00footnotetext: 2020 Mathematics Subject Classification. Primary 30F45, 30H50,
35P10; Secondary 30B40, 53A5500footnotetext: Key words and phrases. Eigenvalue
theory of the invariant Laplacian, holomorphic eigenfunctions, spectral
decomposition, Möbius invariant subspaces

# Spectral theory of the invariant Laplacian
on the disk and the sphere – a complex analysis approach

Michael Heins  M. Heins: Department of Mathematics, University of Würzburg,
Emil Fischer Strasse 40, 97074, Würzburg, Germany.
<EMAIL_ADDRESS>,  Annika Moucha† A. Moucha:
Department of Mathematics, University of Würzburg, Emil Fischer Strasse 40,
97074, Würzburg, Germany<EMAIL_ADDRESS>and
Oliver Roth  O. Roth: Department of Mathematics, University of Würzburg, Emil
Fischer Strasse 40, 97074, Würzburg, Germany<EMAIL_ADDRESS>wuerzburg.de

###### Abstract.

The central theme of this paper is the holomorphic spectral theory of the
canonical Laplace operator of the complement
$\Omega\coloneqq\\{(z,w)\in\widehat{\mathbb{C}}^{2}\colon z\cdot w\neq 1\\}$
of the “complexified unit circle” $\\{(z,w)\in\widehat{\mathbb{C}}^{2}\colon
z\cdot w=1\\}$. We start by singling out a distinguished set of holomorphic
eigenfunctions on the bidisk in terms of hypergeometric ${}_{2}F_{1}$
functions and prove that they provide a spectral decomposition of every
holomorphic eigenfunction on the bidisk. As a second step, we identify the
maximal domains of definition of these eigenfunctions and show that these
maximal domains naturally determine the fine structure of the eigenspaces. Our
main result gives an intrinsic classification of all closed Möbius invariant
subspaces of eigenspaces of the canonical Laplacian of $\Omega$. Generalizing
foundational prior work of Helgason and Rudin, this provides a unifying
complex analytic framework for the real–analytic eigenvalue theories of both
the hyperbolic and spherical Laplace operators on the open unit disk resp. the
Riemann sphere and, in particular, shows how they are interrelated with one
another.

${}^{\dagger}\,$Partially supported by the Alexander von Humboldt Stiftung

##  1\. Introduction

Let $\widehat{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$ denote the Riemann
sphere. The purpose of this paper is to explore the spectral theory of the
complex invariant Laplace operator

$\Delta_{zw}=4(1-zw)^{2}\partial_{z}\partial_{w}\,$

of the complement of the complexified unit circle,

(1.1)
$\Omega:=\widehat{\mathbb{C}}^{2}\setminus\left\\{(z,w)\in\widehat{\mathbb{C}}^{2}\,:\,z\cdot
w=1\right\\}$

by function–theoretic methods. This approach allows a unified study of the
real–analytic spectral theories of the hyperbolic Laplacian
$\Delta_{\mathbb{D}}:=4(1-|z|^{2})^{2}\partial_{z}\partial_{\overline{z}}$ on
the open unit disk $\mathbb{D}:=\\{z\in\mathbb{C}\,:\,|z|<1\\}$ and the
spherical Laplacian
$\Delta_{\widehat{\mathbb{C}}}:=-4(1+|z|^{2})^{2}\partial_{z}\partial_{\overline{z}}$
on the Riemann sphere $\widehat{\mathbb{C}}$ from a complex analytic point of
view and, in addition, it also shows how they are interrelated to one another.
Beyond that, the complex point of view taken in this paper offers several
other useful advantages. In particular, it connects in a natural way the fine
structure of the eigenspaces of the hyperbolic and spherical Laplacians as
described by Helgason [21] and Rudin [37] with the maximal domain of existence
of the corresponding holomorphic eigenfunctions of the invariant Laplacian
$\Delta_{zw}$.

As one instance, we analyze from a complex analysis point of view the building
blocks of each $\lambda$–eigenspace of the hyperbolic Laplacian
$\Delta_{\mathbb{D}}$, which have previously been identified e.g. by Helgason
in [21] using real variable methods. It turns out that these so–called Poisson
Fourier modes naturally extend to holomorphic eigenfunctions of $\Delta_{zw}$
which are (maximally) defined either on $\Omega$ or on one of three
distinguished subdomains of $\Omega$ depending on the choice of the eigenvalue
$\lambda$. This then allows a transparent proof that each holomorphic
eigenfunction of $\Delta_{zw}$ defined on any rotationally invariant subdomain
of $\Omega$ has a unique spectral decomposition in form of a locally uniformly
and absolutely convergent infinite series composed of Poisson Fourier modes.
In the special case of the bidisk $\mathbb{D}^{2}$ and further restriction to
the “diagonal” $\\{(z,\overline{z})\,:\,z\in\mathbb{D}\\}$ we recover the
spectral decomposition of the smooth eigenfunctions of $\Delta_{\mathbb{D}}$
on the unit disk $\mathbb{D}$ as described e.g. in [21] or [8].

As a second instance, we investigate the structure of the closed “Möbius
invariant” subspaces of any fixed eigenspace $X_{\lambda}(\mathbb{D})$ of
$\Delta_{\mathbb{D}}$ from a complex analysis point of view. This topic has
been investigated in detail by Rudin in [37] using purely “real” methods. We
shall see that the distinguished subdomains of $\Omega$ mentioned above
naturally lead to the same distinction between exceptional and non–exceptional
eigenvalues which have been found by Rudin. In Rudin’s work, the exceptional
cases correspond to the eigenvalues $\lambda=4m(m+1)$, $m=0,1,2\ldots$, and
they are characterized by the existence of three non–trivial Möbius invariant
closed subspaces of $X_{\lambda}(\mathbb{D})$, exactly one of which,
$X^{0}_{\lambda}(\mathbb{D})$ say, is finite dimensional. It turns out that a
complex number $\lambda\in\mathbb{C}$ is an exceptional eigenvalue in the
sense of Rudin if and only the invariant Laplacian $\Delta_{zw}$ has a
globally (that is on $\Omega$) defined holomorphic $\lambda$–eigenfunction.
Moreover, in this case the unique finite dimensional invariant subspace
$X^{0}_{\lambda}(\mathbb{D})$ corresponds precisely to the full
$\lambda$–eigenspace of all globally defined $\lambda$–eigenfunctions of the
invariant Laplacian $\Delta_{zw}$, which then, in fact, is invariant under the
full group of all Möbius transformations. For the other two non–trivial
invariant subspaces of the exceptional $\Delta_{\mathbb{D}}$–eigenspace
$X_{\lambda}(\mathbb{D})$ discovered by Rudin as well as the full eigenspace
$X_{\lambda}(\mathbb{D})$ itself we give a similar but more intricate
description in form of Runge–type approximation results in terms of
holomorphic $\lambda$–eigenfunctions defined precisely on one of the
distinguished three subdomains of $\Omega$, see Theorem 2.8.

In the next section we give an account of the main results of this work and
their ramifications for the spectral theory of the hyperbolic and spherical
Laplacian as well as an outline of the structure of the remaining sections.
The accompanying papers [18, 19, 27, 32] are related to other aspects of the
function theory of the set $\Omega$, the complement of the complexified unit
circle, and its applications. Our interest in this set and its inhabitants,
the holomorphic functions on $\Omega$, first arose in connection with previous
work [7, 10, 12, 28, 41, 43] on canonical Wick–type star products in strict
deformation quantization of the unit disk and the Riemann sphere, and from our
desire to understand the somehow mysterious role played by $\Omega$ and in
particular by its function–theoretic properties in this regard. A partial
explanation was given in [19], where it was indicated that invariant
differential operators of Peschl–Minda type, on the one hand, effectively
facilitate and unify the study of the star products on the disk and the
sphere, and on the other hand, are perhaps best understood as operators acting
on the spaces of holomorphic functions on $\Omega$ and its three distinguished
subdomains. We started wondering whether and how the most basic differential
operator acting on $\Omega$, the invariant Laplacian $\Delta_{zw}$, and its
spectral theory possibly fit into this emerging picture. This paper describes
what we have found. In the forthcoming paper [32] of the second–named author
these endeavours will come to full circle: it is shown that there are globally
defined eigenfunctions of the Laplacian $\Delta_{zw}$ which form a Schauder
basis of the Fréchet space $\mathcal{H}(\Omega)$ of all holomorphic functions
on $\Omega$. The results of the present paper then imply that the algebra

$\mathcal{A}(\mathbb{D})=\big{\\{}f:\mathbb{D}\to\mathbb{C}\,\big{|}\,f(z)=F(z,\overline{z})\text{
for all }z\in\mathbb{D}\text{ for some }F\in\mathcal{H}(\Omega)\big{\\}}\,,$

for which the Wick–star product on $\mathbb{D}$ in [28] is constructed, admits
a spectral decomposition precisely into the finite dimensional invariant
subspaces of the exceptional eigenspaces of the hyperbolic Laplace operator
$\Delta_{\mathbb{D}}$ on $\mathbb{D}$ discovered by Rudin [37] many years ago.
This provides an intrinsic characterization of the algebra
$\mathcal{A}(\mathbb{D})$ in terms of the natural hyperbolic geometry of the
unit disk and its canonical invariant Laplacian $\Delta_{\mathbb{D}}$.

