eigenfunction of~$\Dt$ in the disk of radius $r$
around~$z_0$.


\subsection{Mixed eigenfunctions near the diagonal of
$\proj\RR\times\proj\RR$}\label{sect-medb} Parts i) and~iv) of
Proposition~\ref{prop-Dp} show that if $(h,u)$ is a section of $\D_s$
near a point $(z,z)\in \uhp\times\uhp$ of the diagonal or a point
$(\bar z,z)\in \lhp\times\uhp$ of the antidiagonal, then the function
$h$ and $u$ determine each other. Diagonal points
$(\x,\x)\in \proj\RR\times\proj\RR$ are not contained in the set
$\proj\CC\times\uhp$ on which the sheaf $\D_s$ is defined.
Nevertheless, there is a relation between the analytic extendability
of $h$ and~$u$ near such points, which we now study.
\begin{thm}\label{thm-conj}Let $\x\in \proj\RR$. Suppose that $(h,u) $
is a section of $\D_s$ over
$U\cap\bigl( \uhp \times\nobreak\uhp\bigr)$ for some neighborhood~$U$
of~$(\x,\x)$ in~$\proj\CC \times \proj\CC$. Then the following
statements are equivalent:
\begin{enumerate}
\item[a)] The function $y^{s-1}u$ extends real-analytically to a
neighborhood of~$\x$ in~$\proj\CC$.
\item[b)] The function $y^s h$ extends real-analytically to a
neighborhood of $(\x,\x)$ in $\proj\CC\times\proj\CC$.
\item[c)] The function $y^s h$ extends real-analytically to
$U'\cap \bigl( \proj\CC\times\uhp\bigr)$ for some neighborhood~$U'$
of $(\x,\x)$ in $\proj\CC\times\proj\CC$.
\end{enumerate}
\end{thm}

The theorem can be formulated partly in terms of stalks of sheaves. In
particular, the functions $u$ in a) represent elements of the stalk
$(\W\om{1-s})_\x\,$, and the pairs $(y^s h,y^{s-1} u)$ with
$y^{s-1} u$ as in~a) and $y^s h$ as in~b) represent germs in the
stalk $\bigl( \tD_s\bigr)_{(\x,\x)}$. The theorem has the following
consequence:
\begin{cor}\label{cor-conj}For each $\x\in \proj\RR$ the morphism
$C:\D_s \rightarrow p_2^{-1}\E_s$ in Theorem~\ref{thm-DKEexs} induces
a bijection
\[ \limind U \D_s \bigl( U\cap(\proj\RR\times\uhp) \bigr)\;\cong\;
\bigl( \W\om{1-s} \bigr)_\x \,, \]
where $U$ runs over the open neighborhoods of $(\x,\x)$ in
$\proj\CC\times\proj\CC$, and
\[ \bigl( \W\om{1-s} \bigr)_\x\;\cong\; \bigl(
\tD_s\bigr)_{(\x,\x)}\,. \]
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm-conj}]We observe that since
$U\cap\left( \uhp\times\uhp\right)$ intersects the diagonal, the
functions $h$ and $u$ in the theorem determine each other near
$(\x,\x)$ by virtue of parts~i) and~iv) of Proposition~\ref{prop-Dp}.
Hence the theorem makes sense.

Clearly b) $\Rightarrow$ c). We will prove a) $\Rightarrow$ b)
and c) $\Rightarrow$ a). By $G$-equivariance we can assume that
$\x=0$.\smallskip

For a) $\Rightarrow$ b) we write $u=y^{1-s}A$ with $A$ real-analytic
on a neighborhood of $0$ in~$\CC$. We apply Theorem~\ref{thm-ps}. The
power series~\eqref{A-ps} converges for $|z_0|\leq R$,
$|z-\nobreak z_0|<r$ for some $r, R>0$. (Choose $r$ to be the minimum
of $r(z_0)$ in $|z_0|\leq R$ for $R$ small.) The theorem gives us an
analytic function $B$ on the region
$W=\bigl\{(\z,z) \in \CC\times\CC\;:\; |z|<R, |\z-\nobreak z|<r\bigr\}$
such that $(B,A)\in \tD_s(W)$. By the uniqueness clause of
Proposition~\ref{utoh1}, the restriction of $B$ to
$W\cap\bigl(\CC\times\nobreak\uhp\bigr)$ is $y^s h$. Since
$(0,0)\in W$ this gives~b).
\smallskip


For c) $\Rightarrow$ a)
we start with a section $(y^s B,y^{1-s}A)$ of
$\D_s\bigl( U_R^{\vphantom{+}} \times U_R^+\bigr)$ for some $R>0$,
where $U_R = \{ z\in \CC\;:\; |z|<R\}$ and
$U_R^+=U_R^{\vphantom{+}}\cap \uhp$. Then $A\in \A_{1-s}(U^+_R)$. We
apply Theorem~\ref{thm-ps} again, with $z_0\in U_R^+$. By the
uniqueness clause in Proposition~\eqref{utoh1} the function $B$
appearing in~\eqref{Phi-R-L} is the same as the given~$B$ in a
neighborhood of $\bigl\{(z,z)\;:\; z\in
U_R^+\bigr\}\;\cup \; \bigl\{(\bar z,z)\;:\; z\in U_R^+\bigr\}$.
Since $B(\dv,z_0)$ is holomorphic in $U_R$ for each $z_0\in U_R^+$,
the right hand side of \eqref{Phi-R-L} is holomorphic for all $v,w$
with $|z_0+\nobreak v|, |\bar z_0+\nobreak w|<R$.
(The denominator does not produce any poles since the numerator
vanishes whenever the denominator does.) Hence the first statement of
Theorem~\ref{thm-ps} shows that the series \eqref{A-ps} represents
$A(z)$ on the open disk $|z-\nobreak z_0|< R-|z_0|$. For
$|z_0|<\frac 12 R$ this disc contains~$0$, so $A$ is real-analytic
at~$0$.
\end{proof}

In Proposition~\ref{prop-W-supp} we showed that the Poisson transform
of a hyperfunction represents an element of $\W\om{1-s}$ outside the
support of the hyperfunction. With Theorem~\ref{thm-conj} we arrive
at the following more complete result.
\begin{thm}\label{thm-W-supp}Let $I\subset\p\H$ be open, and let
$\al\in \V{-\om}s$. Then $\poiss s \al$ represents an element of
$\W\om{1-s}(I)$ if and only if $I \cap \supp(\al)=\emptyset$.
\end{thm}
\begin{proof}
Proposition~\ref{prop-W-supp} gives the implication $\Leftarrow$\,.
For the other implication suppose that $\poiss s \al$ represents an
element of $\W\om{1-s}(I)$. Let $g_{\text{can}}$ be the canonical
representative of~$\al$, defined in~\S\ref{sect-pm}. Then
$(g_{\text{can}},\poiss s \al)\in \D_s\bigl( (\proj\CC\sm \proj\RR) \times
\uhp\bigr)$ by Theorem~\ref{thm-can-DE} and Definition~\ref{D-def}.
 The implication a)$\;\Rightarrow\;$b) in Theorem~\ref{thm-conj} gives
the analyticity of $y^s\,g_{\text{can}}$ on a neighborhood of
$(\x,\x)$ in $\proj\CC\times\proj\CC$ for each $\x\in I$. It follows
that for $z_0\in \uhp$ sufficiently close to $\x$ the function
$g_{\text{can}}(\dv,z_0)$ is holomorphic at~$\x$. It then follows
from the definition of the mixed hybrid model in~\S\ref{sect-pm} that
$g_{\text{can}}(\dv,z)$ is holomorphic at~$\x$ for all $z\in \uhp$.
Thus $\x$ cannot be in~$\supp(\al)$.
\end{proof}\medskip


Theorem~\ref{thm-conj} is a local statement. We end this subsection
with a generalization of Proposition~\ref{prop-noq}, which shows that
the results of Theorem~\ref{thm-conj} have no global counterpart. For
convenience we use the disk model.
\begin{prop}\label{prop-gle}Let $A\subset \disk$ be an annulus of the
form $r_1<|w|<1$ with $0\leq r_1<1$, and let $V \subset \proj\CC$ be
a connected open set that intersects the region $r_1<|w|<r_1^{-1}$.
Then $\D_s(V\times A)$ does not contain non-zero sections of the form
$(h,u)$ where $u\in \E_s(A)$ represents an element of $\W\om {1-s}$.
\end{prop}
\begin{proof}The proof is similar to that of
Proposition~\ref{prop-noq}. Suppose that
$(h,u) \in \D_s(V\times\nobreak A)$ where $u$ represents an element
of $\W\om{1-s}$. By \eqref{br-potp} the holomorphic function
$\x\mapsto \int_C \bigl[ R^\ci(\x;\dv)^s, u\bigr]$ is identically
zero on some neighborhood $\rho<|\x|<\rho^{-1}$ of the unit circle.
We have the absolutely convergent representation
$u=\sum_n b_n Q_{1-s,n}$ on~$A$ for a sequence $(b_n)$ of complex
numbers. Combining this with the expansion $R^\ci(\x;\dv)^s = \sum_m
\frac{(-\x)^{-m}}{(1-s)_m}\, P_{1-s,m}$ and~\eqref{pq-br}, we obtain
\[ \sum_n b_n \, \frac{(-\x)^n}{(1-s)_m} \=0\]
for all $\x\in \uc$. Hence all $b_n$ vanish, so $u$ and hence also $h$
are zero.
\end{proof}

\begin{cor}\label{cor-bdW}If $V$ is a neighborhood of $\p\disk$
in~$\proj\CC$, then $\D_s\bigl( V \times(V\cap\uhp)\bigr)=\{0\}$.
\end{cor}
\begin{proof}Let $(h,u) \in \D_s\bigl( V \times(V\cap\uhp)\bigr)$.
Corollary~\ref{cor-conj} implies that $u\in \E_s(B\cap\disk)$
represents an element of $\W{1-s}\om$. The neighborhood $V$ contains
an annulus of the form $r_1<|w|<r_1^{-1}$, and
Proposition~\ref{prop-gle} shows that $(h,u)=(0,0)$.
\end{proof}



\subsection{The extended sheaf of mixed eigenfunctions}\label{sect-AD}
In \S\ref{sect-interpolation} we defined an extension $\tD_s$ of the
sheaf of mixed eigenfunctions from $\proj\CC\times\uhp$ to
$\proj\CC\times\proj\CC$. We now prove an analogue of
Theorem~\ref{thm-DKEexs}, the main result on the sheaf~$\D_s$,
for~$\tD_s$.

% We now give an extension to~$\tD_s$ of Theorem~\ref{thm-DKEexs}, 
% which was the main result on the sheaf~$\D_s$.

We denote by $\hol$ the sheaf of holomorphic functions on~$\proj\CC$,
by $p_j: \proj\CC\times\proj\CC\rightarrow\proj\CC$ the projection of
the $j$-th factor ($j=1,2$), and put
$\Dt^\pm = \bigl\{(z,z)\bigr\}_{z\in \proj \CC} \,\cup\,\bigl\{(z,\bar z)\bigr\}_{z\in\CC}$.
We define $\tK_s$ to be the subsheaf of $\tD_s$ whose sections have
the form $(B,0)$.
\begin{thm} The sheaf $\tK_s$ is the kernel of the surjective sheaf
morphism $C:\tD_s \rightarrow p_2^{-1} \A_s$ that sends
$(B,A) \in \tD_s(U)$ ($U\subset \proj\CC\times\proj\CC\,$ open)
to~$A$. The restriction of $\tK_s$ to $\Dt^\pm$ vanishes, and its
restriction to $\left( \proj\CC\times\proj\CC\right)\sm \Dt^\pm$ is
locally isomorphic to~$p_1^{-1}\hol$.
\end{thm}
This theorem gives us the exact sequence
\[ 0 \longrightarrow \tK_s \longrightarrow \tD_s \stackrel C
\longrightarrow p_2^{-1} \A_{1-s} \longrightarrow 0\]
generalizing the exact sequence in Theorem~\ref{thm-DKEexs}.
\begin{proof}By $G$-equivariance we can work on open
$U\subset \CC\times\CC$. The differential equations~\eqref{AD-eq}
imply that sections $(B,0)$ of $\tD_s$ on~$U$ have the form
$B(\z,z) = \ph(\z)\, \allowbreak\left(\z-z\right)^s \,\allowbreak
 \left( \z-\bar z\right)^s$ for some function~$\ph$. The analyticity
of~$B$ implies that $\ph=0$ near points of $\Dt^\pm$, and the
holomorphy of $B$ in its first variable implies that $\ph$ is
holomorphic. Thus $\tK_s$ is locally isomorphic to $\p_1^{-1}\hol$
outside $\Dt^\ast$ and its stalks at points of~$\Dt^\ast$ vanish.