##  2\. Overview and main results

In order to place the results of this paper into a broader context we begin by
recalling in greater detail the striking distinction between exceptional and
non–exceptional eigenvalues of the hyperbolic Laplace operator
$\Delta_{\mathbb{D}}$ and its relevance for the study of the invariant
subspaces of the $\Delta_{\mathbb{D}}$–eigenspaces which has been discovered
by Rudin [37]. The Fréchet space of all twice continuously (real)
differentiable functions $f:\mathbb{D}\to\mathbb{C}$ equipped with the
standard compact–open topology is denoted by $C^{2}(\mathbb{D})$. For each
$\lambda\in\mathbb{C}$ we denote by $X_{\lambda}(\mathbb{D})$ the vector space
of all $\lambda$–eigenfunctions $f\in C^{2}(\mathbb{D})$ of the hyperbolic
Laplacian, that is,

$X_{\lambda}(\mathbb{D})=\left\\{f\in
C^{2}(\mathbb{D})\,:\Delta_{\mathbb{D}}f=\lambda f\text{ on
}\mathbb{D}\right\\}\,.$

It is known that each such $\Delta_{\mathbb{D}}$–eigenspace
$X_{\lambda}(\mathbb{D})$ is a closed infinite dimensional subspace of
$C^{2}(\mathbb{D})$. The hyperbolic Laplacian $\Delta_{\mathbb{D}}$ is
invariant under the full group of all conformal automorphisms (biholomorphic
maps) $T$ of the unit disk $\mathbb{D}$ in the sense that

$\Delta_{\mathbb{D}}(f\circ T)=\left(\Delta_{\mathbb{D}}f\right)\circ
T\,\quad\text{ for all }f\in C^{2}(\mathbb{D})\,.$

In order to emphasize that this group consists entirely of Möbius
transformations, we call it the Möbius group of $\mathbb{D}$ and denote it by
$\mathcal{M}(\mathbb{D})$. A closed subspace $Y$ of $C^{2}(\mathbb{D})$ is
called Möbius invariant if $f\circ\psi\in Y$ for all $f\in Y$ and all
$\psi\in\mathcal{M}(\mathbb{D})$.222Möbius invariant spaces are called
$\mathcal{M}$–spaces in [37]. We will reserve the symbol $\mathcal{M}$ for
some other purpose. It is called non–trivial if $Y\not=\\{0\\}$ and $Y\not=X$.

######  Theorem 2.1 (Rudin [37])

Let $\lambda\in\mathbb{C}$.

  * (NE)

If $\lambda\not=4m(m+1)$ for $m=0,1,2,\ldots$, then $X_{\lambda}(\mathbb{D})$
has no non–trivial Möbius invariant subspaces.

  * (E)

If $\lambda=4m(m+1)$ for some $m=0,1,2,\ldots$, then $X_{\lambda}(\mathbb{D})$
has precisely three distinct non–trivial Möbius invariant subspaces. There is
exactly one non–trivial Möbius invariant subspace of $X_{\lambda}(\mathbb{D})$
which is finite dimensional; its dimension is $2m+1$.

The alternative (E) in Theorem 2.1 will be called the exceptional case and the
unique finite dimensional Möbius invariant subspace of
$X_{\lambda}(\mathbb{D})$ will be denoted by $X^{0}_{\lambda}(\mathbb{D})$.
The alternative (NE) will be referred to as the non–exceptional case.

One of the main results of the present paper is a complete analogue of Theorem
2.1 with $\Delta_{\mathbb{D}}$ replaced by the differential operator
$\Delta_{zw}$, see Theorem 2.5. Apart from being potentially interesting in
its own right, it provides a concrete function–theoretic description of the
exceptional eigenspaces in Theorem 2.1. This also adds a conceptual component
to Rudin’s handling of the invariant eigenspaces of the hyperbolic Laplace
operator $\Delta_{\mathbb{D}}$.

The characteristic feature of our approach is to look for holomorphic
solutions $F$ of the eigenvalue equation

(2.1) $\Delta_{zw}F=\lambda F\,$

defined on a subdomain $D$ of $\Omega$ which we wish to choose as large as
possible depending on the eigenvalue $\lambda$. These maximal domains of
existence (see Definition 2.4) of the $\lambda$–eigenfunctions turn out to be
the only essential ingredients which are needed to give a complete description
of the (invariant) $\lambda$–eigenspaces of the operator $\Delta_{zw}$ and its
offsprings $\Delta_{\mathbb{D}}$ and $\Delta_{\widehat{\mathbb{C}}}$.

In order to state our main results we have to adapt the notation which we have
introduced above for the hyperbolic Laplacian to the case of the differential
operator $\Delta_{zw}=4(1-zw)^{2}\partial_{z}\partial_{w}$. Instead of working
in the Fréchet space $C^{2}(\mathbb{D})$ we now fix a subdomain $D$ of the set
$\Omega:=\\{(z,w)\in\widehat{\mathbb{C}}\,:\,z\cdot w\not=1\\}$, and work in
the Fréchet space $\mathcal{H}(D)$ of all complex–valued holomorphic functions
defined on $D$ (again equipped with the topology of locally uniform
convergence, this time on $D$). Our goal is determine the holomorphic
solutions $F:D\to\mathbb{C}$ of the eigenvalue equation (2.1), i.e. we are
interested in the $\Delta_{zw}$–eigenspaces

$X_{\lambda}(D):=\left\\{F\in\mathcal{H}(D)\,:\,\Delta_{zw}F=\lambda F\text{
on }D\right\\}\,,\qquad\lambda\in\mathbb{C}\,.$

With regard to Rudin’s theorem (Theorem 2.1) a particularly natural choice for
the domain $D$ is the bidisk $\mathbb{D}^{2}:=\mathbb{D}\times\mathbb{D}$
since it is easy to see that for every $F\in X_{\lambda}(\mathbb{D}^{2})$ the
“restriction” of $F$ to the “diagonal”
$\\{(z,\overline{z})\,:\,z\in\mathbb{D}\\}$, that is,
$f(z):=F(z,\overline{z})$, $z\in\mathbb{D}$, yields an eigenfunction $f\in
C^{2}(\mathbb{D})$ of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$ for the
eigenvalue $\lambda$. In fact, the following result shows that every $f\in
X_{\lambda}(\mathbb{D})$ arises in this fashion, that is, it has an
“extension” to an eigenfunction $F$ of $\Delta_{zw}$ which is holomorphic on
the bidisk $\mathbb{D}^{2}$:

######  Theorem 2.2 (Smooth eigenfunctions of $\Delta_{\mathbb{D}}$ on
$\mathbb{D}$ vs. holomorphic eigenfunctions of $\Delta_{zw}$ on
$\mathbb{D}^{2}$)

Let $\lambda\in\mathbb{C}$ and $f\in C^{2}(\mathbb{D})$ such that
$\Delta_{\mathbb{D}}f=\lambda f$ in $\mathbb{D}$. Then there is a uniquely
determined function $F\in\mathcal{H}(\mathbb{D}^{2})$ such that
$\Delta_{zw}F=\lambda F$ on $\mathbb{D}^{2}$ and $f(z)=F(z,\overline{z})$ for
all $z\in\mathbb{D}$. Moreover, the induced bijective linear map

$X_{\lambda}(\mathbb{D})\to X_{\lambda}(\mathbb{D}^{2})$

is continuous.

######  Remark 1.

If $f\in C^{2}(\mathbb{D})$ solves $\Delta_{\mathbb{D}}f=\lambda f$ in
$\mathbb{D}$, then $f$ is real–analytic ([38, Theorem 4.2.5], so there is
trivially a holomorphic function $\tilde{F}$ defined on some open neighborhood
$U\subseteq\mathbb{D}^{2}$ of the diagonal
$\\{(z,\overline{z})\in\mathbb{D}^{2}\,:\,z\in\mathbb{D}$} such that
$\tilde{F}(z,\overline{z})=f(z)$ for all $z\in\mathbb{D}$ satisfying
$(z,\overline{z})\in U$. In view of a well–known variant of the identity
principle (see Lemma 3.1) there is only one such holomorphic extension
$\tilde{F}:U\to\mathbb{C}$ of $f\in C^{2}(\mathbb{D})$ to $U$. The point of
Theorem 2.2 is that $\tilde{F}$ has a holomorphic extension (at least) to the
bidisk $\mathbb{D}^{2}$.

Theorem 2.2 gives rise to the following definition.

###### Definition 2.3

Let $\lambda\in\mathbb{D}$. Then the continuous bijective linear mapping

$\mathcal{R}_{h}:X_{\lambda}(\mathbb{D}^{2})\to
X_{\lambda}(\mathbb{D})\,,\qquad\mathcal{R}_{h}(F)(z):=F(z,\overline{z})\quad(z\in\mathbb{D})\,$

is called the hyperbolic restriction map. Its continuous inverse

$\mathcal{E}_{h}:={\left(\mathcal{R}_{h}\right)}^{-1}:X_{\lambda}(\mathbb{D})\to
X_{\lambda}(\mathbb{D}^{2})\,$

is called the hyperbolic extension map from $X_{\lambda}(\mathbb{D})$ to
$X_{\lambda}(\mathbb{D}^{2})$.

The hyperbolic restriction and extension mappings provide the bridge between
the holomorphic spectral theory of the invariant Laplacian $\Delta_{zw}$ and
the smooth spectral theory of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$.
In particular, one can study the spectral properties of $\Delta_{\mathbb{D}}$
on $\mathbb{D}$ from the viewpoint of complex analysis on the bidisk
$\mathbb{D}^{2}$. Based on Theorem 2.2, we can now proceed to associate the
fine structure of the eigenspaces $X_{\lambda}(\mathbb{D})$ with the maximal
domains of existence of holomorphic eigenfunctions in
$X_{\lambda}(\mathbb{D}^{2})$, a concept which is defined as follows, cf. [14,
p. 97].

###### Definition 2.4

Let $F$ be a holomorphic function on the bidisk $\mathbb{D}^{2}$. A subdomain
$D\subseteq\Omega$ that contains the bidisk $\mathbb{D}^{2}$ is called a
maximal domain of existence of $F$ if the function $F$ has a holomorphic
extension to $D$ but to no strictly larger subdomain of $\Omega$.

For our purposes, this definition is natural in several respects. First, the
condition $D\supseteq\mathbb{D}^{2}$ obviously comes from Theorem 2.2. Second,
the condition $D\subseteq\Omega$ is natural in view of the fact that every
function which is holomorphic on a subdomain of $\widehat{\mathbb{C}}^{2}$
that is strictly larger than $\Omega$ is necessarily constant (see Theorem 5.3
in [18]). In particular, the largest possible maximal domain of existence of
any eigenfunction of $\Delta_{zw}$ in $\mathcal{H}(\mathbb{D}^{2})$ is
$\Omega$. Notably, this includes the constant eigenfunctions. At first sight,
such a linguistic subtlety might seem irritating, but it facilitates the
statement of many of our results.

With this concept at hand, we can now give a function–theoretic
characterization of the exceptional eigenvalues of the hyperbolic Laplacian
$\Delta_{\mathbb{D}}$ and the corresponding unique finite–dimensional
non–trivial Möbius invariant subspaces $X_{\lambda}^{0}(\mathbb{D})$ of the
$\Delta_{\mathbb{D}}$–eigenspaces $X_{\lambda}(\mathbb{D})$. In addition, the
following theorem provides an equivalent condition in terms of existence of
globally defined eigenfunctions of the spherical Laplacian
$\Delta_{\widehat{\mathbb{C}}}$.

######  Theorem 2.5 (Exceptional smooth eigenfunctions $\Delta_{\mathbb{D}}$
on $\mathbb{D}$ vs. holomorphic eigenfunctions of $\Delta_{zw}$ on $\Omega$
vs. smooth eigenfunctions of $\Delta_{\widehat{\mathbb{C}}}$ on
$\widehat{\mathbb{C}}$)

Let $\lambda\in\mathbb{C}$. Then the following conditions are pairwise
equivalent:

  * (i)

There exists a function $F\in X_{\lambda}(\mathbb{D}^{2})$ with $\Omega$ as
maximal domain of existence.

  * (ii)

There exists a function $g\in C^{2}(\widehat{\mathbb{C}})$ such that
$\Delta_{\widehat{\mathbb{C}}}g=\lambda g$ on $\widehat{\mathbb{C}}$.

  * (iii)

$\lambda$ is an exceptional eigenvalue of $\Delta_{\mathbb{D}}$, i.e.
$\lambda=4m(m+1)$ for some non–negative integer $m$.

If one of these conditions is in place and $\lambda=4m(m+1)$, then $\dim
X_{\lambda}(\Omega)=2m+1$ and the following statements hold:

  * (a)

(Function–theoretic description of $X_{\lambda}^{0}(\mathbb{D})$)

The domain $\Omega$ is the maximal domain of existence for every $F\in
X_{\lambda}^{0}(\mathbb{D}^{2})$, i.e.

$X^{0}_{\lambda}(\mathbb{D})=\mathcal{R}_{h}\left(X_{\lambda}(\Omega)\right)\,.$

  * (b)

(Function–theoretic description of the smooth eigenfunctions of
$\Delta_{\widehat{\mathbb{C}}}$ on $\widehat{\mathbb{C}}$)

(2.2) $\left\\{g\in
C^{2}(\widehat{\mathbb{C}})\,:\,\Delta_{\widehat{\mathbb{C}}}g=\lambda g\text{
on }\widehat{\mathbb{C}}\right\\}=\big{\\{}\widehat{\mathbb{C}}\ni z\mapsto
F(z,-\overline{z})\,:\,F\in X_{\lambda}(\Omega)\big{\\}}\,.$

######  Remark 2.

  * (a)

Note that $X_{0}^{0}(\mathbb{D}^{2})$ and $X_{0}(\Omega)$ consist precisely of
the constant functions.

  * (b)

In view of Theorem 2.5, an eigenvalue $\lambda$ of $\Delta_{\mathbb{D}}$ is
exceptional if and only if there exists a holomorphic eigenfunction of
$\Delta_{zw}$ on $\mathbb{D}^{2}$ with the largest possible maximal domain of
existence, the set $\Omega$.

  * (c)

In Theorem 2.5 (a) we think of $X_{\lambda}(\Omega)$ as a subspace of
$X_{\lambda}(\mathbb{D}^{2})$. It is a closed subspace of
$X_{\lambda}(\mathbb{D}^{2})$ (in the topology of
$\mathcal{H}(\mathbb{D}^{2})$) because it is finite dimensional.

  * (d)

(Spherical restriction map)
If $\lambda\in\mathbb{C}$ is an exceptional eigenvalue, then the spherical
restriction map

$\mathcal{R}_{s}:X_{\lambda}(\Omega)\to
C^{2}(\widehat{\mathbb{C}})\,,\qquad\mathcal{R}_{s}(F)(z):=F(z,-\overline{z})\qquad(z\in\widehat{\mathbb{C}})$

is well–defined. By Theorem 2.5 (b), $\mathcal{R}_{s}$ is a bijection from
$X_{\lambda}(\Omega)$ onto the $\lambda$–eigenspace $\big{\\{}g\in
C^{2}(\widehat{\mathbb{C}})\,:\,\Delta_{\widehat{\mathbb{C}}}g=\lambda g\text{
on }\widehat{\mathbb{C}}\big{\\}}$ of $\Delta_{\widehat{\mathbb{C}}}$, which
is clearly continuous.

With Theorem 2.5 we have reached two of our goals, a conceptual
characterization of exceptional eigenvalues and the finite dimensional
non–trivial Möbius invariant subspaces of the eigenspaces
$X_{\lambda}(\mathbb{D})$ of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$.
Next, we address the infinite dimensional non–trivial Möbius invariant
subspaces of the $\Delta_{\mathbb{D}}$–eigenspaces $X_{\lambda}(\mathbb{D})$.
This turns out to be more difficult, and first requires clarification of the
invariance properties of the Laplacian $\Delta_{zw}$. Implicitly, the
underlying difficulty is already present in Theorem 2.5, and can be seen as
follows. Let $\lambda\in\mathbb{C}$ be an exceptional eigenvalue of
$\Delta_{\mathbb{D}}$. Then the finite–dimensional Möbius invariant subspace
$X^{0}_{\lambda}(\mathbb{D})$ is invariant under all automorphisms of
$\mathbb{D}$. However, the corresponding eigenspace $X_{\lambda}(\Omega)$ for
$\Delta_{zw}$, which consists of functions holomorphic on $\Omega$, is,
loosely speaking, invariant under a much larger group of automorphisms. In
fact, we first note that the invariant Laplacian $\Delta_{zw}$ of $\Omega$ is
not invariant under all biholomorphic automorphisms of $\Omega$ in the sense
that the invariance condition

(2.3) $\Delta_{zw}\left(F\circ T\right)=\left(\Delta_{zw}F\right)\circ
T\qquad\text{ for all }F\in\mathcal{H}(\Omega)\,$

holds for all $T\in{\sf Aut}(\Omega)$. The reason is simply that the
automorphism group ${\sf Aut}(\Omega)$ is much too large, see [18]. However,
$\Delta_{zw}$ is invariant under the subgroup $\mathcal{M}$ of ${\sf
Aut}(\Omega)$ defined by

(2.4)
$\mathcal{M}:=\bigcup\limits_{\psi\in\mathcal{M}(\widehat{\mathbb{C}})}\left\\{(z,w)\mapsto\left(\psi(z),\frac{1}{\psi(1/w)}\right)\,,\,(z,w)\mapsto\left(\psi(w),\frac{1}{\psi(1/z)}\right)\right\\}\,,$

where we write $\mathcal{M}(\widehat{\mathbb{C}})$ for the group of all Möbius
transformations $\psi:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$. This is
easy to prove by direct verification. (Conversely, one can show that every
$T\in{\sf Aut}(\Omega)$ for which the invariance property (2.3) holds does
necessarily belong to the subgroup $\mathcal{M}$, see [18, Theorem 5.2], but
we do not need such a result in this paper.) Since $\mathcal{M}$ is “induced”
by the set $\mathcal{M}(\widehat{\mathbb{C}})$ of all Möbius transformations,
we call $\mathcal{M}$ the Möbius group of $\Omega$. Note that
$\mathcal{M}(\widehat{\mathbb{C}})$ is strictly bigger than
$\mathcal{M}(\mathbb{D})$333$\mathcal{M}(\widehat{\mathbb{C}})$ is real six
dimensional, while $\mathcal{M}(\mathbb{D})$ is real three dimensional., so
the Möbius group $\mathcal{M}$ of $\Omega$ is strictly larger than the Möbius
group $\mathcal{M}(\mathbb{D})$ of $\mathbb{D}$. Now, while
$X^{0}_{\lambda}(\mathbb{D})$ is invariant under each element of
$\mathcal{M}(\mathbb{D})$, the set $X_{\lambda}(\Omega)$ is invariant under
each element of $\mathcal{M}$, simply because $X_{\lambda}(\Omega)$ is the
entire $\lambda$–eigenspace of the Laplacian $\Delta_{zw}$ on $\Omega$ and
$\Delta_{zw}$ is invariant with respect to the Möbius group $\mathcal{M}$.

In view of this discussion, it is now clear that a suitable concept of
invariance for the eigenspaces $X_{\lambda}(\mathbb{D}^{2})$ of $\Delta_{zw}$
on the bidisk $\mathbb{D}^{2}$ has to be based on the group

$\mathcal{M}(\mathbb{D}^{2}):={\sf Aut}(\mathbb{D}^{2})\cap\mathcal{M}\,,$

which we call the Möbius group of the bidisk $\mathbb{D}^{2}$. It consists
precisely of all automorphisms of the bidisk $\mathbb{D}^{2}$ which have the
invariance property (2.3). In fact, it is not difficult to show that

(2.5)
$\mathcal{M}(\mathbb{D}^{2})=\bigcup\limits_{\psi\in\mathcal{M}(\mathbb{D})}\left\\{(z,w)\mapsto\left(\psi(z),\frac{1}{\psi(1/w)}\right)\,,\,(z,w)\mapsto\left(\psi(w),\frac{1}{\psi(1/z)}\right)\right\\}\,.$

Clearly, each $\Delta_{zw}$–eigenspace $X_{\lambda}(\mathbb{D}^{2})$ is
invariant with respect to $\mathcal{M}(\mathbb{D}^{2})$, that is, whenever
${F\in X_{\lambda}(\mathbb{D}^{2})}$ and $T\in\mathcal{M}(\mathbb{D}^{2})$,
then $F\circ T\in X_{\lambda}(\mathbb{D}^{2})$. Each closed subspace of
$\mathcal{H}(\mathbb{D}^{2})$ which is invariant under the Möbius group
$\mathcal{M}(\mathbb{D}^{2})$ will be called a Möbius invariant subspace.
Theorem 2.2 implies immediately the following result.