Let $(h,u)$ be a section of $\tD_s$ over some open
$U\subset \CC\times\CC$. Denote by $D_{(a)}$ and $D_{(b)}$ the
expressions in the left hand sides of~\eqref{AD-eq}. A computation
shows that
\[ \bigl( (\z-\bar z)\p_{\bar z } + s \bigr)\, D_{(a)} -
\bigl((\z-z)\p_z + s\bigr)\, D_{(b)}\]
is $\frac 1{2i} \, (\z-\nobreak z) \, (\z-\nobreak \bar z)$ times
$(z-\nobreak \bar z) A_{z\bar z} - (1-s) \ A_z + (1-s)\, A_{\bar z}$.
The vanishing of the latter is the differential equation
defining~$\A_{1-s}$. So $A$ is a section of $\A_{1-s}$ on
$p_2 U \sm\Dt^\ast$. By analyticity it is in $\A_{1-s}(p_2 U)$. Hence
$C:(B,A)\mapsto A$ determines a sheaf morphism between the
restrictions of $\tD_s$ and $p_2^{-1}\A_{1-s}$ on $\CC\times\CC$, and
by $G$-equivariance on $\proj\CC\times\proj\CC$.


To prove the surjectivity of $C$ we have construct for each
$(\z_0,z_0)\in \CC\times\CC$ and each $A\in \A_s(U)$ for some
neighborhood $U$ of~$z_0$ a section $(B,A)$ of $\tD_s$ on a possibly
smaller neighborhood of $(\z_0,z_0)$. This suffices by
$G$-equivariance.

For $(\z_0,z_0)\in \Dt^\ast$ this construction is carried out
in~\eqref{Bsum}. Let $(\z_0,z_0)\not\in \Dt^\ast$. The integral
in~\eqref{h-int-c} suggests that we should consider the differential
form
\begin{align*}
\om &\= y^s \, \bigl[ \bigl(R(\z;z_1)/R(\z;z)\bigr)^s, y_1^{1-s} \,
A(z_1)\bigr]_{z_1}
\\
&\= \Bigl( \frac{ (\z-z)(\z-\bar z)}{(\z-z_1)(\z-\bar z_1)} \Bigr)^s\,
\biggl( \frac{s(\z-\bar z_1)}{2i (\z-z_1)} \, A(z_1)\, dz_1
+ \bigl( \frac i2(1-s) A(z_1) + y_1\, A_{\bar z}(z_1)\bigr)\, d\bar
z_1 \biggr)\,.
\end{align*}
Choosing continuous branches of
$\bigl( \frac{ (\z-z)(\z-\bar z)}{(\z-z_1)(\z-\bar z_1)} \bigr)^s$
near $(\z_0,z_0)$ we obtain $B(\z,z) = \int_{z_0}^z \om$ such that
$(B,A)$ satisfies \eqref{AD-eq} near $(\z_0,z_0)$, which can be
checked by a direct computation, and follows from the proof of
Theorem~\ref{thm-DKEexs} if $z_0\in \uhp$.
\end{proof}
\noindent\emph{Remark. }We have define $\tD_s$ in such a way that the
restriction of $\tD_s$ to $\proj\CC\times\uhp$ is isomorphic
to~$\D_s$. Let $c:(\z,z)\mapsto (\z,\bar z)$. An isomorphism
$\tD_s \rightarrow c^{-1} \tD_s$ is obtained by
$\tilde B(\z,z) = B(\z,\bar z)+y A(\bar z)$,
$\tilde A(z) = A(\bar z)$. So the restriction of $\tD_s$ to
$\proj\CC\times\lhp$ is isomorphic to $c^{-1}\D_s$. New in the
theorem is the description of $\tD_s$ along $\proj\CC\times
\proj\RR$. In points $(\x,\x)$ with $\x\in \proj\RR$ the surjectivity
of $C$ is the step a)\;$\Rightarrow$\; b) in Theorem~\ref{thm-conj}.


\subsection{Boundary germs for the sheaf $\D_s$}\label{sect-gbg-D}
In Subsection \ref{sect-medb} we considered sections of $\D_s$ that
extend across~$\p\H$ and established a local relation between these
sections and the sheaf $\W\om{1-s}$. In this subsection we look
instead at the sections of $\D_s$ along the inverse image
$p_1^{-1} \proj\RR$, where $p_1: \proj\CC\times \uhp \to\proj\CC$ is
the projection on the first component. The proofs will be omitted or
sketched briefly.

A first natural thought would be to consider the inductive limit
$\limind{} \D_s\bigl(U \cap(\proj\CC\times\uhp)\bigr)$, where $U$
runs through the collection of all neighborhoods of
$\proj\RR\times\proj\RR$ in $\proj\CC\times\proj\CC$, but
Corollary~\ref{cor-bdW} shows that this space is zero. Instead, we
define
\be\label{Dh-def} \dd_s \= \limind{} \D_s\bigl(U
\sm(\proj\RR\times\uhp \bigr)\bigr)\,,
\qquad \hh_s \= \limind{} \D_s\bigl( U\bigr)\,,
\ee
where the open sets $U$ run over either
\begin{enumerate} \item[(a)] the collection of open neighborhoods of
$\proj\RR \times\uhp$ in $\proj\CC\times\uhp\,$, or
\item[(b)] the larger collection of open neighborhoods of
$\proj\RR \times\uhp'$ in $\proj\CC\times\uhp'$ with $\uhp'$ the
complement of some compact subset of $\uhp\,$.
\end{enumerate}
It turns out that the direct limits in \eqref{Dh-def} are the same for
both choices. Clearly $\dd_s$ contains $\hh_s$ and the group $G$ acts
on both spaces. The canonical model $\can_s$ is a subspace of the
space $\dd_s$.

In Theorem~\ref{thm-DKEexs} we considered the sheaf morphism
$C:\D_s \rightarrow p_2^{-1}\E_s$ that sends a pair $(h,u)$ to its
second coordinate~$u=u(\z,z)$, which is locally constant in~$\z$ and
a $\ld_s$-eigenfunction in~$z$. This morphism induces a surjective
map $C:\hh_s\to\E_s$ whose kernel is the space $\vomrig s$ introduced
in~\S\ref{sect-pm}. It also induces a map (still called~$C$)
from the larger space $\dd_s$ to $\E_s\oplus\E_s$ by sending $(h,u)$
to the pair $(u_+,u_-)$, where $u_\pm(\,\cdot\,)=u(\z_\pm,\,\cdot\,)$
for any $\z_\pm\in\uhp^\pm$. This map is again surjective and its
kernel is the space $\hrig_s$ studied in~\S\ref{sect-pm}. Moreover,
the results of that subsection show that the kernels of these two
maps $C$ are related by the exact sequence
$$ 0\lra \vomrig s \lra \hrig_s \stackrel{\poiss s}\lra \E_s \lra 0\,, $$
where the Poisson map $\poiss s$ is given explicitly by
$$ \poiss s h(z,z_1) \= \frac1\pi \intCC \biggl( \frac{R(\z;z_1)}{R(\z;z)}\biggr)^s\,
h(\z,z_1)\, R(\z;z)\,d\z\qquad(z,\,z_1\in\uhp), $$
(eq.~\eqref{Poiss-flab}). Here $C_+$ (resp.~$C_-$) is a closed path in
$\uhp$ (resp.~$\lhp$) encircling $z$ and $z_1$ (resp.~$\bar z$ and
$\bar z_1$) and the right-hand side is independent of~$z_1$. Now
consider an element of $\hh_s$ represented by the pair
$(h,u) \in \D_s \bigl(U\sm(\proj\RR\times\uhp)\bigr)$ for some open
neighborhood $U$ of $\proj\RR\times\uhp$ in $\proj\CC\times\uhp$, and
define $\poiss s h(z,z_1)$ by the same formula, where $C_\pm$ are now
required to lie in the neighborhood
$\bigl\{\z\in\proj\CC\mid(\z,z_1)\in U\bigr\}$ of $\proj\RR$ and to
be homotopic to $\proj\RR$ in this neighborhood. The right-hand side
is still independent of the choice of contours $C_\pm$, and is also
independent of the choice of representative $(h,u)$ of
$[(h,u)]\in\nobreak\dd_s$, but it is no longer independent of $z_1$.
Instead, we have that the function $\poiss s h(\dv,z_1)$ belongs to
$\E_s$ for each fixed $z_1\in\uhp$ and that its dependence on~$z_1$
is governed by
  \be d_{z_1} \bigl( \poiss s (h,u)(z,z_1)\bigr) \= \bigl[
  p_s(\dv,z),\;u_+-u_-\bigr] \ee
with the Green's form as in~\eqref{Grf} and the point pair invariant
$p_s(\dv,\dv)$ as in~\eqref{pppi}.

We therefore define a space $\pE_s$ consisting of pairs $(f,v)$ where
$v$ belongs to $\E_s$ and $f:\uhp\times\uhp\rightarrow\CC$ satisfies
% and an element $v\in \E_s$ a   \emph{$p_s$-family of $\ld_s$-eigenfunctions} with \emph{direction} $v\in \E_s$ if
\beqabc
\begin{align}
f(\dv,z_1)&\;\in \E_s\text{ for each }z_1\in \uhp\,,\\
d\bigl( f(z,\dv) \bigr) &\= [p_s(\dv,z),v]\text{ on $\uhp$ for each
$z\in \uhp$}\,.
\end{align}
\eeqabc
The group $G$ acts on this space by composition (diagonally in the
case of $f$). By the discussion above we can define an equivariant
and surjective map $\ppoiss s\,:\,\hh_s\to\pE_s$ with kernel $\hh_s$
by $[(h,u)] \mapsto \bigl( \poiss s h,u_+-u_-\bigr)$. Finally, the
space $\pE_s$ is mapped to $\E_s$ by $(f,v)\mapsto v$ with kernel
$\E_s$ (because $f(\,\cdot\,,z_1)$ is constant if $v=0$). (In fact,
the space $\pE_s $ is isomorphic to $\E_s\times\E_s$ as a vector
space, though not as a $G$-module, by the map sending $(f,v)$ to
$\bigl(f(\,\cdot\,,i),v\bigr)$.) Putting all these maps together, we
can summarize the interaction of the morphisms $C$ and $\poiss s$ by
the following commutative diagram with exact rows and columns:
%\begin{figure}[ht]
\[ \xymatrix@R=.5cm{
&0 \ar[d]
&0 \ar[d]
&0 \ar[d]
\\
0 \ar[r]
&\vomrig s \ar[r] \ar[d]
&\hh_s \ar[r]^(.4)C \ar[d]
&\E_s \ar[r] \ar[d]^{\mathrm{diag}}
&0
\\
0 \ar[r]
&\hrig_s \ar[r] \ar[d]^{\poiss s}
&\dd_s \ar[r]^(.4)C \ar[d]^{\ppoiss s}
& \E_s\oplus\E_s \ar[r] \ar[d]^{(1,-1)}
& 0
\\
0 \ar[r]
&\E_s \ar[d] \ar[r]
& \pE_s \ar[d] \ar[r]^(.42){\mathrm{pr}_2}
& \E_s \ar[d] \ar[r]
&0
\\
& 0
&0
&0 } \]
\bigskip



\section{Boundary splitting of eigenfunctions}\label{sect-split}
In the introduction we mentioned that eigenfunctions often have the
local form
$y^s\times\nobreak(\text{analytic})\,+\, y^{1-s}\times(\text{analytic})\;$
near points of $\RR$. Here we consider this pheno\-menon more
systematically in both the analytic context (\S\ref{sect-bsplan}) and
the differentiable context
(\S\ref{sect-bspldi}). This will lead in particular to a description
of both $\E_s^\om=\poiss s(\V\om s)$ and
$\E_s^\infty=\poiss s(\V\infty s)$ in terms of boundary behavior.