###### Corollary 2.6

Let $\lambda\in\mathbb{C}$ and $Y$ a subspace of $X_{\lambda}(\mathbb{D})$.
Then $Y$ is Möbius invariant if and only if $\mathcal{E}_{h}(Y)$ is a Möbius
invariant subspace of $X_{\lambda}(\mathbb{D}^{2})$.

We can now give a function–theoretic characterization of the Möbius invariant
subspaces of $X_{\lambda}(\mathbb{D}^{2})$ and thereby, in view of Corollary
2.6, the Möbius invariant subspaces of $X_{\lambda}(\mathbb{D})$. The
following subdomains of $\Omega$ play the essential role for this purpose:

$\displaystyle\Omega_{+}$
$\displaystyle\coloneqq\Omega\setminus\big{\\{}(z,\infty)\colon
z\in\hat{\mathbb{C}}\big{\\}}\,.$ $\displaystyle\Omega_{-}$
$\displaystyle\coloneqq\Omega\setminus\big{\\{}(\infty,w)\colon
w\in\hat{\mathbb{C}}\big{\\}}\,.$ $\displaystyle\Omega_{*}\,$
$\displaystyle:=\left\\{(z,w)\in\mathbb{C}^{2}\,:\,zw\in\mathbb{C}\setminus[1,\infty)\right\\}\,.$

Note that each of these three subdomains of $\Omega$ contains the bidisk
$\mathbb{D}^{2}$. Moreover,

$\Omega=\Omega_{+}\cup\Omega_{-}\cup\\{(\infty,\infty)\\}$

and

$\mathcal{H}(\Omega)=\mathcal{H}(\Omega_{+})\cap\mathcal{H}(\Omega_{-})\,.$

Figure 1 provides a schematic view of $\Omega$ and its distinguished subsets.

Figure 1. Schematic picture of the sets $\Omega_{*}$, $\Omega_{+}$,
$\Omega_{-}$ and $\Omega$ (from left to right) with points at infinity. Here,
$\mathbb{D}$ is identified with the diagonal
$\\{(z,\overline{z})\,:\,z\in\mathbb{D}\\}$ and $\widehat{\mathbb{C}}$ with
the rotated diagonal $\\{(z,-\overline{z})\,:\,z\in\widehat{\mathbb{C}}\\}$.

We note in passing that the subdomains $\Omega_{+}$ and $\Omega_{-}$ arise
naturally in the study of the Fréchet space structure of $\mathcal{H}(\Omega)$
(see [18]) and also for studying invariant differential operators of
Peschl–Minda type acting on $\mathcal{H}(\Omega)$ (see [19]). The following
result shows that they are also useful for describing the Möbius invariant
subspaces of the eigenspaces of the invariant Laplacian $\Delta_{zw}$. We use
the following terminology:

###### Definition 2.7

Let $U\subseteq V$ be subdomains of some complex manifold, and let $Y\subseteq
X$ be subsets of $\mathcal{H}(U)$. We say that “$X\cap\mathcal{H}(V)$ is dense
in $Y$” if $X\cap\mathcal{H}(V)\subseteq Y$ and if every function in $Y$ can
be approximated locally uniformly on $U$ by functions in $X$, which have a
holomorphic extension to $V$.

We are now, finally, in a position to formulate the main result of this paper.

###### Theorem 2.8

Let $\lambda\in\mathbb{C}$ and let $Y$ be a non–trivial Möbius invariant
subspace of $X_{\lambda}(\mathbb{D}^{2})$. Then one and only one of the
following four alternatives holds.

  * ($\text{E}_{0}$ )

$Y=X_{\lambda}(\mathbb{D}^{2})\cap\mathcal{H}(\Omega)$ and $\dim Y<\infty$.

  * ($\text{E}_{+}$)

$X_{\lambda}(\mathbb{D}^{2})\cap\mathcal{H}(\Omega_{+})$ is dense in $Y$.

  * ($\text{E}_{-}$)

$X_{\lambda}(\mathbb{D}^{2})\cap\mathcal{H}(\Omega_{-})$ is dense in $Y$.

  * (NE)

$X_{\lambda}(\mathbb{D}^{2})\cap\mathcal{H}(\Omega_{*})$ is dense in $Y$. In
this case $Y=X_{\lambda}(\mathbb{D}^{2})$.

In addition,

  * (i)

($\textrm{E}_{0}$ ), ($\textrm{E}_{+}$ ) and ($\textrm{E}_{-}$ ) cannot occur
if $\lambda\not=4m(m+1)$ for every non–negative integer $m$.

  * (ii)

($\textrm{E}_{0}$ ) holds if and only if $\dim Y<\infty$. In this case, $\dim
Y=2m+1$ and $\lambda=4m(m+1)$ for some non–negative integer $m$.

  * (iii)

None of the density statements (NE ), ($\textrm{E}_{+}$ ) and
($\textrm{E}_{-}$ ) may be improved to equalities.

######  Remark 3.

Recall that two domains $U\subseteq V$ in $\mathbb{C}^{n}$ form a Runge pair
$(U,V)$ if every function in $\mathcal{H}(U)$ can be approximated locally
uniformly in $U$ by functions in $\mathcal{H}(V)$. Identifying domains
$U\subseteq V$ as Runge pairs is a fundamental and in many cases challenging
problem in complex analysis. Note that in our terminology $(U,V)$ is a Runge
pair if and only if $\mathcal{H}(U)\cap\mathcal{H}(V)$ is dense in
$\mathcal{H}(U)$. For subspaces $Y\subseteq X$ of $\mathcal{H}(U)$ it is
tempting to call $(Y,X,\mathcal{H}(V))$ a Runge triple, if
$X\cap\mathcal{H}(V)$ is dense in $Y$. Then for every non–trivial infinite
dimensional Möbius invariant subspace $Y$ of $X_{\lambda}(\mathbb{D}^{2})$
exactly one of the triples

$\left(Y,X_{\lambda}(\mathbb{D}^{2}),\mathcal{H}(\Omega_{+})\right)\,,\qquad\left(Y,X_{\lambda}(\mathbb{D}^{2}),\mathcal{H}(\Omega_{-})\right)\,,\qquad\left(Y,X_{\lambda}(\mathbb{D}^{2}),\mathcal{H}(\Omega_{*})\right)$

is a Runge triple. Theorem 2.8 can therefore be regarded as a Runge–type
approximation theorem for the Möbius invariant spaces of eigenfunctions of the
invariant Laplacian $\Delta_{zw}$.

The plan of the paper is as follows. We introduce some basic concepts and
notation in a preliminary Section 3. In Section 4 and Section 5 we develop the
general spectral theory of the invariant Laplacian $\Delta_{zw}$ on $\Omega$
in analogy to the well–established spectral theory of the hyperbolic Laplacian
$\Delta_{\mathbb{D}}$ on $\mathbb{D}$. In a sense, we rather closely follow
the presentation Berenstein and Gay [8, Section 1.6] have given for the
spectral theory of the hyperbolic Laplacian, but we have made an effort either
to provide even more rigorous proofs or to give precise references to the
literature for all auxiliary results which are needed. In contrast to [8] we
completely work in the holomorphic setting. On the one hand, this makes it
possible to take advantage of many efficient tools from complex analysis which
are not available otherwise. On the other hand, we need to incorporate from
the beginning the maximal domain of existence of eigenfunctions; an issue
which does not even show up when working “only” on the unit disk. Here, our
approach requires some finer analysis of the building blocks of the
eigenfunctions, namely certain hypergeometric functions and their integral
representations in terms of Poisson Fourier modes.

In Section 6 we prove Theorem 2.2 and show that it is in some sense best
possible by providing an explicit example. This implies that the smooth
spectral theory of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$ on the disk
$\mathbb{D}$ and the holomorphic spectral theory of the Laplacian
$\Delta_{zw}$ on the bidisk $\mathbb{D}^{2}$ are essentially equivalent. In
the same spirit we relate the smooth spectral theory of the spherical
Laplacian $\Delta_{\widehat{\mathbb{C}}}$ with the holomorphic spectral theory
of the Laplacian $\Delta_{zw}$ on $\Omega$ as well as the exceptional
eigenvalues of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$, see Theorem
6.2. Section 7 is devoted to a study of the transformation behavior of the
Poisson Fourier modes under precompositions with elements of the Möbius group
$\mathcal{M}$. These results are needed for Section 8 where we prove our main
results, Theorem 2.5 and Theorem 2.8. By and large, we follow Rudin’s [37]
treatment of invariant subspaces of eigenfunctions of the hyperbolic
Laplacian, but again completely working in the holomorphic setting; we briefly
comment on the similarities and differences between our and Rudin’s approach
in Remark 13. We close the paper with Section 9 which connects the Poisson
Fourier modes to the invariant differential operators of Peschl–Minda type
studied in [19].

Four final preliminary remarks are in order. First, treating the hyperbolic
eigenvalue equation $\Delta_{\mathbb{D}}f=\lambda f$ and the spherical
eigenvalue equation $\Delta_{\widehat{\mathbb{C}}}f=\lambda f$ as special
cases of the more general complex eigenvalue equation $\Delta_{zw}F=\lambda F$
has been a recurrent theme in the literature for a long time. To mention but a
few of the many references, we refer for instance to the papers [4, 5, 6] and
their bibliographies. What seems to be new is the systematic study of the
maximal domains of existence of the holomorphic solutions of
$\Delta_{zw}F=\lambda F$ and its ramifications for the study of the invariant
subspaces of the $\Delta_{\mathbb{D}}$–eigenspaces. As a second remark, we
should point out that Rudin’s work [37] is in fact concerned with the
invariant Laplace operator on the unit ball of $\mathbb{C}^{n}$, while our
focus is exclusively on the complex one–dimensional case $n=1$. Thirdly, even
though we are superficially dealing with holomorphic functions of two complex
variables, we only need very few and only elementary facts from the theory of
several complex variables. Finally, Helgason [20, 21, 22] has systematically
studied invariant differential operators and their eigenvalue problem in the
setting of homogeneous spaces. In contrast to our holomorphic approach, he
used entirely real methods. This Lie theoretic approach has since been
generalized significantly. While providing a comprehensive list of references
would go beyond what we can achieve here, we would like to mention [23, 26],
who generalized the theory to higher dimensional symmetric spaces, and Maaß
[29], who initiated the vast and fruitful research of Maaß wave forms.

##  3\. Notation and Preliminaries

We denote the open unit disk in $\mathbb{C}$ by $\mathbb{D}$, the bidisk
$\mathbb{D}^{2}:=\mathbb{D}\times\mathbb{D}$ and the Riemann sphere by
$\widehat{\mathbb{C}}$. The open disk of radius $r>0$ centered at the origin
is denoted by $\mathbb{D}_{r}$. Moreover, we write
$\mathbb{N}\coloneqq\\{1,2,\ldots\\}$ for the set of positive integers,
$\mathbb{N}_{0}\coloneqq\mathbb{N}\cup\\{0\\}$ and $\mathbb{Z}$ for the set of
all integers. For an open subset $U$ of $\widehat{\mathbb{C}}$ or
$\widehat{\mathbb{C}}^{2}$, we write $\partial U$ for its boundary and
$\overline{{U}}$ for its closure. The set of all twice resp. infinitely (real)
differentiable functions $f:U\to\mathbb{C}$ is denoted $C^{2}(U)$ resp.
$C^{\infty}(U)$, and we write $\mathcal{H}(U)$ for set of all holomorphic
functions $f:U\to\mathbb{C}$.
The set
$\Omega=\widehat{\mathbb{C}}^{2}\setminus\\{(z,w)\in\widehat{\mathbb{C}}^{2}\,:\,z\cdot
w\not=1\\}$ is a complex manifold of complex dimension $2$ and an open
submanifold of $\widehat{\mathbb{C}}^{2}$. In order to describe the complex
structure of $\Omega$ only two charts are necessary, the standard chart

$\Omega\cap(\mathbb{C}\times\mathbb{C})\to\mathbb{C}^{2}\,,\qquad(u,v)\mapsto(u,v)$

and the flip chart

$\Omega\cap\left(\widehat{\mathbb{C}}\setminus\\{0\\}\times\widehat{\mathbb{C}}\setminus\\{0\\}\right)\to\mathbb{C}^{2}\,,\qquad(u,v)\mapsto(1/v,1/u)\,.$

In these local coordinates the invariant Laplace operator $\Delta_{zw}$ of the
complex manifold $\Omega$ is then given by

$\Delta_{zw}f(z,w)=4(1-zw)^{2}\partial_{z}\partial_{w}f(z,w)\,,$

and it is easily seen that $\Delta_{zw}$ is a well–defined object. Here,
$\partial_{z}$ and $\partial_{w}$ denote the Wirtinger derivatives with
respect to $z$ and $w$, respectively.

One of the few elementary results from the theory of functions of several
complex variables we need in this paper is the following simple lemma.

######  Lemma 3.1 (“Two variable identity principle” (p. 18 in [35]))

Let $U$ be a subdomain of $\mathbb{C}^{2}$ which contains a point of the form
$(z,\overline{{z}})$ resp. $(z,-\overline{{z}})$. Let $f:U\to\mathbb{C}$ be a
holomorphic function such that

$\\{f(z,\overline{z})\,:\,(z,\overline{z})\in
U\\}=\\{0\\}\quad\text{resp.}\quad\\{f(z,-\overline{z})\,:\,(z,-\overline{z})\in
U\\}=\\{0\\}\,.$

Then $f\equiv 0$.

Finally, we briefly recall some standard terminology from linear algebra. We
denote by $\mathrm{span}\,\,M$ the collection of all finite linear
combinations of elements of a subset $M$ of a given vector space $X$. The
vector spaces that occur in this paper are spaces of smooth or holomorphic
functions defined on some open subset $U$ of $\Omega$ which we equip with the
standard topology of uniform convergence on compact subsets of $U$. If $M$
denotes a set of smooth or holomorphic functions on $U$, we denote by
$\operatorname{clos}_{U}M$ the closure of $M$ with respect to locally uniform
convergence on $U$.

##  4\. Homogeneous eigenfunctions and Poisson Fourier modes

By making a separation of variables approach Rudin [37] showed that every
$\lambda\in\mathbb{C}$ is an eigenvalue of $\Delta_{\mathbb{D}}$. The
analogous result is true for $\Delta_{zw}$: let $n\in\mathbb{Z}$ and suppose
$f_{n}:D\to\mathbb{C}$ is a holomorphic function defined on a domain
$D\subseteq\mathbb{C}^{2}$ containing the origin $(0,0)$. Further assume that
$f_{n}$ is $n$–homogeneous, i.e.,

(4.1) $f_{n}(\eta z,w/\eta)=\eta^{n}f_{n}(z,w)\qquad\text{for all
}\eta\in\partial\mathbb{D}$

and for all $(z,w)\in\mathbb{C}^{2}$ belonging to the bidisk
$\mathbb{D}_{r}^{2}=\mathbb{D}_{r}\times\mathbb{D}_{r}$ for some (and hence
all) $r>0$ such that $\mathbb{D}_{r}^{2}\subseteq D$. A consideration of the
power series expansion of $f_{n}$ at $(0,0)$ in $\mathbb{D}_{r}^{2}$ implies
in a straightforward way that there is a holomorphic function
$y_{n}:\mathbb{D}_{r^{2}}\to\mathbb{C}$ such that

(4.2) $f_{n}(z,w)=y_{n}(zw)z^{n}\quad\text{ if }n\geq 0\quad\text{ and }\quad
f_{n}(z,w)=y_{n}(zw)w^{|n|}\quad\text{ if }n\leq 0\,$

for all $(z,w)\in\mathbb{D}_{r}^{2}$. It is, then, easy to see that
$\Delta_{zw}f_{n}=\lambda f_{n}$ on $U$ if and only if

(4.3)
$4\left(1-t\right)^{2}\left[ty_{n}^{\prime\prime}(t)+(|n|+1)y_{n}^{\prime}(t)\right]=\lambda
y_{n}(t)$

for all $t\in\mathbb{D}_{r^{2}}$. A power series ansatz shows that there is at
most one solution $y_{n}$ of (4.3) which is holomorphic in a neighborhood of
$t=0$ and normalized such that $y_{n}(0)=1$. In order to find this solution,
we convert (4.3) into a hypergeometric differential equation as follows. We
choose $\mu\in\mathbb{C}$ such that $\lambda=4\mu(\mu-1)$ and let
$\hat{y}_{n}(t):=(1-t)^{-\mu}y_{n}(t)$. Then (4.3) is equivalent to

(4.4)
$t(1-t)\hat{y}_{n}^{\prime\prime}+\big{[}c-(a+b+1)t\big{]}\hat{y}_{n}^{\prime}-ab\hat{y}_{n}=0$

with

$a=\mu,\,b=\mu+|n|,\,c=|n|+1\,.$

It is well–known (see [33, §15.10]) that the only solution $\hat{y}_{n}$ of
(4.4) which is holomorphic at $t=0$ and normalized at $t=0$ by
$\hat{y}_{n}(0)=1$ is the hypergeometric series

${}_{2}F_{1}\left(a,b;c;t\right):=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}}\frac{t^{k}}{k!}\,,\quad|t|<1,$

where

(4.5)
$(\alpha)_{k}:=\prod\limits_{j=0}^{k-1}(\alpha+j),\,\qquad\alpha\in\mathbb{C},\,k\in\mathbb{N}_{0}$

denotes the _(rising) Pochhammer symbol_. This procedure leads to the
conclusion that

(4.6) $y_{n}(t)=(1-t)^{\mu}{}_{2}F_{1}\left(\mu,\mu+|n|;|n|+1;t\right)$

is the unique solution of (4.3) which is holomorphic in $t=0$ and normalized
by $y_{n}(0)=1$. Note that there are in fact two complex numbers
$\mu\in\mathbb{C}$ such that $\lambda=4\mu(\mu-1)$. However, as we have seen,
both necessarily lead to the same holomorphic solution $y_{n}$ of (4.3) with
$y_{n}(0)=1$.

######  Remark 4.

Note that if $\mu$ is one solution to $\lambda=4\mu(\mu-1)$, then $1-\mu$ is
the other one. As we have seen, both numbers induce the same function (4.6).
This also follows from the transformation formula (see [1, Eq. 15.3.3])

$\displaystyle{}_{2}F_{1}\left(a,b;c;t\right)$
$\displaystyle=(1-t)^{c-a-b}{}_{2}F_{1}\left(c-a,c-b;c;t\right)\,.$

In the following it will often be convenient to choose $\mu$ such that ${\sf
Re}\,\mu\geq 1/2$.

By a standard fact about hypergeometric functions ([1, 15.3.1]) the function
$y_{n}$ has a holomorphic extension at least to the slit plane

$\mathbb{C}\setminus[1,\infty)\,.$

Returning to (4.2) with this choice of $y_{n}$, we therefore see that

(4.7)
$F^{\mu}_{n}(z,w):=(1-zw)^{\mu}{}_{2}F_{1}\left(\mu,\mu+|n|;|n|+1;zw\right)\cdot\begin{cases}z^{n}&\text{
if }n\geq 0\,,\\\ w^{|n|}&\text{ if }n\leq 0\,,\end{cases}$

is holomorphic at least on the domain

$\Omega_{*}:=\left\\{(z,w)\in\mathbb{C}^{2}\,:\,zw\in\mathbb{C}\setminus[1,\infty)\right\\}$

and provides the unique $n$–homogeneous solution of $\Delta_{zw}f=\lambda f$
on $\Omega_{*}$ up to a multiplicative constant.