As stated in the introduction, results concerning the boundary
behavior of elements of $\E_s$ are known (also for more general
groups; see, eg.,~\cite{KKMOOT} and~\cite{BS}). However, our approach
is more elementary and also includes several formulas that do not
seem to be in the literature and that are useful for certain
applications (such as those in~\cite{BLZ-p}.

% includes results that do not seem to be in the literature.  
% includes several results that do not seem to be in the literature and is much more explicit, making
% However, our presentation is self-contained, and the results do not seem to be identical to the ones in the literature.



\subsection{Analytic case}\label{sect-bsplan}
In Proposition~\ref{propsplit} we showed that the space $\F s$ of
boundary germs is the direct sum of $\E_s$ (the functions that extend
to the interior) and $\W\om s$ (the functions that extend across the
boundary). We now look at the relation of these spaces with
$\E^\om_s$, the image in $\E_s$ of $\V\om s$ under the Poisson
transformation.

If $s\neq\frac12$ we denote by $\F s^\om$ the direct sum of $\W\om s$
and $\W\om{1-s}$. (That this sum is direct is obvious since for
$s\neq \frac12$ an eigenfunction~$u$ cannot have the behavior
$y^s \times(\text{analytic})$ and at the same time
$y^{1-s}\times(\text{analytic})$ near points of~$\RR$.)
For $s=\frac12$ we will define $\F{1/2}^\om$ as a suitable limit of
these spaces in the following sense. If $s\neq \frac12$, an element
of $\F s^\om$ is locally (near $x_0\in \RR)$ represented by a linear
combination of $y^s$ and $y^{1-s}$ with coefficients that are
analytic in a neighborhood of~$x_0$. Replacing $y^s$ and $y^{1-s}$ by
$\frac12\left(y^s+y^{1-s}\right)$ and
$\frac1{2s-1}\left(y^s-y^{1-s}\right)$, we see that an element of
$\F{1/2}^\om$ should
(locally)
have the form $A(z)\,y^{1/2}\, \log y+B(z)\,y^{1/2}$ with $A$ and $B$
analytic at~$x_0$. We therefore define $\F{1/2}^\om$ (now using disk
model coordinates to avoid special explanations at~$\infty$) as the
space of germs in~$\F{1/2}$ represented by
\be\label{F1/2} f(w) \= (1-|w|^2)^{1/2}\;\Bigl(
A(w)\log(1-|w|^2)+B(w)\Bigr)\,,\ee
with $A$ and $B$ real analytic on a neighborhood of $\uc$ in~$\CC$. We
have a $G$-equivariant exact sequence
\be\label{Fes} 0 \lra \W\om{1/2} \lra \F{1/2}^\om \stackrel \tau\lra
\W\om{1/2}\lra 0
\ee
where $\tau$ sends $f$ in~\eqref{F1/2} to~$A$. The surjectivity
of~$\tau$ is a consequence of the following proposition, which we
will prove below. This proposition shows that for all $s$ with
$0<\re s<1$ the space $\F s^\om$ is isomorphic as a $G$-module to the
sum of two copies of~$\V\om s$.
\begin{prop}\label{prop-F1/2}The exact sequence \eqref{Fes} splits
$G$-equivariantly.
\end{prop}

The splittings $\F s = \E _s \oplus \W\om s = \E_s \oplus \W\om{1-s}$
show that non-zero elements of~$\E_s$ cannot belong to $\W\om s$ or
$\W\om{1-s}$. The following theorem shows that they \emph{can} be in
$\F s^\om$, and that this happens if and only if they belong
to~$\E^\om_s$.
\begin{thm}\label{thmEF}Let $0<\re s<1$. Then
\[ \E^\om_s = \E_s \cap \F s^\om \,,\]
and
$\F s ^\om\= \E^\om_s \oplus \W\om s = \E^\om_s \oplus \W\om{1-s}$.
\end{thm}
So for $s\neq \frac12$, the space $\F s^\om$ is the direct some of
each two of the three isomorphic subspaces $\E^\om_s$, $\W\om s$ and
$\W\om{1-s}$. For $s=\frac12$ two of these subspaces coincide.
\medskip

We discuss the cases $s\neq \frac12$ and $s=\frac12$ separately.
\begin{prop}\label{propEF}Let $s\neq \frac12$. For each
$\ph \in \V\om s$ we have
\be\label{poiss-bdspl} \poiss s \ph \= c(s) \,\tpoiss s \ph + c(1-s)\,
\tpoiss {1-s} I_s \ph\,,\ee
where, with $\bb(s)$ as in \eqref{btdef-p}, the factor $c(s)$ is given
by
\be c(s) \= \frac{\tan\pi s}{\pi} \, \bb(s) \=
\frac1{\sqrt\pi}\,\frac{\Gf(\frac12-s)}{\Gf(1-s)}\,. \ee
\end{prop}
\begin{proof}
Since $\ph$ is given by a Fourier expansion which converges absolutely
uniformly on the paths of integration in the transformation occurring
in~\eqref{poiss-bdspl}, it is sufficient to prove this relation in
the spacial case $\eph_{s,n}$ ($n\in \ZZ$). We have
$\poiss s \eph_{s,n} = (-1)^n \, \frac{\Gf(s)}{\Gf(s+n)} P_{\!s,n}$
and $\tpoiss s \eph_{s,n} = (-1)^n
\, \frac{\Gf(s+\frac12)}{\sqrt\pi\, \Gf(s+n)} \, Q_{s,n}$. See
\S\ref{sect-Pta}. The relations \eqref{pqrel} and \eqref{Iseq} give
the lemma for $\ph=\eph_{s,n}$ for all $n\in \ZZ$.
\end{proof}

\noindent\emph{Remark. } One can also give a direct (but more
complicated) proof of \eqref{poiss-bdspl} for arbitrary
$\ph \in \V\om s$, without using the basis
$\bigl\{ \eph_{s,n} \bigr\}$, by writing all integral transforms
explicitly and moving the contours suitably.
\medskip

The proof of Theorem~\ref{thmEF} (for $s\neq \frac12$)
follows from Proposition~\ref{propEF}. The inclusion
$\E^\om_s \subset \F s^\om$ is a consequence of the more precise
formula~\eqref{poiss-bdspl}. For the reverse inclusion we write an
arbitrary $u\in  \F s^\om$ in the form $c(s)\,\tpoiss s \ph + v$ with
$v\in \W\om{1-s}$ and $\ph\in \V\om s$. If $u\in \E_s$, then
$u-\poiss s \ph =
v-c(1\!-\nobreak\!s) \tpoiss{s-1}I_s\ph \in \E_s \cap  \W\om{1-s}=\{0\}$,
so $u=\poiss s \ph \in \E^\om_s$. This completes the proof.\medskip

We can summarize this discussion and its relation with the Poisson
transformation in the following commutative diagram of $G$-modules
and canonical $G$-equivariant morphisms
\[ \xymatrix@R=1.4cm{
& \V\om s \ar@/^/[rr]^{I_s} \ar@/_/[dl]_{\tpoiss s}
\ar[rd]^(.55){\poiss s}
&&\V\om{1-s} \ar@/^/[ll]^{I_{1-s}} \ar[dl]_(.55){\poiss {1-s}}
\ar@/^/[dr]^{\tpoiss{1-s}}
\\
\W\om s \ar@/_/[ru]_{\rho_s} \ar@{{}-{}}[rr]^\cong
&& \E_s^\om=\E_{1-s}^\om \ar@{{}-{}}[rr]^\cong
&& \W\om{1-s} \ar@/^/[lu]^{\rho_{1-s}} } \]
together with the fundamental examples (and essential ingredient in
the proof): {\small\[ \xymatrix@R=1.4cm@C=0cm{
& R(\dv;z_0)^s \ar@{<->}[rr] \ar@{<->}[dl] \ar@{<->}[dr]
&& R(\dv;z_0)^{1-s} \ar@{<->}[dl] \ar@{<->}[dr]
\\
b(s)q_s(\dv,z_0) \ar@{<->}[rr]
&& \txt{$p_s(\dv,z_0)=p_{1-s}(\dv,z_0)=$\\
$\pi^{-1}\tan\pi s\,\bigl(q_s(\dv,z_0)+q_{1-s}(\dv,z_0)\bigr)
$}
&& b(1-s)q_{1-s}(\dv,z_0) \ar@{<->}[ll] } \]}
\bigskip

We now turn to the case $s=\frac12$. We have to prove
Proposition~\ref{prop-F1/2} and Theorem~\ref{thmEF} in this case.

To construct a splitting $\s:\W\om{1/2}\lra \F{1/2}^\om$ of the exact
sequence~\eqref{Fes} we put $\s Q_{1/2,n} =
-\frac{\pi^2}2\, P_{\!1/2,n} \in \E_s$ for $n\in \ZZ$. Since
$P_{\!1/2,n} \in \poiss {1/2} \V\om{1/2}$, we have $\s
Q_{n,1/2}\in \E^\om_s$. Further, $\tau\s Q_{1/2,n}=Q_{1/2,n}$ by
\eqref{qasb} and~\eqref{p-bd-as}. The $Q_{1/2,n} \in \W\om{1/2}$ with
$n\in \ZZ$ generate a dense linear subspace of $\W\om{1/2}$ for the
topology of $\V\om{1/2}$ transported to $\W\om{1/2}$ by
$\tpoiss {1/2} :\V\om{1/2}\lra \W\om{1/2}$. Hence there is a
continuous linear extension $\s:\W\om{1/2}\lra \E^\om_{1/2}$. The
generators $\EE^+$ and $\EE^-$ of the Lie algebra of~$G$ act in the
same way on the system $\bigl(Q_{s,n}\bigr)_n$ as on on the system
$\bigl(P_{\!s,n}\bigr)_n$.
(See \S\ref{sect-Lie}, and use case~\textbf{G} in Table~\ref{tab-Pta}
of \S\ref{sect-Pta} and case~\textbf{a} in Table~\ref{tab-bg} of
\S\ref{sect-bg}.) So $\s$ is an infinitesimal $G$-morphism, and,
since $G$ is connected, a $G$-morphism. The splitting
$\s:\W\om{1/2}\rightarrow \F{1/2}^\om$ also gives the surjectivity
of~$\tau$, and hence the exactness of the sequence \eqref{Fes}.