Next, we relate the $n$–homogeneous eigenfunction $F^{\mu}_{n}$ of
$\Delta_{zw}$ to a complexified version of the Poisson kernel of the unit
disk. This slight change of perspective will turn out to be important in the
sequel. In fact, a well–known integral representation formula for the
hypergeometric function ${}_{2}F_{1}\left(\mu,\mu+|n|;|n|+1;zw\right)$, see
[11, Section 2.5.1, Formula (10), p. 81], shows that for every
$n\in\mathbb{Z}$,

(4.8)
$\binom{\mu+|n|-1}{|n|}F^{\mu}_{n}(z,w)=\frac{(1-zw)^{\mu}}{2\pi}\int\limits_{0}^{2\pi}\frac{e^{int}}{\left(1-ze^{-it}\right)^{\mu}\left(1-we^{it}\right)^{\mu}}\,dt\,,$

with the _generalized binomial coefficients_
$\binom{\mu}{n}\coloneqq\frac{(\mu-n+1)_{n}}{n!}$ for $\mu\in\mathbb{C}$,
$n\in\mathbb{N}_{0}$. On the right–hand side of this identity one can
recognize the $(-n)$–th Fourier coefficient of (a suitably defined power of
order $\mu$ of) the _generalized Poisson kernel_

(4.9) $P:\mathbb{D}^{2}\times\partial\mathbb{D}\rightarrow\mathbb{C}\,,\quad
P(z,w;\xi):=\frac{1-zw}{(1-z/\xi)(1-w\xi)}\,.$

Note that if $w=\overline{{z}}\in\mathbb{D}$, then $P(z,\overline{z};\xi)$ is
the standard Poisson kernel of the unit disk. These considerations motivate
the following definition.

######  Definition 4.1 (Poisson Fourier mode, PFM)

Let $\mu\in\mathbb{C}$ and $n\in\mathbb{Z}$. Then the function

(4.10) $P_{n}^{\mu}:\mathbb{D}^{2}\rightarrow\mathbb{C}\,,\quad
P_{n}^{\mu}(z,w):=\frac{\left(1-zw\right)^{\mu}}{2\pi}\int\limits_{0}^{2\pi}\frac{e^{-int}}{\left(1-ze^{-it}\right)^{\mu}\left(1-we^{it}\right)^{\mu}}\,dt\,.$

is called the _$n$ –th Poisson Fourier mode (PFM) of order $\mu$_.
Here, $a^{\mu}$ is defined for $a\in\mathbb{C}\setminus(-\infty,0]$ as
$\exp\left(\mu\log a\right)$, where $\log$ denotes the principal branch of the
logarithm.

######  Remark 5.

If $w=\overline{z}\in\mathbb{D}$, then $P^{\mu}_{n}(z,\overline{z})$ is the
$n$–th Fourier coefficient of the $\mu$–power of the (real–valued and, in
fact, non–negative) Poisson kernel of the unit disk $\mathbb{D}$. Further, if
$\mu=m\in\mathbb{Z}$, then $P^{m}_{n}(z,w)$ also is the $n$–th Fourier
coefficient of the $m$–power of the generalized Poisson kernel from (4.9).

We can now reformulate (4.8) in terms of PFMs as follows:

(4.11)
$P^{\mu}_{n}=(-1)^{n}\binom{-\mu}{|n|}F^{\mu}_{-n}=\binom{\mu+|n|-1}{|n|}F^{\mu}_{-n}\,.$

In particular, $P^{\mu}_{n}$ has a holomorphic extension from $\mathbb{D}^{2}$
to $\Omega_{*}$, which we continue to denote by $P^{\mu}_{n}$.

In analogy with (4.1) we call a subdomain $D\subseteq\Omega$ _rotationally
invariant_ if $(\eta z,w/\eta)\in D$ for all $(z,w)\in D$ and all
$\eta\in\mathop{}\\!\partial\mathbb{D}$. Summarizing our considerations leads
to the following complete description of the $n$–homogeneous eigenfunctions of
$\Delta_{zw}$:

###### Theorem 4.2

Let $D$ be a rotationally invariant subdomain of $\Omega$ containing the
origin and $n\in\mathbb{Z}$. Suppose that $f\in\mathcal{H}(D)$ is an
$n$–homogeneous solution to $\Delta_{zw}f=\lambda f$ for some
$\lambda\in\mathbb{C}$ of the form $\lambda=4\mu(\mu-1)$ with ${\sf
Re}\,\mu\geq 1/2$. Then there is a constant $c\in\mathbb{C}$ such that

$f(z,w)=c\,P_{-n}^{\mu}(z,w),\qquad(z,w)\in D\cap\Omega_{*}\,.$

In particular, $f$ has a holomorphic extension to $D\cup\Omega_{*}$. Moreover,
the following dichotomy holds:

  * (NE)

(Non–exceptional case)
If $\lambda\not=4m(m+1)$ for all $m\in\mathbb{N}_{0}$, then $\Omega_{*}$ is
the maximal domain of existence of $f$.

  * (E)

(Exceptional case)
If $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$, then the maximal domain of
existence of $f$ is

$\begin{array}[]{lcl}\Omega_{+}&&n<-m\,;\\\\[5.69054pt] \Omega_{-}&\text{ if
}&n>m\,;\\\\[5.69054pt] \Omega&&\\!|n|\leq m\,.\end{array}$

###### Proof.

It remains to prove that for any nonconstant $n$–homogeneous eigenfunction
$F^{\mu}_{n}$ of $\Delta_{zw}$ in $\mathcal{H}(\Omega_{*})$ the dichotomy
“(NE) vs. (E)” holds.

(i) Let $\mu\not\in\mathbb{N}$ and assume that $\Omega_{*}$ is not the maximal
domain of existence of $F^{\mu}_{n}$ which is contained in $\Omega$, see
Definition 2.4. Then $F^{\mu}_{n}$ has a holomorphic extension to some point
$(z_{0},w_{0})\in\Omega$ such that $z_{0}w_{0}\in\mathbb{R}\cup\\{\infty\\}$
and $z_{0}w_{0}>1$. We first consider the case $n\geq 0$. By definition of
$F^{\mu}_{n}$, see (4.7), and in view of [1, 15.3.4], we have

(4.12)
$F^{\mu}_{n}(z,w)=(1-zw)^{\mu}{}_{2}F_{1}\left(\mu,\mu+n;|n|+1;zw\right)z^{n}={}_{2}F_{1}\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right)z^{n}\,,$

from which we infer that

$t\mapsto{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;t\right)$

is holomorphic in a neighborhood of the point
$x_{0}:=z_{0}w_{0}/(z_{0}w_{0}-1)\geq 1$. However, see [33, 15.2.3],

(4.13) $\begin{split}\lim\limits_{y\to
0^{+}}&\big{[}{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;x+iy\right)-{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;x-iy\right)\big{]}\\\
&\quad=\frac{2\pi
i}{\Gamma(\mu)\Gamma(1-\mu)}\left(x-1\right)^{|n|}{}_{2}F_{1}\left(|n|+1-\mu,|n|+\mu;|n|+1;1-x\right)\end{split}$

for all $x>1$. By our assumption, the left–hand side of (4.13) has to vanish
for all $x\in\mathbb{R}$ in some open interval $(x_{0},x_{0}+\varepsilon)$
with $\varepsilon>0$ and hence the same is true for the right–hand side. Since
the right–hand side is a holomorphic function of $x$ on the domain
$\mathbb{C}\setminus(-\infty,0]$ this is clearly only possible if
$\mu\in\mathbb{N}$. The remaining case $n\leq 0$ follows from the case $n\geq
0$ and $F^{\mu}_{-n}(z,w)=F^{\mu}_{n}(w,z)$.

(ii) Let $\mu\in\mathbb{N}$ and $|n|\leq\mu-1$. If $n\geq 0$, then

$F^{\mu}_{n}(z,w)=\frac{G^{\mu}_{n}(zw)}{(1-zw)^{\mu-1}}z^{n}\,,$

where

(4.14) $\displaystyle G^{\mu}_{n}(zw)$
$\displaystyle:=(1-zw)^{2\mu-1}{}_{2}F_{1}\left(\mu,\mu+|n|;|n|+1;zw\right)$
$\displaystyle\hskip
3.1298pt={}_{2}F_{1}\left(1-\mu,|n|+1-\mu;|n|+1;zw\right)$

is a polynomial in $zw$ of degree $\mu-n-1\geq 0$, see [1, 15.3.3 and 15.1.1].
Hence $F^{\mu}_{n}$ is the product of $z^{n}$ and a rational function in $zw$
of numerator degree $\mu-n-1$ and of denominator degree $\mu-1$ with pole only
at the point $1$. Therefore, $F^{\mu}_{n}\in\mathcal{H}(\Omega)$. Since
$F^{\mu}_{n}(z,w)=F^{\mu}_{-n}(w,z)$, this implies
$F^{\mu}_{n}\in\mathcal{H}(\Omega)$ also for $1-\mu\leq n\leq 0$.

(iii) Let $\mu\in\mathbb{N}$ and $|n|\geq\mu$. Since
$F^{\mu}_{n}(z,w)=F^{\mu}_{-n}(w,z)$ and
$\mathcal{H}(\Omega_{+})\cap\mathcal{H}(\Omega_{-})=\mathcal{H}(\Omega)$ we
may assume $n\geq\mu$. In this case, the function $G^{\mu}_{n}$ in (4.14) is a
polynomial in $zw$ of degree $\mu-1$, and thus $F^{\mu}_{n}$ is the product of
$z^{n}$ times a rational function in $zw$ of numerator degree $\mu-1$ and of
denominator degree $\mu-1$, and is therefore holomorphic on $\Omega_{-}$.
Moreover, $F^{\mu}_{n}$ has no holomorphic extension to a point
$(\infty,w_{0})$ with $w_{0}\in\widehat{\mathbb{C}}\setminus\\{0\\}$ since in
view of (4.14) and (4.12)

$\displaystyle\lim\limits_{z\to\infty}\frac{G^{\mu}_{n}(zw_{0})}{(1-zw_{0})^{\mu-1}}$
$\displaystyle=\lim\limits_{z\to\infty}{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;\frac{zw_{0}}{zw_{0}-1}\right)={}_{2}F_{1}\left(\mu,1-\mu;|n|+1;1\right)$
$\displaystyle=\frac{\Gamma(n+1)\Gamma(n)}{\Gamma(n+1-\mu)\Gamma(n+\mu)}\in(0,\infty)\,,$

by [1, 15.1.20], so $|F^{\mu}_{n}(z,w_{0})|\to\infty$ as $z\to\infty$. This
implies that $\Omega_{-}$ is the maximal domain of existence of $F^{\mu}_{n}$.
∎

###### Corollary 4.3

For every $\lambda\in\mathbb{C}$ and each $n\in\mathbb{Z}$ there is an
$n$–homogeneous holomorphic solution of $\Delta_{zw}f=\lambda f$ on the domain
$\Omega_{*}$. More precisely, if $\lambda=4\mu(\mu-1)\in\mathbb{C}$ with ${\sf
Re}\,\mu\geq 1/2$, then every such solution has the form $cP^{\mu}_{-n}$ for
some $c\in\mathbb{C}$.

We close this section by collecting some elementary properties of the PFM
$P^{\mu}_{n}$ which will be needed in the sequel.

######  Remark 6 (Elementary properties of Poisson Fourier modes).

Let $\mu\in\mathbb{C}$, $n\in\mathbb{N}_{0}$ and $z,w\in\mathbb{D}$.

  1. (a)

The Poisson Fourier modes are related to the hypergeometric function
${}_{2}F_{1}$ via

(4.15)
$\displaystyle\begin{split}P_{n}^{\mu}(z,w)&=(-1)^{n}\binom{-\mu}{n}(1-zw)^{\mu}w^{n}{}_{2}F_{1}\left(\mu,\mu+n;n+1;zw\right)\\\
&=(-1)^{n}\binom{-\mu}{n}w^{n}{}_{2}F_{1}\left(\mu,1-\mu;n+1;\frac{zw}{zw-1}\right)\,.\end{split}$

This is (4.7) together with (4.11) resp. (4.12).

  2. (b)

Using the series representation of ${}_{2}F_{1}$ functions we see that

(4.16)
$P_{n}^{\mu}(z,w)=(-1)^{n}(1-zw)^{\mu}\sum_{k=0}^{\infty}\binom{-\mu}{k+n}\binom{-\mu}{k}z^{k}w^{k+n}\,.$

If $-\mu=m\in\mathbb{N}_{0}$, then $P_{\pm n}^{-m}=0$, whenever $n>m$.
Otherwise,

(4.17)
$P_{n}^{-m}(z,w)=(-1)^{n}\sum_{k=0}^{m-n}\binom{m}{k+n}\binom{m}{k}\frac{z^{k}w^{k+n}}{(1-zw)^{m}}\,.$

  3. (c)

The PFM are symmetric in the sense that $P_{n}^{\mu}(z,w)=P_{-n}^{\mu}(w,z)$.
This allows us to simplify our proofs in the following: we will often prove
identities for $P^{\mu}_{n}$ only which then implies the corresponding result
for $P^{\mu}_{-n}$.

  4. (d)

Remark 4 implies

(4.18) $\binom{\mu-1}{n}P_{\pm n}^{\mu}=\binom{-\mu}{n}P_{\pm n}^{1-\mu}\,.$

######  Remark 7 (Invariant representative functions and the finite
dimensional invariant eigenspaces).

We consider $\mathbb{D}$ as a symmetric space
$\mathcal{M}(\mathbb{D})/\partial\mathbb{D}$ over its automorphism group
$\mathcal{M}(\mathbb{D})$ with $\partial\mathbb{D}\cong\operatorname{U}(1)$
acting by rotations. This yields an alternative way of deriving the
restrictions to $\mathbb{D}$ of those Poisson Fourier modes from (4.17), which
are defined on all of $\Omega$, using finite dimensional representation
theory. Such considerations are very much in the spirit of [21]. The group
$\mathcal{M}(\mathbb{D})$ is isomorphic to the projective split unitary group

$\operatorname{PSU}(1,1)=\Big{\\{}\mbox{$\scriptsize\begin{pmatrix}a&b\\\
\overline{{b}}&\overline{{a}}\end{pmatrix}$}\colon\lvert a\rvert^{2}-\lvert
b\rvert^{2}=1\Big{\\}}\Big{/}\Big{\\{}\pm\mbox{$\scriptsize\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$}\Big{\\}}$

as a real Lie group. By Schur’s Lemma every irreducible representation
$\pi\colon\mathcal{M}(\mathbb{D})\longrightarrow\operatorname{GL}_{n}(\mathbb{C})$
as invertible $(n\times n)$–matrices induces eigenfunctions of the Laplacian
on $\operatorname{PSU}(1,1)$ via

(4.19)
$\pi_{k,\ell}\colon\operatorname{PSU}(1,1)\longrightarrow\mathbb{C},\quad\pi_{k,\ell}(g)\coloneqq\pi(g)_{k,\ell}\,,\qquad
1\leq k,\ell\leq n,$

which are known as _representative functions_ or _matrix elements_. For a
systematic discussion we refer to the textbook [9, Chapter III]. The
corresponding eigenvalues may be computed from representation theoretic data,
see e.g. [16, Prop. 10.6]. Note that in the literature, many results are
formulated for general invariant differential operators or the corresponding
Casimir elements and joint eigenfunctions thereof instead of the Laplacian,
which is always invariant and thus constitutes a special case. However, the
disk $\mathbb{D}$ is a two–point homogeneous space, so all
$\mathcal{M}(\mathbb{D})$–invariant differential operators are polynomials in
the Laplacian, see [20, Theorem 11]. Adapting [16, Example 4.10], one may
parametrize the irreducible representations of $\operatorname{PSU}(1,1)$ as
follows: let $m\in\mathbb{N}_{0}$ and

$V_{m}\coloneqq\operatorname{span}\big{\\{}z^{k}w^{m-k}\in\mathbb{C}[z,w]\;\big{|}\;0\leq
k\leq m\big{\\}}$

the vector space of polynomials of total degree $m$ with $\dim(V_{m})=m+1$.
Then

$\pi_{m}\colon\operatorname{SU}(1,1)\longrightarrow\operatorname{GL}(V_{m}),\quad\pi_{m}\mbox{$\scriptsize\begin{pmatrix}a&b\\\
\overline{{b}}&\overline{{a}}\end{pmatrix}$}p(z,w)\coloneqq
p\big{(}\overline{{a}}z-bw,-\overline{{b}}z+aw\big{)}$

for $m\in\mathbb{N}$ constitutes a complete list of the irreducible finite
dimensional representations of $\operatorname{SU}(1,1)$, see also [16, Prop.
4.11 & Sec. 4.6]. Moreover, $\pi_{m}$ descends to the quotient
$\operatorname{PSU}(1,1)$ if and only if $m$ is even, providing a description
of all irreducible finite dimensional representations of
$\operatorname{PSU}(1,1)$. A computation then yields explicit formulas for
(4.19), which completes the eigenvalue theory of the Laplacian on
$\operatorname{PSU}(1,1)$. Finally, parametrizing a copy of the rotation group
$\operatorname{U}(1)\subseteq\operatorname{PSU}(1,1)$ by
$\big{(}\begin{smallmatrix}i\eta&0\\\
0&-i\overline{{\eta}}\end{smallmatrix}\big{)}$ with
$\eta\in\partial\mathbb{D}$ yields that exactly the representative functions
$(\pi_{2m})_{m,d}$ with $1\leq d\leq 2m+1$ are invariant under the action of
$\operatorname{U}(1)$ and thus pass to the quotient
$\mathbb{D}\cong\operatorname{PSU}(1,1)/\operatorname{U}(1)$. By a
computation, $(\pi_{2m})_{m,d}=P_{m-d}^{-m}$ with the exceptional Poisson
Fourier modes from (4.17). Note that this only recovers the finite dimensional
invariant eigenspace, i.e. the case $(E_{0})$ in Theorem 2.8. It would be
interesting to study whether this approach generalizes to the other invariant
subspaces by incorporating representations on infinite dimensional spaces.

##  5\. Spectral decomposition of eigenspaces

In this section we show that for every rotationally invariant domain
$D\subseteq\Omega$ containing the origin each holomorphic eigenfunction of the
invariant Laplacian $\Delta_{zw}$ on $D$ has a unique representation as a
Poisson Fourier series, a doubly infinite series with Poisson Fourier modes as
building blocks. We shall also see that if $D=\Omega$ this series corresponds
to a finite sum, and when $D$ is one of the distinguished domains $\Omega_{+}$
or $\Omega_{-}$, then the series is one–sided infinite.

###### Theorem 5.1

Let $D$ be a rotationally invariant subdomain of $\Omega$ containing the
origin, and let $f\in\mathcal{H}(D)$ be such that $\Delta_{zw}f=\lambda f$ for
some $\lambda\in\mathbb{C}$ of the form $\lambda=4\mu(\mu-1)$ with ${\sf
Re}\,\mu\geq 1/2$. Then there are uniquely determined coefficients
$c_{n}\in\mathbb{C}$ such that

(5.1)
$f=\sum\limits_{n=-\infty}^{\infty}c_{n}P_{n}^{\mu}:=\sum_{n=0}^{\infty}c_{n}P_{n}^{\mu}+\sum_{n=1}^{\infty}c_{-n}P_{-n}^{\mu}\,.$

Here, both series converge absolutely and locally uniformly in $D$.