Since $\s Q_{1/2,n}$ belongs to $\E^\om_{1/2}$, we have
$\s(\W\om{1/2}) \subset \E^\om_{1/2}$. Since $\E^\om_{1/2}$ is an
irreducible $G$-module this inclusion is an equality. This gives
$\F{1/2}^\om=\E^\om_{1/2}\oplus \W\om{1/2}$ and
$\E^\om_{1/2} \subset \E_{1/2} \cap \F{1/2}^\om$. The reverse
inclusion the follows by the same argument as for
$s\neq\frac12$.\qed\medskip

\noindent\emph{Remark. }The case $s=\frac12$ could also have been done
with explicit elements. For each $s$ with $0<\re s<1$ and each
$n\in \ZZ$, the subspace $\F{s,n}^\om$ of $\F s^\om$ in which the
elements $\matr{\cos\th}{\sin\th}{-\sin\th}{\cos\th}$ act as
multiplication by $e^{2in\th}$ has dimension~$2$. In the family
$s\mapsto \F{s,n}^\om$ there are three families of non-zero
eigenfunctions $  s\mapsto P_{s,n}\in \E^\om_s$,
$s\mapsto Q_{s,n}\in \W\om s$ and $s\mapsto Q_{1-s,n}\in \W\om{1-s}$.
For $s\neq \frac12$ each of these functions can be expressed as a
linear combination of the other two, as given by~\eqref{pqrel}, which
is at the basis of our proof of Proposition~\ref{propEF}. At
$s=\frac12$, the elements $Q_{s,n}$ and $Q_{1-s,n}$ coincide. This is
reflected in the singularities at $s=\frac12$ in the
relation~\eqref{pqrel}. The families $s\mapsto P_{s,n}$ and
$s\mapsto Q_{s,n}$ provide a basis of $\F s^\om$ for all~$s$,
corresponding to the decomposition
$\F s^\om = \E^\om_s \oplus \W\om s$. Relation \eqref{pqrel} implies
\[ P_{1/2,n} \= \frac{-2}{\pi^2} \, \frac d {ds} Q_{s,n}
\Bigr|_{s=1/2}\,,\]
which explains the logarithmic behavior at the boundary.

\subsection{Differentiable case}\label{sect-bspldi}
In the previous subsection we described the boundary behavior of
elements of $\E^\om_s = \poiss s \V\om s$ in terms of convergent
expansions. In the differentiable case the spaces $\W p s$ consist of
boundary jets, not of boundary germs. So a statement like that in
Theorem~\ref{thmEF} seems impossible. Nevertheless, we have the
following generalization of Proposition~\ref{propEF}:
\begin{prop}
\begin{enumerate}
\item[i)] Let $p\in \NN$, $p\geq 2$, and $s\neq \frac12$. For each
$\ph \in \V p s$ there are $b\in \G^p_s$ representing
$c(s)\, \tpoiss s \ph \in \W p s$ and $a\in \G^{p-1}_{1-s}$
representing $c(1-\nobreak s)\, \tpoiss{1-s} I_s\ph \in \W{p-1}{1-s}$
such that
\be\label{spl-p} \poiss s \ph(w) \= b(w)+a(w) + \oh\bigl(
(1-|w|^2)^{p-s}\bigr)
\qquad
(|w|\uparrow1)
\,. \ee
\item[ii)] Let $s\neq \frac12$. For each $\ph \in \V\infty s$ there
are $b\in \G^\infty_s$ and $a\in \G^\infty_{1-s}$ representing
$c(s) \,\tpoiss s \ph$ and $c(1-\nobreak s)\,\tpoiss{1-s}\ph$,
respectively, such that for each $N\in \NN$
\be \poiss s \ph(w) \;\sim\; b(w) + a(w) + \o\bigl( (1-|w|^2)^N
\bigr)\qquad
(|w|\uparrow1)\,. \ee
\end{enumerate}
\end{prop}
\begin{proof}The proof of Proposition~\ref{propEF} used the fact that
the $\eph_{s,n}$ generate a dense subspace of $\V\om s$ and that the
values of Poisson transforms and transverse Poisson transforms are
continuous with respect to this topology. That reasoning seems hard
to generalize when we work with boundary jets. Instead, we use the
explicit Lemma~\ref{lem-smoothw}.\smallskip


A given $\ph \in \V p s$ can be written as a sum of elements in
$\V p s$ each with support in a small interval in $\p\H$. With the
$G$-action this reduces the situation to be considered to
$\ph \in C^p_c(I)$ where $I$ is a finite interval in~$\RR$.
Proposition~\ref{prop-W-supp} shows that $\poiss s \ph$ represents an
element of $\W\om{1-s}\bigl( \proj\RR\sm\nobreak I\bigr)$. So we can
restrict our attention to $\poiss s \ph (z)$ with $z$ near $I$, and
work in the line model.

We take $\al$ and $\bt$ as in Lemma~\ref{lem-smoothw} with $N=p$. Then
\begin{align*} \poiss s \ph(z) &\=
\pi^{-1}\,y^{1-s}\int_{-\infty}^\infty (t^2+y^2)^{s-1}\, \ph(t+x)\,
dt\\
&\= \frac1\pi \, y^s \int_{-\infty}^\infty (t^2+1)^{s-1}\,\ph(x+yt)\,
dt \= y^s A(z) + y^{1-s} B(z)\,,\end{align*}
with
\[ B(z) \= \pi^{-1} \int_{-\infty}^\infty \bt(t)\, \ph(x+yt)\,
dt\,,\qquad A(z) \= \pi^{-1} y^{2s-1} \int_{-\infty}^\infty
\al^{(p)}(t)\, \ph(x+yt)\, dt\,.\]

We consider $B(z)$ and $A(z)$ for $x\in I$ and $0<y\leq 1$. The decay
of $\bt$ implies that
\[ B(z) \= \frac1\pi \sum_{n=0}^p \frac{\ph^{(n)}(x)}{n!} y^n
\int_{-\infty}^\infty t^n\, \bt(t)\, dt \,+\, \o\bigl(y^p\bigr)\,. \]
In \eqref{bti} we have computed the integrals. We arrive at
\be B(z) \= c(s) \sum_{k=0}^{[p/2]} \frac{(-1/4)^k
\Gf(s+\frac12)}{k!\; \Gf(k+s+\frac12)} \,\ph^{(2k)}(x)\, y^{2k} +
\o(y^p)\,. \ee
A comparison with \eqref{anrel1} shows that $y^s\,B(z)$ has the
asymptotic behavior near~$I$ of representatives of
$c(s) \, \tpoiss s \ph$.


In the second term we apply $p$-fold integration by parts.
\[ A(z) \= (-1)^p \pi^{-1} y^{2s-1+p} \int_{-\infty}^\infty \al(t)
\,\ph^{(p)}(x+yt)\, dt\,.\]
For fixed $\ph$, this expression is a holomorphic function of $s$ on
the region $\re s>0$. In the computation we shall work with $\re s$
large.

The function $h\mapsto (1+\nobreak h)^{s-1}$ has a Taylor expansion at
$h=0$ of any order $R$, with a remainder term $\oh(h^{R+1})$ that is
uniform for $h\geq 0$. This implies that $\al(t)$ has an expansion of
the form $\al(t) =
\sum_{n=0}^R b_n |t|^{p+2s-2-2n} + \oh(|t|^{p+2s-2R-4}\bigr)$,
uniformly for $t\in \RR\sm\{0\}$. We take $R=[p/2]$, and use the
 relation $\p_t^p \al(t) = (1+t^2)^{s-1}-\bt(t)$ and the decay of
$\bt(t)$ to conclude that
\be \al(t) \= \sum_{0\leq n\leq p/2} \binom{s-1}n \frac{(\sign t)^p
|t|^{2s-2n+p-2}}{(2s-2n-1)_p} + \oh\bigl( |t|^{2s-3}\bigr)\,. \ee
We compute this with $\re s>1$. The error term contributes to~$A(z)$:
\be y^{2s-1+p} \int_{t=-\infty}^\infty \oh\bigl(|t|^{2s-3}\bigr)\,
\ph^{(p)}(x+yt)\, dt \= \oh\bigl( y^{p+1} \bigr)\,. \ee
(We have replaced $t$ by $t/y$ in the integral.)
The term of order~$n$ contributes:
\begin{align*}
&\frac{(-1)^p y^{2n}}\pi \binom{s-1}n
\frac{y^{-2s-p+2n+1}}{(2s-2n-1)_p} \int_{-\infty}^\infty (\sign t)^p
|t|^{2s+p-2n-2} \,\ph^{(p)}(x+t)\, dt
\displaybreak[0]
\\
&\quad\= \frac{(-1)^p y^{2n}\, \Gf(s) \,\Gf(2s-2n-1)}{\pi\, n!\;
\Gf(s-n)\, \Gf(2s-2n-1+p)} \; (-1)^{p-2n} (2s-1)_{p-2n} \\
&\quad\qquad\hbox{} \cdot
\int_{-\infty}^\infty |t|^{2s-2} \ph^{(2n)}(x+t)\,
dt\qquad(\text{partial integration $p-2n$ times})\,.
\end{align*}
In \eqref{Iln} we see that the holomorphic function
$\int_{-\infty}^\infty  |t|^{2s-2} \ph^{(2n)}(x+t)\, dt $ continued
to the original value of $s$ gives us $\bb(s-\nobreak\frac12) \,
(I_s\ph)^{(2n)}(x)$, provided $2n<p$. We have
$I_s \V p s \subset \V{p-1}{1-s}$, but not necessarily
$I_s \ph \in \V p s$. For even $p$ we move the contribution
$\oh(y^p)$ to the error term. The terms of order $n<p/2$ give
\begin{align*}
&\frac{ y^{2n}\, \Gf(s)\, \Gf(2s-2n-1)
}{ \pi\, n! \Gf(s-n)\, \Gf(2s-1)
}\; \frac{\sqrt\pi\, \Gf(s-\frac12)}{\Gf(s)}\; (I_s \ph)^{(2n)}(x)\\
&\quad\= \frac{\tan\pi(1-s)}{\sqrt\pi}\,
\frac{\Gf(1-s)}{\Gf(\frac32-s)}\; \frac{(-1/4)^k
\,\Gf(\frac32-s)}{n!\; \Gf(\frac32-s+n)}\,
(I_s\ph)^{(2n)}(x)\,y^{2k}\,.
\end{align*}
Thus we arrive at
\be A(z) \= c(1-s) \sum_{0\leq n < p/2} \frac{(-1/4)^n
\,\Gf(\frac32-s)}{n!\; \Gf(\frac32-s+n)}\,
(I_s\ph)^{(2n)}(x)y^{2k}
+ \oh\bigl( y^{2\bigl[ \frac{p+1}2\bigr]} \bigr)\,. \ee
Again we have arrived at the expansion a representative of
$\W{p-1}{1-s}$ should have according to~\eqref{anrel1}. This
completes the proof of part~i).

In view of Definition~\ref{GN-def} the estimate \eqref{spl-p} holds
for all representatives $b\in \G^p_s$ and $a\in \G^p_{1-s}$ of
$c(s)\, \tpoiss s \ph$, respectively
$c(1-\nobreak s)\, \tpoiss{1-s}\ph$. In particular, for $\ph \in
\V\infty s$, this estimate holds for each $p\in \NN$, $p\geq 2$, for
representatives $b_\infty\in\G^\infty_s$ of $c(s)\,\tpoiss s \ph$ and
$a_\infty \in \G^\infty_{1-s}$ of $c(1-\nobreak s)\,\tpoiss{1-s}\ph$.
This implies part~ii) of the proposition.
\end{proof}
\smallskip

\appendix
\newcounter{tnm}
\setcounter{equation}{0}
\renewcommand\thesection{A}
\renewcommand\theequation{A.\arabic{equation}}
\renewcommand\equationstandard{A.\arabic{equation}}
\renewcommand\thesubsection{A.\arabic{subsection}}
\section*{Appendix: Examples and explicit
formulas}\label{sect-examples}
We end by giving a collection of definitions and formulas that were
needed in the main body of the paper or that illustrate its results.
In particular, we describe a number of examples of eigenfunctions of
the Laplace operator (in~\ref{sect-spf}), of Poisson transforms
(in~\ref{sect-Pta}), of transverse Poisson transforms
(in~\ref{sect-bg}), and of explicit potentials of the Green's form
$\{u,v\}$ for various special choices of $u$ and~$v$
(in~\ref{sect-pot}), as well as some formulas for the action of the
Lie algebra of~$G$ (in~\ref{sect-Lie}).


\subsection{Special functions and equivariant elements of
$\E_s$}\label{sect-spf}Let $H\subset G$ be one of the subgroups
$N=\Bigl\{n(x)\;:\; x\in \RR\Bigr\}$, \
$A=\Bigl\{a(y)\;:\;y>0\Bigr\}$ \
or~$K=\bigl\{k(\th)\;:\; \th\in \RR/\ZZ\bigr\}$ with
\be \label{nakd}
 n(x) \= \matc 1x01\,,\qquad a(y)\=\matc{\sqrt y}00{1/\!\!\sqrt
y}\,,\qquad k(\th) \= \matr{\cos\th}{\sin\th}{-\sin\th}{\cos\th}\,.
\ee
For each character $\ch$ of $H$ we determine the at most
two-dimensional subspace $\E_{s,\ch}^H $ of $\E_s$ transforming
according to this character.

\subsubsection{Equivariant eigenfunctions for the unipotent group
$N$}\label{sect-Bes}
The characters of $N$ are $\ch_\al:n(x) \mapsto e^{i\al x}$ with
$\al\in \RR$. If $u \in \E_{s,\al}^N$ (we write $\E_{s,\al}^N$
instead of $\E_{s,\ch_\al}^N$), then $u(z) = e^{i\al x} f(y)$, where
$f$ satisfies the differential equation
\be y^2 f''(y) \= (s^2-s+\al^2 y^2)f(y)\,. \ee
This can also be applied to $\E_{s,\al}^N(U)$ for any connected
$N$-invariant \alsochanged{$U$ inserted, $\uhp$ instead of
$\H$}subset $U$ of~$\uhp$. For the trivial character, i.e., $\al=0$,
this leads to the basis $z\mapsto y^s$, $z\mapsto y^{1-s}$ of
$\E_{s,0}^N$ if $s\neq \frac12$, and $z\mapsto y^{1/2}$,
$z\mapsto y^{1/2}\log y$ if $s=\frac12$. For non-zero~$\al$ we have:
\beqabc\begin{align}
\label{k-def}
k_{s,\al}(z) &\= \sqrt y\, K_{s-1/2}(|\al|y) \, e^{i\al x}\\
\nonumber
&\= \frac{2^{s-\frac32}\,\Gf(s)}{\sqrt\pi\, |\al|^{s-\frac12}}\,
e^{i\al x} \int_{-\infty}^\infty e^{i\al t}\,\frac{y^s\,
dt}{(y^2+t^2)^s}\,,\\
\label{i-def}
i_{s,\al}(z) &\= \frac{\Gf(s+\frac12)}{|\al/2|^{s-\frac12}} \,\sqrt
y\, I_{s-1/2}(|\al|y) \, e^{i\al x}\,,
\end{align}
\eeqabc
with the modified Bessel functions
\be\label{IK} I_u(t) \= \sum_{n=0}^\infty \frac { (t/2)^{u+2n}}{n!\;
\Gf(u+1+n)}\,,\qquad K_u(t) \= \frac\pi 2\;
\frac{I_{-u}(t)-I_u(t)}{\sin\pi u}\,. \ee
The element $i_{\al,s}$ \alsochanged{}represents a boundary germ in
$\W\om s(\RR)$. The normalization of $i_{s,\al}$ is such that the
restriction $\rho_s i_{\al,s} (x)=e^{i\al x}$ in the line model.

The elements $k_{s,\al}$ and $i_{s,\al}$ form a basis of
$\E^N_{s,\al}$ for all $s$ with $0<\re s<1$. For $s\neq \frac12$
another basis is $i_{s,\al}$ and $i_{1-s,\al}$. The element
$k_{s,\al}$ is invariant under $s\mapsto 1-s$, and
\be k_{s,\al} \= \frac{\Gf(\frac12-s)}{|\al|^{1/2-s}2^{s+1/2}}\,
i_{s,\al} + \frac{\Gf( s-\frac12)}{|\al|^{s-1/2} 2^{3/2-s}}\,
i_{1-s,\al} \ee
gives (for $s\neq \frac12$) a local boundary splitting as an element
of $\W\om s(\RR) \oplus \W\om{1-s}(\RR)$.

For the trivial character, $k_{s,\al}$ may be replaced by
\be\label{ell-def} \ell_s(z) \= \frac{y^s-y^{1-s}}{2s-1}
\quad\text{for }s\neq\frac12\,,\qquad \ell_{1/2}(z) = y^{1/2}\, \log y
\,. \ee

\subsubsection{Equivariant eigenfunctions for the compact group
$K$}\label{sect-Leg}The characters of $K$ are $k(\th) \mapsto
e^{in\th}$ with $n\in \ZZ$ and $k(\th)$ as in~\eqref{nakd}. If
$u(re^{i\th}) = f(r)e^{i n \th}$ is in $\E^K_{s,n}(U)$, with a
$K$-invariant subset $U\subset \uhp$, then $f$ satisfies the
differential equation
\be\label{r-F-dv}-\frac14 (1-r^2)^2 \bigl( f''(r)+r^{-1}f'(r) -
n^2r^{-2}f(r)
\bigr) \= s(1-s) f(r)\,. \ee
For general annuli in~$\H$, the solution space has dimension~$2$, with
basis
\be\label{Leg-d}
\begin{aligned}
P_{\!s,n}(r e^{i\th}) &\= P_{\!1-s,n}(re^{i\th}) \= P^n_{\!s-1}
\biggl( \frac{1+r^2}{1-r^2} \biggr)\, e^{in\th}\,,\\
Q_{s,n}(r e^{i\th}) &\= Q^n_{s-1} \biggl( \frac{1+r^2}{1-r^2} \biggr)
\,e^{in\th}\,,
\end{aligned}
\ee
with the Legendre functions
\be\label{Leg}
\begin{aligned}
P^m_{\!s-1}&\biggl( \frac{1+r^2}{1-r^2}\biggr)
\= \frac{\Gf(s+m)}{|m|!\; \Gf(s-|m|)} \, r^{|m|} \hypg21{
1-s,s;1+|m|;\frac{r^2}{r^2-1}}\\
&\= \frac{\Gf(s+m)}{|m|!\; \Gf(s-|m|)} \, r^{|m|} \, (1-r^2)^s\,
\hypg21{s,s+|m|;1+|m|;r^2}\\
&\= \frac{\Gf(s+m)}{|m|!\; \Gf(s-|m|)} \, r^{|m|} \, (1-r^2)^{1-s}\,
\hypg21{1-s,1-s+|m|;1+|m|;r^2} \,,\\
Q^m_{s-1}&\biggl( \frac{1+r^2}{1-r^2}\biggr) \= \frac{(-1)^m}2 \,
\frac{\Gf(s)\Gf(s+m)}{\Gf(2s)}\,
\frac{(1-r^2)^s}{r^m}\hypg21{s-m,s;2s;1-r^2}
\\
&\= \frac{(-1)^m}2 \, \frac{\Gf(s)\Gf(s+m)}{\Gf(2s)}\,\frac{(1-r^2)^s}
{r^{2s-m}}\hypg21{s-m,s;2s;1-r^{-2}} \,,
\end{aligned}
\ee
and the hypergeometric function $F={}_{2}F_{\!1}$ given for $|z|<1$ by
\be \hypg21{a,b;c;z}\=\sum_{n\geq 0} \frac{(a)_n (b)_n }{(c)_n}
\,\frac{z^n}{n!}\,,\qquad\text{with }\quad
(a)_n \= \prod_{j=0}^{n-1} (a+j)\,. \ee
(See \cite{EMOT-p}, 2.1, 3.2 (3), 3.3.1 (7), 3.3.1 (1), 3.2 (36) and
2.9
(2) and~(3).)
The space $\E^K_{s,n}(\H)$ is spanned by $P_{\!s,n}$ alone, since
$Q_{s,n}(r)$ has a singularity as $r\downarrow0$:
\be\label{qas0} Q_{s,n}(r) \=\begin{cases}
-\log r \, \bigl(1+r\cdot(\text{analytic in $r$})\bigr) +
(\text{analytic in $r$})&\text{ if }n=0\,,\\
\frac12 \, (|n|-1)!\; \frac{\Gf(s+n)}{\Gf(s+|n|)} r^{-|n|}(1+
r\cdot(\text{analytic in $r$})\bigr)&\\
\qquad\hbox{} + \log r \cdot (\text{analytic in $r$})
&\text{ otherwise\,.}
\end{cases}
\ee
See \cite{EMOT-p}, 3.9.2 (5)--(7) for the leading terms, and 2.3.1 for
more information. Directly from \eqref{Leg} we find for
$r\downarrow0$
\be P_{\!s,n}(r) = \frac{\Gf(s+n)}{|n|!\; \Gf(s-|n|)} \,r^{|n|} \,
\bigl(1+ r\cdot(\text{analytic in $r$})\bigr)\,. \ee

The solution $Q_{s,n}$ is special near the boundary $\uc$ of $\disk$.
As $r\uparrow1$:
\be\label{qasb} Q_{s,n}(r) \= (-1)^n\, \frac{\sqrt\pi
\,\Gf(s+n)}{\Gf(s+\frac12)}\; 2^{-2s}
(1-r^2)^s \bigl(1+ (1-r)\cdot(\text{analytic in $1-r$})\bigr)\,. \ee
Thus, $Q_{s,n} \in \W\om s$, and $\rho_s Q_{s,n}(\x) \= (-1)^n \,
\frac{\sqrt\pi\, \Gf(s+n)}{\Gf(s+\frac12)} \, \x^n$ on~$\uc$.

For $s\neq \frac12$ we have:
\be\label{pqrel} P_{\!s,n} \= \frac1\pi \tan\pi s \, \bigl( Q_{s,n} -
Q_{1-s,n}\bigr)\,. \ee
(The formula in \cite{EMOT-p}, 3.3.1, (3) gives this relation with a
minus sign in front of~$\frac1\pi$.) This relation confirms that
$P_{\!1-s,n}=P_{\!s,n}$, and forms the basis of the boundary
splitting in~\eqref{poiss-bdspl}. It shows that in the asymptotic
expansion of $P_{\!s,n}(r)$ as $r\uparrow0$ there are non-zero terms
with $(1-\nobreak r)^s$ and with $(1-\nobreak r)^{1-s}$. At
$s=\frac12$, we have as $r\uparrow 1$
\be \label{p-bd-as}
P_{\!1/2,n}(re^{i\th)} \= - \frac{(-1)^n\,
\Gf(\frac12+n)}{\pi^{3/2}}\,
(1-r^2)^{1/2}\,\log(1-r^2)+ \oh(1)\,. \ee
So $P_{\!s,n}$ is not in $\W\om s(I)$ for any interval $I\subset \uc$.



\subsubsection{Equivariant eigenfunctions for the torus
$A$}\label{sect-A} The characters of $A$ are of the form $a(t)\mapsto
t^{i\al}$ with $\al\in \RR$. We use the coordinates
$z=\rho e^{i\phi}$ on $\uhp$, for which $a(t)$ acts as
$(\rho,\phi)\mapsto
(t\rho,\phi)$. If $u(\rho e^{i\phi})f(\cos\phi)$ is in $\E^A_{s,\al}$,
then $f$ satisfies on $(-1,1)$ the differential equation
\be -(1-t^2)^2 f''(t) +t(1-t^2)f'(t) +\bigl( \al^2(1-t^2) -s(1-s)
\bigr)f(t)=0\,. \ee
This leads to the following basis of the space $\E^A_{s,\al}$:
\be\label{fpm-frlt}
\begin{aligned}
f^+_{s,\al}(\rho e^{i\phi}) &\= \rho^{i\al} (\sin \phi)^s
\hypg21{\frac{s+i\al}2,\frac{s-i\al}2;\frac12;\cos^2\phi}\,,\\
f^-_{s,\al}(\rho e^{i\phi}) &\= \rho^{i\al} \cos\phi \,(\sin\phi)^s
\hypg21{\frac{s+i\al+1}2,\frac{s-i\al+1}2;\frac32;\cos^2\phi}\,.
\end{aligned}
\ee
The $+$ or $-$ indicates the parity under $z\mapsto -\bar z$. In
particular:
\be\label{fpmdif} f^+_{s,\al}(i)\=1\,,\quad\frac{\p f^+_{s,\al}}{\p
\phi} (i)\=0\,,\quad f^-_{s,\al}(i)\=0\,,\quad \frac{\p
f^-_{s,\al}}{\p \phi}(i) \= -1\,.\ee
Relation
(2), \S2.9 in \cite{EMOT-p} shows that $f^+_{1-s,\al}=f^+_{s,\al}$ and
$f^-_{1-s,\al}=f^-_{s,\al}$.

For the boundary behavior it is better to employ the Kummer relation
(33) in \S2.9 of \cite{EMOT-p}, and express these functions in terms
of
\be\label{frl} \rho^{i\al} (\sin \phi)^s
\hypg21{\frac{s+i\al}2,\frac{s-i\al}2; s+\frac12;\sin^2\phi}\,.\ee
In applying this Kummer relation one has to choose
$\sqrt{\sin^2\phi}$. This produces a singularity, and the expression
in \eqref{frl} describes an element of $\E_s\left( \uhp \sm i\RR_+
\right)$. Denote by $f^R_{s,\al}$ the restriction to
$0<\phi<\frac\pi2$ and by $f^L_{s,\al}$ the restriction to
$\frac\pi2<\phi<\pi$. The Kummer relation implies the following
equalities:
\be\label{fRL}
\begin{aligned}
f^R_{s,\al} &\= \frac{\sqrt\pi\, \Gf(s+\frac12)}{\Gf(\frac{s+i\al+1}2)
\Gf(\frac{s-i\al+1}2)} \,f^+_{\al,s} -
\frac{2\sqrt\pi\,\Gf(s+\frac12)}
{\Gf(\frac{s+i\al}2)\Gf(\frac{s-i\al}2)}\,f^-_{\al,s}\,,\\
f^L_{s,\al} &\= \frac{\sqrt\pi\, \Gf(s+\frac12)}{\Gf(\frac{s+i\al+1}2)
\Gf(\frac{s-i\al+1}2)} \,f^+_{\al,s} +
\frac{2\sqrt\pi\,\Gf(s+\frac12)}
{\Gf(\frac{s+i\al}2)\Gf(\frac{s-i\al}2)}\, f^-_{\al,s}\,.
\end{aligned}
\ee
Thus we see that $f^R_{s,\al}$ and $f^L_{s,\al}$ extend as elements of
$\E_s$, that $f^R_{s,\al} $ represents an element of
$ \W\om s(\RR_+)$ with, in the line model,
 $\rho_s f^R_{s,\al}(x) = x^{i\al}$, and that $f^L_{s,\al}$ represents
an element of $ \W\om s(\RR_-)$ with $\rho_s f^L_{s,\al}(x) =
(-x)^{i\al}$. Inverting the relation in \eqref{fRL} one finds, for
$s\neq \frac12$, the following expressions for $f^+_{s,\al}$ and
$f^-_{s,\al}$ as a linear combination of $f^R_{s,\al}$ and
$f^R_{1-s,\al}$,
\be
\begin{aligned}
f^+_{s,\al} &\=
\frac{\sqrt\pi\,\Gf(\frac12-s)}{\Gf(\frac{1-s+i\al}2)\Gf(\frac{1-s-i\al}2)}
f^R_{s,\al} + \frac{\sqrt\pi\,\Gf(s-\frac12)}{\Gf(\frac{s+i\al}2)
\Gf(\frac{s-i\al}2)} f^R_{1-s,\al}\,,\\
f^-_{s,\al}&\= \frac{\sqrt\pi\,
\Gf(\frac12-s)}{2\Gf(1-\frac{s+i\al}2)\Gf(1-\frac{s-i\al}2)}
f^R_{s,\al} + \frac{\sqrt\pi\,
\Gf(s-\frac12)}{2\Gf(\frac{s+i\al+1}2)\Gf(\frac{s-i\al+1}s)}
f^R_{1-s,\al}\,,
\end{aligned}
\ee
and similarly of $f^L_{s,\al}$ and $f^L_{1-s,\al}$, showing that each
of these elements belongs to the direct sums
$\W\om s(\RR_+)\oplus \W\om{1-s}(\RR_+)$ and $\W\om s(\RR_-)\oplus
\W\om{1-s}(\RR_-)$, but not to $\W\om s(I)\oplus \W\om{1-s}(I)$ for
any neighborhood $I$ of $0$ or $\infty$ in~$\proj\RR$; in other
words, just as for the Bessel functions $i_{s,\al}$ and $k_{s,\al}$,
we have a local but not a global boundary splitting.


\subsection{Poisson transforms}\label{sect-Pta}Almost all of the
special elements in~\S\ref{sect-spf} belong to $\E_s$, and hence are
the Poisson transform of some hyperfunction by Helgason's
Theorem~\ref{thmHel-p}. Actually in all cases except one the function
has polynomial growth, and hence are the Poisson transform of a
distribution (Theorem~\ref{thm-John}). In Table~\ref{tab-Pta} and the
discussion below we give explicit representations of these
eigenfunctions as Poisson transforms of distributions and/or
hyperfunctions.
\begin{table}[ht]
\setlength{\doublerulesep}{\arrayrulewidth}
\renewcommand\arraystretch{1.4}
\setcounter{tnm}{0}
\renewcommand\thetnm{{\textbf{\Alph{tnm}}}}
\newcommand\rgl[3]{\stepcounter{tnm}\text{\thetnm}& #1& #2& \text{#3}
\\ \hline}
\[
\begin{array}{|c|c|c|c|} \hline
& u \in \E_s & \poiss s^{-1} u \in \V{-\om}s & \text{model} \\
\hline\hline
\rgl{y^{1-s}}{\dt_{s,\infty}:\ph^\prj \mapsto \ph^\prj(\infty)}{proj.}
%
\rgl{\frac{\Gf(\frac12-s)}{\sqrt\pi\, \Gf(1-s)}\,
y^s}{\begin{array}{c}\bold 1_s\;=\;\text{ integration against
}1\text{ for } \re s<\frac12\,,\\
\text{ with meromorphic continuation}\end{array}}{line}
%
\rgl{ \begin{array}{c} -2\frac{\Gf(\frac32-s)}{\sqrt\pi\, \Gf(1-s)}
\ell_s(z)\\
\text{$\ell_s$ as in \eqref{ell-def} }\end{array}
}{\begin{array}{c}\ph \mapsto\frac{-1}{2} \int_{-\infty}^\infty(\ph(t)
- \frac{\ph_\infty}{\sqrt{1+t^2}})\, dt\\
\text{ with }\ph_\infty = \lim_{t\rightarrow\infty}|t|^{2s-2}\ph(t)
\end{array}}{line}
%
\rgl{R(t;z)^{1-s}\quad(t\in \RR)} {\dt_{s,t} : \ph \mapsto
\ph(t)}{line}
%
\rgl{\begin{array}{c}\frac{2^{s+\frac12}|\al|^{\frac12-s}}{\sqrt\pi\,
\Gf(1-s)} k_{s,\al}(z)\\
(\al\in \RR\sm\{0\})\end{array}}{\begin{array}{c}\text{integration
against $e^{i\al t}$ for $\re s<\frac12$,}\\
\text{or integration of $-\ph'$ against $\frac{e^{i\al
x}}{i\al}$}\\\text{ for $0<\re s<1$}
\end{array}}{line}
%
\rgl{\begin{array}{c}i_{1-s,\al}(z)
\\
(\al\in \RR\sm\{0\})
\end{array}
}{\begin{array}{c}\text{support $\{\infty\} $; representative near
$\infty$:}
\\ - \frac i2 \tau\,(1+\tau^{-2})^s \hypg11{1;2-2s; i \al
\tau}\end{array}
}{proj. }
%
\rgl{ \frac{(-1)^n\Gf(s)}{\Gf(s+n)} \,p_{s,n}}{\eph_{s,n}}{circle}
%
\rgl{p_s(w',\dv)}{R^\ci(\dv;w')^s}{circle}
%

\rgl{ \frac{\Gf(1+i\al-s)\Gf(1-i\al-s)}{\pi\, \Gf(2-2s)} f^L_{1-s,\al}
}{\text{integration against $x^{i\al-s}$ on $\RR_+$}}{line}
%
\rgl{\frac{\Gf(1+i\al-s)\Gf(1-i\al-s)}{\pi\, \Gf(2-2s)} f^R_{1-s,\al}
}{\text{integration against $(-x)^{i\al-s}$ on $\RR_-$}}{line}
\hline
%
\end{array}
\]
\caption{Poisson representation of elements of $\E_s$.}
\label{tab-Pta}
\end{table}

\newcommand\Pcase[1]{\medskip\par\noindent\textbf{#1. } \ignorespaces}
\Pcase{A}In \eqref{dtsi} we have shown that $y^{1-s}$ is the Poisson
transform of the distribution~$\dt_{s,\infty}$. See \eqref{dtsi} for
an explicit description of $\dt_{s,\infty}$ as a hyperfunction.

\Pcase{B}The description of $y^s$ as a Poisson transform takes more
work. For $\re s<\frac12$ the linear form
$\bold 1_s:\ph \mapsto \frac1\pi \int_{-\infty}^\infty \ph(t)\, dt$
is continuous on $\V0{1-s}$, in the line model. Note that the
constant function~$1$ is not in~$\V\om s$, since it does not satisfy
the asymptotic behavior \eqref{asy-p} at~$\infty$. Application of
\eqref{poiss-l} gives the Poisson transform $\poiss s  \bold 1_s$
indicated in the table.

To describe $\bold 1_s$ as a hyperfunction in the line model (and also
to continue it in~$s$) we want to give representatives $g_\RR$ and
$g_\infty$ of $\bold 1_s$ on $\RR$ and $\proj\RR\sm\{0\}$, related by
$g_\infty(\z) = \z^{-2s} \, g_\RR(-1/\z)$ up to a holomorphic
function on a neighborhood of~$\RR\sm\{0\}$.

Formula \eqref{proj-repr} gives a representative $g^\prj$ in the
projective model:
\[ g^\prj(\z) \= \frac1{2\pi i}\int_{-\infty}^\infty \frac{\z+i}{t-\z}
\,
(t-i)^{s}(t+i)^{s-1}\, dt\qquad(\z\in \proj\CC\sm\proj\RR)\,.\]
(The factor $(t^2+1)^{s-1}$ comes from passage between models.)
This function extends both from $\uhp$ and from $\lhp$ across the real
axis. An application of Cauchy's formula shows that the difference of
both extension is given by $(\z^2+\nobreak 1)^s$, corresponding to
the function~$1$ in the line model. See~\eqref{line-proj}.

To get a representative near~$\infty$ we write
\be\label{co-int} \bigl( 1+ e^{2\pi i s} \bigr) \,g^\prj(\z) \=
\frac1{2\pi i} \int_C \frac{\z+i}{(z-\z)(z+i)}\, (z^2+1)^s\, dz\,,\ee
where $C$ is the contour shown below. The factor $(z^2+\nobreak 1)^s$
is multivalued on the contour
\twocolwithpictr{and is fixed by choosing
$\arg\bigl(z^2+\nobreak1\bigr)\in[0,2\pi)$. On the part of the
contour just above $(0,\infty)$ the argument of $z^2+1$ is
approximately zero, and just below $(0,\infty)$ the argument is
approximately $2\pi$. Near $(-\infty,0)$ the argument is
approximately $2\pi$ just above the real line and approximately $0$
below the real line. We take the contour so large that
$\z\in \uhp\cup\lhp$ is inside one of the loops of~$C$. If we let the
contour grow, the arcs in the upper and lower half-planes give a
contribution $o(1)$. In the limit, for $\re s<\frac12$, we are left
with twice
\smallskip
}{\setlength\unitlength{.8cm}
\begin{picture}(6,6)(-3,-3)
\put(0,-1){\line(0,1){2}}
\put(-3,0){\line(1,0){6}}
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\put(0,-1){\circle*{.1}}
\put(.1,.9){$i$}
\put(.1,-1.1){$-i$}
\thicklines
\put(-2.5,.3){\line(1,0){2.2}}
\put(-2.5,-.3){\line(1,0){2.2}}
\put(2.5,.3){\line(-1,0){2.2}}
\put(2.5,-.3){\line(-1,0){2.2}}
\put(-.3,.3){\line(1,-1){.6}}
\put(-.3,-.3){\line(1,1){.6}}
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\put(.5,2.3){\vector(-1,0){.7}}
\put(.5,-2.3){\vector(-1,0){.7}}
\end{picture}
} the integral along $(0,\infty)$ and along $(-\infty,0)$, both once
with the standard value and and once with $e^{2\pi i s}$ times the
standard value. This gives the equality~\eqref{co-int}, and the
continuation of $g^\prj$ as a meromorphic function of~$s$.


Now consider $\z\in \uhp^\pm$ with $|\z|>1$. Moving the path of
integration across~$\z$, we obtain with Cauchy's theorem that
$\bigl( 1+\nobreak e^{2\pi i s} \bigr) \,g^\prj(\z)$ is equal to
$\pm (\z^2+\nobreak 1)^s$ plus a holomorphic function of~$\z$ on a
neighborhood of~$\infty$. The term $\pm (\z^2+\nobreak 1)^s$ obeys
the choice of the argument discussed above. To bring it back to the
standard choice of arguments in $(-\pi,\pi]$, we write it as
$\z^{2s} \bigl( 1+\nobreak\z^{-2}\bigr)^s$ for $\z\in \uhp$ and as
$- (-\z)^{2s} \bigl( 1+\nobreak\z^{-2}\bigr)^s$ for $\z\in\lhp$. The
factor $(1+\nobreak\z^{-2}\bigr)^s$ is what we need to go back to the
line model with~\eqref{line-proj}. Thus we arrive at the following
representatives in the line model.
\be g_\RR(\z) \= \begin{cases}
1&\text{ on }\uhp\,,\\
0&\text{ on }\lhp\,;
\end{cases}\qquad g_\infty(\z) \= \pm \z^{-2s}\, \bigl( 1+e^{\mp2\pi i
s})^{-1}\text{ on }\uhp^\pm\,. \ee
Finally one checks that $g_\RR(\z) - (\z^2)^{-s} g_\infty(-1/\z)$
extends holomorphically across both $\RR_+$ and $\RR_-$, thus showing
that the pair $(g_\RR,g_\infty)$ determines the hyperfunction
$\bold 1_s$. These representatives also show that $\bold 1_s$ extends
meromorphically in $s$, giving $\bold 1_s\in \V{-\om}s$ for all
$s\neq \frac12$ with $0<\re s<1$.
\medskip

For the relation between the cases \textbf A and \textbf B, we use
\eqref{P-I-p} to get
\[\poiss s I_{1-s} \dt_{1-s,\infty} (z)
\= \poiss {1-s} \dt_{1-s,\infty} (z) \= y^s\,.\]
The fact that the Poisson transformation is an isomorphism
$\V{-\om}s \lra \E^s$ implies
\be \bold 1_s \= \frac{\Gf(\frac12-s)}{\sqrt\pi\,\Gf(1-s)} \, I_s
\dt_{s,\infty}\,. \ee

\Pcase{C}For $\re s<\frac12$ we have
\[ \bigl\langle\ph,\,\bold 1_s \bigr\rangle\= \frac1\pi
\int_{-\infty}^\infty \Bigl( \ph(t) - \ph^\prj(\infty)
(1+t^2)^{s-1} \Bigr)\, dx + \frac{\Gf(\frac12-s)}{\sqrt\pi\,\Gf(1-s)}
\dt_{s,\infty}(\ph^\prj)
\,.\]
So the distribution $L_s$ given by
\[L_s: \ph\mapsto \frac1\pi \int_{-\infty}^\infty \bigl( \ph(t) -
\ph^\prj(\infty)(1+t^2)^{s-1}\bigr)\, dt\,,\]
which is well defined for $\re s<1$, is equal to
$\bold 1_s - \frac{\Gf(\frac12-s)}{\sqrt\pi\,\Gf(1-s)}\,
\dt_{s,\infty}$ for $\re s<\frac12$. The results of the cases
\textbf{A} and \textbf{B} give the expression of $\poiss s L_s$ as a
multiple of $\ell_s$ defined in~\eqref{ell-def}. Going over to the
line model, we obtain the statement in the table.

\Pcase{D}This is simplify the definition of the Poisson transformation
in~\eqref{Potr} and~\eqref{Pd} applied to the delta distribution
at~$t$. It also follows from Case~\textbf{A}, using the
$G$-equiv\-ari\-ance.

The latter method involves a transition between the models. We explain
some of the steps to be taken. In the projective model,
$\dt_{s,t}^\prj : \ph^\prj\mapsto (1+\nobreak t^2)^{s-1}\, \ph^\prj(t)$.
We have
\[ \bigl\langle \dt_{s,t}^\prj \bigr|_{2s} \matr t{-1}10 ,
\ph^\prj\bigr\rangle \= \bigl\langle \dt_{s,t}^\prj,
\ph^\prj\bigr|_{2-2s} \matr01{-1}t \bigr\rangle \= \cdots \=
\dt_{s,\infty}^\prj \bigl( \ph^\prj\bigr)\,. \]
Hence
\[ \poiss s \bigl(\dt_{s,t} \bigr)(z) \= \poiss s \Bigl(
\dt_{s,\infty}\bigr|\matr 01{-1}t \Bigr)(z) \= \bigl(\poiss s
\dt_{s,\infty} \bigr) \bigl(1/(t-z)\bigr) \= \Bigl(\frac
y{|t-z|^2}\Bigr)^{1-s}\,.\]

\Pcase{E}For $\al\neq 0$ we need no complicated contour integration.
When $\re s<\frac12$ the distribution $\ph \mapsto
\frac1\pi\int_{-\infty}^\infty \ph(t) e^{i\al t}\, dt$ in the line
model, is equal to $\ph \mapsto \frac{-1}{\pi i \al}
\int_{-\infty}^\infty \ph'(t)e^{i\al t}\, dt$. The latter integral
converges absolutely for $\re s<1$.

\Pcase F Since $i_{s,\al}$ has exponential growth we really need a
hyperfunction. The representative in the table does not behave well
near~$0$. However it is holomorphic on a deleted neighborhood
of~$\infty$, and represents a hyperfunction on $\proj\RR\sm \{0\}$ in
the projective model. We extend it by zero to obtain a hyperfunction
on~$\proj\RR$.

The path of integration $\int_{C_+}-\int_{C_-}$ can be deformed into a
large circle $|\tau|=R$, such that we can replace $\tau$ by $\tau-x$
in the integration. We obtain
\begin{align*}\frac1\pi&\int_{|\tau|=R } \frac{i\tau}{-2}
(1+\tau^{-2})^s \hypg11{1;2-2s;i\al\tau} \biggl( \frac
{y(1+\tau^2)}{(\tau-z)(\tau-\bar z)}\biggr)^{s-1}\,
\frac{d\tau}{1+\tau^2}
\\ &\=\frac1{2\pi i} y^{1-s} \int_{|\tau|=R} \bigl(1+\frac
x\tau\bigr)^{1-2s} \bigl(1+\frac {y^2}{\tau^2}
\bigr)^{s-1}\hypg11{1;2-2s;i\al(\tau+x)}\, \frac{d\tau}\tau\,.
\end{align*}
Expand the factors $\bigl(1+\frac x\tau\bigr)^{1-2s}$ and
$\bigl(1+\frac {y^2}{\tau^2} \bigr)^{s-1}$ and the hypergeometric
function into power series and carry out the integration term by
term. In the resulting sum we recognize the power series of $e^{i\al
x}$, and after some standard manipulations with gamma factors, also
the expansion of the modified Bessel function $I_{1/2-s}(|\al|y)$.

\Pcase G See the discussion after Theorem~\ref{thmHel-p}.

\Pcase H See \eqref{P-R}.

\Pcase{I {\rm and} J }Integration against $x\mapsto x^{i\al-s}$ on
$(0,\infty)$ and against $x\mapsto
(-x)^{i\al-s}$, in the line model, defines distributions. For
$\rho e^{i\phi}\in \uhp$, the Poisson integral leads to
\[ \frac{\rho^{i\al}(\sin\phi)^{1-s}}\pi \int_0^\infty t^{i\al-s}
\bigl( t^2+1+2C t\bigr)^{s-1}\, dt\,, \]
with $C=\mp \cos\phi$. Let us consider this for small values of $C$,
i.e., for points near $i\RR_+$ in~$\uhp$. Expanding the integrand in
powers of $C$ gives a series in which one may separate the even and
odd terms, and arrive at
\begin{align*}
&\frac{\rho^{i\al}\sqrt{1-C^2}^{1-s}}{2\pi \Gf(1-s)} \biggl( \Gf\bigl(
\frac{1-i\al-s}2\bigr) \,\Gf\bigl(\frac{1+i\al-s}2\bigr)
\,\hypg12{\frac{1-i\al-s}2,\frac{1+i\al-s}2;\frac12;C^2}\\
&\qquad\hbox{}
- 2C \, \Gf\bigl( 1-\frac{i\al+s}2\bigr) \,\Gf\bigl(
1+\frac{i\al-s}2\bigr)\,
\hypg21{1-\frac{i\al+s}2, 1+\frac{i\al-s}2;\frac32;C^2}
\biggr)\,.
\end{align*}
Now take $C=-\cos\phi$, respectively $C=\cos\phi$, and conclude that
we have a multiple of $f^L_{1-s,\al}$, respectively $f^R_{1-s,\al}$.



\subsection{Transverse Poisson transforms}\label{sect-bg}In
Table~\ref{tab-bg} we give examples of pairs $u=\tpoiss s \ph$,
$\ph=\rho_s u$, where $\ph \in \V \om s (I)$ for some $I\subset\p\H$.
\begin{table}[ht]
\[
\renewcommand\arraystretch{1.4}
\setcounter{tnm}{0}
\renewcommand\thetnm{{\alph{tnm}}}
\newcommand\rgl[4]{\stepcounter{tnm}\textbf{\thetnm} & #1& #2& #3&
\text{#4} \\ \hline}
\setlength{\doublerulesep}{\arrayrulewidth}
\begin{array}{|c|c|c|c|c|}
\hline
&u= \tpoiss s \ph\in \W\om s(I)& \ph= \rho_s u \in \V\om s(I)& I&
\text{model}\\ \hline\hline\hline
\rgl{Q_{s,n}}{(-1)^n\,\frac{\sqrt\pi\, \Gf(s+n)}{\Gf(s+\frac12)}\, \,
\eph_{s,n} }{ \uc }{ circle}
%
\rgl{q_s(\dv,w')}{\frac{\sqrt\pi\, \Gf(s)}{2^{2s}\,\Gf(s+1/2)}
\,\frac{(1-|w'|^2)^s}{|\x-w'|^{2s}}}{\uc}{circle}
%
\rgl{y^s}{1}{\RR}{line}
%
\rgl{R(t;z)^s\quad (t\in \RR)}{|t-x|^{-2s} }{\RR \sm\{t\}}{line }
%
\rgl{R(\z;z)^s\quad(\z\in
\CC\sm\RR)}{(\z-x)^{2s}\quad(\text{multivalued})
}{\RR}{line}
%
\rgl{i_{s,\al}}{e^{i\al x}}{\RR}{line}
%
\rgl{f_{s,\al}^R}{x^{i\al-s}}{(0,\infty)}{line}
%
\rgl{f_{s,\al}^L}{(-x)^{i\al-s}}{(-\infty,0)}{line}
\end{array}\]
\caption{Transverse Poisson representations of boundary germs}
\label{tab-bg}
\end{table}

In Cases \textbf{c}, \textbf{d}, \textbf{f}, \textbf{g} and \textbf{h}
in the table the eigenfunction $u$ is in $\E_s=\E_s(\H)$, hence it is
also a Poisson transform. If we write $u=\poiss {1-s}\al$, then
entries \textbf{A}, \textbf{D}, \textbf{F}, \textbf{J} and
\textbf{I}, respectively, in Table~\ref{tab-Pta}
(with the $s$ replaced by $1-s$ in most cases) show that the support
of $\al$ is the complement of the set $I$ in~$\p\H$ for each of these
cases, illustrating Theorem~\ref{thm-W-supp}.



\subsection{Potentials for Green's forms}\label{sect-pot}
If $u,v\in \E_s(U)$ for some $U \subset\H$, then the Green's forms
$\{u,v\}$ and $[u,v]$ are closed. So if $U$ is simply connected there
are well defined potentials of $[u,v]$ and $\{u,v\}$ in $ C^\om(U)$,
related according to~\eqref{Grf}. We list some examples of potentials
$F$ of $\{u,v\}$ in Table~\ref{tab-Grpot}. Then
$\frac1{2i}F+\frac12uv$ is a potential of the other Green's
form~$[u,v]$.
\begin{table}[ht]{\small
\[
\renewcommand\arraystretch{1.3}
\setcounter{tnm}{0}
\renewcommand\thetnm{{\arabic{tnm}}}
\newcommand\rgl[4]{\stepcounter{tnm}\text{\bf\thetnm} & #1& #2& #3& #4
\\ \hline}
\setlength{\doublerulesep}{\arrayrulewidth}
\begin{array}{|c|c|c|c|c|}
\hline
&u& v& F \text{ such that }dF=\{u,v\}& \text{domain}\\ \hline\hline
\rgl{y^s}{y^{1-s}}{(2s-1)x}{\uhp}
%
\rgl{y^{1/2}}{y^{1/2}\log y}{-x}{\uhp}
%
\rgl{\begin{array}{c}R(t;z)^s\\t\in \RR\end{array}}
{ \begin{array}{c}R(p;z)^{1-s}\\p\in \RR\sm\{t\}\end{array}}
{ \frac{(t-x)(p-x)+y^2}{y(p-t)} R(t;z)^s R(p;z)^{1-s}}{\uhp}
%
\rgl{\begin{array}{c}R(t;z)^s\\t\in
\RR\end{array}}{R(t;z)^{1-s}}{\frac s{t-z}+ \frac{s-1}{t-\bar z}-
iR(t;z)}{\uhp}
%
\rgl{y^s}{\begin{array}{c}R(t;z)^{1-s}\\t\in\RR\end{array}}
{ -(iy^s+(t-z)y^{s-1})
\,R(t;z)^{1-s}}{\uhp}
%
\rgl{y^s}{R(t;z)^s}{ -F_s((x-t)/y),\text{ $F_s$ as in \eqref{Fsdef}}
}{\uhp}
%
\rgl{\begin{array}{c}R(t;z)^s\\t\in \RR\end{array}}
{\begin{array}{c}R(p;z)^s\\p\in\RR\sm\{t\}\end{array}}
{\bigl( (p-t)^2\bigr)^{-s} F_s\bigl( \frac{(p-x)(t-x)+y^2}{y(p-t)}
\bigr)}{\uhp}
%
\rgl{ k_{s,\al}}{i_{s,\al}}{ \frac{i \,\Gf(s+1/2)
}{ 2^{3/2-s} \al |\al|^{s-1/2} }\, e^{2i\al x} }{ \uhp }
%
\rgl{ P_{\!s,0}(re^{i\th}) }{ Q_{s,0}(re^{i\th})} { -\th }{
\begin{array}{c}U \subset \disk\sm\{0\}\vspace{-.2cm}\\
\text{simply}\vspace{-.2cm}\\\text{connected}\end{array}
}
%
\rgl{\begin{array}{c} P_{\!s,n}(re^{i\th})\\
n\in \ZZ\sm\{0\}\end{array}}
{ Q_{s,n}(re^{i\th})} { -\frac{(-1)^n \, \Gf(s+n)}{2in \, \Gf(s-n)}\,
e^{2in\th} }{\disk\sm\{0\}}
%
\rgl{\begin{array}{c}P_{s,-n}(re^{i\th})\\
n\in \ZZ\sm\{0\}\end{array}}
{Q_{s,n}(re^{i\th)}} {\begin{array}{l}
-2in \int P_{s,-m}(r)\,Q_{s,n}(r)\, \frac{dr}r\\
\hbox{\ \ }-(-1)^n \th
\end{array}
} {
\begin{array}{c}U \subset \disk\sm\{0\}\vspace{-.2cm}\\
\text{simply}\vspace{-.2cm}\\\text{connected}\end{array}}
%
\rgl{\begin{array}{c}P_{\!s,m}(r e^{i \th})\\ m\in \ZZ\end{array}}
 {\begin{array}{c}Q_{s,n}(re^{i\th})\\n\in \ZZ\sm{\{-m\}}\end{array}}
  { \begin{array}{l}
 e^{i(m+n)\th)} r \,\bigl( Q_{s,n}(r)\,\p_r P_{\!s,m}(r)\\
 \hbox{\ \ }
- P_{\!s,m}(r)\,\p_r Q_{s,n}(r) \bigr)/i(m+n)\end{array} } {
\disk\sm\{0\}}
\end{array}
\]} \caption{Potentials for Green's forms}\label{tab-Grpot}
\end{table}
We found most of these potentials by writing down $\{u,v\}$, guessing
$F$, and checking our guess.

Case~\textbf{3} is essentially \eqref{goodlemma}. In Case~\textbf{6}
we needed the following function:
\be \label{Fsdef}
F_s(r) \= 2s\int_r^\infty (1+q^2)^{-s-1}\, dq\,, \ee
For \textbf{7} we have used that $(\im gz) ^s = R(t;z)^s$ and
$R(0;gz)^s=|p-\nobreak t|^{2s}R(p;z)^s$ with
$g=\matc{\frac{-1}{p-t}}{\frac p{p-t}}{-1}t$ with $t,p\in \RR$. So
\textbf{6} leads to the potential in~\textbf{7} if $p\neq t$ are
real. We write $\bigl( (p-t)^2\bigr)^{-s}$ and not
$|p-\nobreak t|^{-2s}$ to allow holomorphic continuation in $p$
and~$t$. For Case~\textbf{8} we use that if $u(z)=e^{i\al x}f(y)$ and
$v(z)=e^{i\al x}g(y)$, then
\[ \{u,v\} \= e^{2i\al x}(f'g-fg')\, dx \,,\]
and that the Wronskian $fg'-f'g$ is constant if $u,v\in \E_s$. Cases
\textbf{9}--\textbf{12} are obtained in a similar way. In \textbf{9}
and~\textbf{11} the potentials are multivalued if $U$ is not simply
connected.

Cases \textbf{3}--\textbf{5} are valid on $\uhp$ if $t$ and $p$ are
real. Otherwise $\{u,v\}$ and $F$ are multivalued with branch points
at $t$, and at~$p$ in~\textbf{3}. We have to chose the same branch in
$\{u,v\}$ and~$F$. Also in \textbf{7} the branches have to be chosen
consistently. In \textbf{4} there are singularities at $t=z$ and
$t=\bar z$, but $\{u,v\}$ and $F$ are univalued.


\subsection{Action of the Lie algebra}\label{sect-Lie}The real Lie
algebra of $G$ has $\HH=\rmatr100{-1}$, $\VV=\rmatc0110$,
$\WW=\rmatr01{-1}0$ as a basis. Any $\YY$ in the Lie algebra acts on
$\V\infty s$ by $f|_{2s}\,\YY = \p_t f|_{2s}\,e^{t\YY}\bigr|_{t=0}$.
Note that for right actions we have
$f\,|\,[\YY_1,\YY_2]=(f\,|\,\YY_2)\,|\,\YY_1-(f\,|\,\YY_1)\,|\,\YY_2$.

In the projective model:
\be
\begin{aligned}
f|_{2s}\,\HH(\tau)&\=
\biggl(2s\frac{1-\tau^2}{1+\tau^2}+2\tau\p_\tau\biggr)f(\tau)\,,\\
f|_{2s}\,\VV(\tau)&\=\biggl(-4s\frac{\tau}{1+\tau^2}+(1-\tau^2)\p_\tau
\biggr)f(\tau)\,,\\
f|_{2s}\,\WW(\tau) &\= (1+\tau^2)\p_\tau f(\tau)\,.
\end{aligned}
\ee
For the elements $\EE^+=\HH+i\VV$ and $\EE^-=\HH-i\VV$ in the
complexified Lie algebra we find:
\be
\begin{aligned}
f|_{2s}\,\EE^+(\tau) \= \biggl(
-2s\frac{\tau+i}{\tau-i}-i(\tau+i)^2 \p_\tau\biggr)f(\tau)\,,\\
f|_{2s}\,\EE^-(\tau) \= \biggl(
-2s\frac{\tau-i}{\tau+i}+i(\tau-i)^2\p_\tau\biggr)f(\tau)\,.
\end{aligned}
\ee
In particular
\be \label{Lie-ephisn} \eph_{s,n}|_{2s}\,\WW \= 2in\,
\eph_{s,n}\,,\qquad \eph_{s,n} |_{2s}\,\EE^\pm \=
-2(s\mp n)\, \eph_{s,n\mp1}\,. \ee
By transposition these formulas are also valid on hyperfunctions.

The Lie algebra generates the universal enveloping algebra, which also
acts on $\V\infty s$. The center of this algebra is generated by the
Casimir operator
$\om=-\frac14 \EE^+\EE^- + \frac14 \WW^2 -\frac i2 \WW$. It acts on
$\V{}s$ as multiplication by $s(1-\nobreak s)$.
\medskip

For the action of $G$ by left translation on functions on~$\uhp$:
\be
\begin{aligned}
\WW&\=(1+z^2)\p_z + (1+\bar z^2)\p_{\bar z}\,,\qquad \EE^\pm\= \mp
i(z\pm i)^2 \p_z \mp i(\bar z\pm i)^2\p_{\bar z}\,,\\
\om&\= (z-\bar z)^2\p_z\p_{\bar z} \= \Dt\,,
\end{aligned}
\ee
and on~$\disk$:
\be
\begin{aligned}
\WW&\= 2iw\p_w-2i\bar w\p_{\bar w}\,,&\quad \EE^+&\= 2\p_w-2\bar
w^2\p_{\bar w}\,,\\
\EE^-&\= -2w^2\p_w+2\p_{\bar w}\,,& \om&\=-(1-|w|^2)^2\,\p_w\p_{\bar
w}\,.
\end{aligned}
\ee
A counterpart of \eqref{Lie-ephisn} is\alsochanged{Correction}
\be
\begin{aligned}
P_{\!s,n}\,|\,\WW&\= 2in\, P_{\!s,n}\,,
&\quad P_{\!s,n}\,|\,\EE^+&\= 2(s-n)(s+n-1)\,P_{\!s,n-1}\,,\\
P_{\!s,n}\,|\,\EE^-&\=2\,P_{\!s,n+1}\,,
&\quad Q_{s,n}\,|\,\EE^+&\= 2(s-n)(s+n-1)\,Q_{s,n-1}\,,\\
Q_{s,n}\,|\,\EE^-&\=2\,Q_{s,n+1}\,,
&\quad Q_{s,n}\,|\,\WW&\= 2in\, Q_{s,n}\,.
\end{aligned}
\ee



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    References                                           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\raggedright
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% E. van den Ban, and H. Schlichtkrull, 
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\end{thebibliography}

\bigskip
\bigskip

\noindent{\sc Mathematisch Instituut Universiteit Utrecht, 3508 TA
Utrecht, Nederland}\\
\texttt{r.w.bruggeman@uu.nl}

\bigskip
\noindent{\sc Massachusetts Institute of Technology, Cambridge,
Massachusetts, 02139, USA}\\
\texttt{jlewis@math.mit.edu}

\bigskip
\noindent{\sc Max-Planck-Institut f\"ur Mathematik, 53111 Bonn,
Deutschland} \\
\noindent{\sc and Coll\`ege de France, 75005 Paris, France} \\
\texttt{don.zagier@mpim-bonn.mpg.de}

\end{document}