###### Proof.

(i) For each $n\in\mathbb{Z}$ and all $(z,w)\in D$ we consider

$f_{n}(z,w):=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}}f\left(\eta
z,\frac{w}{\eta}\right)\eta^{-(n+1)}\,d\eta\,.$

Since $D$ is rotationally invariant, $f_{n}$ is well–defined. Clearly, $f_{n}$
is holomorphic on $D$ and $n$–homogeneous. By Theorem 4.2, there are complex
numbers $c_{n}\in\mathbb{C}$ such that

$f_{n}(z,w)=c_{n}P^{\mu}_{-n}(z,w)\,.$

(ii) We fix $(z,w)\in D$ and choose positive constants $r<1<R$ such that
$(\eta z,w/\eta)\in D$ for all $r<|\eta|<R$. This is possible as $D$ is a
rotationally invariant domain. Then $\eta\mapsto f(\eta z,w/\eta)$ is
holomorphic in the annulus $r<|\eta|<R$ and therefore has a representation as
the Laurent series

$f\left(\eta
z,\frac{w}{\eta}\right)=\sum\limits_{n=-\infty}^{\infty}f_{n}(z,w)\eta^{n}\,,$

which converges locally uniformly in $r<|\eta|<R$. In particular,

$f\left(z,w\right)=\sum\limits_{n=-\infty}^{\infty}f_{n}(z,w)\,,\qquad(z,w)\in
D\,.$

This series converges in fact uniformly on each compact set $K\subseteq D$. In
order to see this, let $K$ be such a compact subset of $D$. We can then choose
positive constants $r_{1}<1<R_{1}$ such that ${K_{1}:=\\{(\eta
z,w/\eta)\,:\,(z,w)\in K,\,r_{1}\leq|\eta|\leq R_{1}\\}}$ is a compact subset
of $D$, and we let ${M_{1}:=\max\\{|f(z,w)|\,:(z,w)\in K_{1}\\}}$. Note that

$f_{n}(z,w)=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}_{\rho}}f\left(\eta
z,\frac{w}{\eta}\right)\,\eta^{-(n+1)}\,d\eta\,$

for all $n\in\mathbb{Z}$, all $(z,w)\in K$ and every $\rho\in[r,R]$. We
therefore get

$|f_{n}(z,w)|\leq M_{1}\cdot\rho^{-n}\,\quad\text{ for all }(z,w)\in K\text{
and all }\rho\in[r_{1},R_{1}]\,.$

In particular, $|f_{n}(z,w)|\leq M_{1}r_{1}^{-n}$ for all $n<0$ and
$|f_{n}(z,w)|\leq M_{1}R_{1}^{-n}$ for all $n\geq 0$, and this ensures the
absolute and uniform convergence of the two series

$\sum\limits_{n=-\infty}^{-1}f_{n}(z,w)\quad\text{ and
}\quad\sum\limits_{n=0}^{\infty}f_{n}(z,w)$

on the compact set $K$.

(iii) In view of Step (ii) the coefficients $f_{n}(z,w)$ are exactly the
Laurent coefficients of ${\eta\mapsto f(\eta z,w/\eta)}$ in an annulus
containing the unit circle and are thus uniquely determined by $f$. Hence,
$f_{n}(z,w)=c_{n}P_{-n}^{\mu}(z,w)$ shows that the coefficients $c_{n}$ are
uniquely determined by $f$. ∎

In fact, the previous proof provides the following more precise information.

###### Corollary 5.2

Let $D$ be a subdomain of $\Omega$ containing the origin, and let
$f\in\mathcal{H}(D)$ be such that $\Delta_{zw}f=\lambda f$ for some
$\lambda\in\mathbb{C}$.

  * (i)

If $D=\Omega$, then $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$ and there
are uniquely determined coefficients $c_{n}\in\mathbb{C}$ such that

$f=\sum\limits_{n=-m}^{m}c_{n}P^{m+1}_{n}\,.$

  * (ii)

If $D=\Omega_{+}$, then $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$ and
there are uniquely determined coefficients $c_{n}\in\mathbb{C}$ such that

$f=\sum\limits_{n=-\infty}^{m}c_{n}P^{m+1}_{n}\,.$

  * (iii)

If $D=\Omega_{-}$, then $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$ and
there are uniquely determined coefficients $c_{n}\in\mathbb{C}$ such that

$f=\sum\limits_{n=-m}^{\infty}c_{n}P^{m+1}_{n}\,.$

###### Proof.

(i) If $f\in\mathcal{H}(\Omega)$, then the functions

$f_{n}(z,w):=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}}f\left(\eta
z,\frac{w}{\eta}\right)\eta^{-(n+1)}\,d\eta\,$

are holomorphic and $n$–homogeneous on $\Omega$ and

$f_{n}(z,w)=c_{n}P^{\mu}_{-n}(z,w)\,$

with $\mu\in\mathbb{C}$ such that $\lambda=4\mu(\mu-1)$ and ${\sf Re}\,\mu\geq
1/2$. Hence, $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$ since otherwise
$P^{\mu}_{-n}$ is not holomorphic on $\Omega$ by Theorem 4.2. If
$\lambda=4m(m+1)$, then for each $|n|>m$ the function $P^{m+1}_{-n}$ is not
holomorphic on $\Omega$ again by Theorem 4.2 which forces $c_{n}=0$ for those
$n$. Parts (ii) and (iii) follow in the same way. ∎

Since all PFM $P^{\mu}_{n}$ are holomorphic on $\Omega_{*}$, it is natural to
inquire whether the series (5.1) in Theorem 5.1 converges on some bigger
domain than $D$. In Section 6 we shall see that in general this is not the
case.

##  6\. Comparison with the eigenvalue equation of the Laplacian on the unit
disk and the Riemann sphere

In this section we relate the spectral theory of the Laplacian on the unit
disk $\mathbb{D}$ resp. the Riemann sphere $\widehat{\mathbb{C}}$ with the
spectral theory of the invariant Laplacian on $\Omega$ which we have developed
so far. In particular, we show that all eigenfunctions of
$\Delta_{\mathbb{D}}$ on $\mathbb{D}$ resp. $\Delta_{\widehat{\mathbb{C}}}$ on
$\widehat{\mathbb{C}}$ do have holomorphic extensions to eigenfunctions of
$\Delta_{zw}$ on the bidisk $\mathbb{D}^{2}$ resp. $\Omega$. Our approach is
similar to the one employed in [8, Section 1.6] which deals exclusively with
the hyperbolic Laplacian $\Delta_{\mathbb{D}}$ on the unit disk $\mathbb{D}$.
However, we need some fine properties of hypergeometric functions, in addition
to those which have been employed in [8]. We begin with the following lemma
which is crucial for our approach.

###### Lemma 6.1

Let $\mu\in\mathbb{C}$ with ${\sf Re}\,\mu>0$. Then

(6.1) $\lim\limits_{n\to\infty}{}_{2}F_{1}\left(\mu,1-\mu;n+1;\omega\right)=1$

locally uniformly for $\omega\in\mathbb{C}\setminus[1,\infty)$.

###### Proof.

Fix $m\in\mathbb{N}_{0}$ such that $m>{\sf Re}\,\mu-2$ and let
$n\in\mathbb{N}_{0}$. Then by [11, Formula (11), p. 76] or [30, p. 84] there
are complex numbers $A_{k}$, $k=1,\ldots,m$, such that

${}_{2}F_{1}\left(\mu,1-\mu;n+1;\omega\right)=1+\sum\limits_{k=1}^{m}\frac{A_{k}}{\left(n+1\right)\ldots\left(n+k\right)}\frac{\omega^{k}}{k!}+\rho_{m+1}(n,\omega)\,,$

where

(6.2)
$\rho_{m+1}(n,\omega):=\tilde{\gamma}_{n}\int\limits_{0}^{1}\int\limits_{0}^{1}t^{1-\mu+m}(1-t)^{n+\mu-1}\left(1-st\omega\right)^{-\mu-1-m}(1-s)^{m}\,ds\,dt\,\cdot\,\omega^{m+1}$

and

$\tilde{\gamma}_{n}:=\frac{\Gamma(n+1)\Gamma(\mu+m)}{\Gamma(1-\mu)\Gamma(\mu+n)\Gamma(\mu)\,m!}\,.$

Note that our choice of the non–negative integer $m$ guarantees that the
integral in (6.2) converges. To prove (6.1) it therefore suffices to show that
$\rho_{m+1}(n,\omega)\to 0$ uniformly on every compact subset $K$ of
$\mathbb{C}\setminus[1,\infty)$ as $n\to\infty$. Fix such a compact set $K$.
Obviously,

$M_{K}:=\min\limits_{\omega\in K,0\leq s,t\leq 1}|1-st\omega|>0\,,$

and since $-{\sf Re}\,\mu-1-m<0$, we have

$\displaystyle|\rho_{m+1}(n,\omega)|$
$\displaystyle\leq|\tilde{\gamma}_{n}|\int\limits_{0}^{1}t^{1-{\sf
Re}\,\mu+m}(1-t)^{n+{\sf Re}\,\mu-1}\,dt\frac{|\omega|^{m+1}}{M_{K}^{m+1+{\sf
Re}\,\mu}}$ $\displaystyle=|\tilde{\gamma}_{n}|\frac{\Gamma(2+m-{\sf
Re}\,\mu)\,\Gamma(n+{\sf
Re}\,\mu)}{\Gamma(2+m+n)}\frac{|\omega|^{m+1}}{M_{K}^{m+1+{\sf Re}\,\mu}}$
(6.3) $\displaystyle=\gamma^{*}\frac{\Gamma(n+{\sf
Re}\,\mu)}{\Gamma(n+1)}\frac{\Gamma(n+1)}{|\Gamma(n+\mu)|}\frac{\Gamma(n+1)}{\Gamma(2+m+n)}\frac{|\omega|^{m+1}}{M_{K}^{m+1+{\sf
Re}\,\mu}}$

where

$\gamma^{*}:=\frac{|\Gamma(\mu+m)|\,\Gamma(2+m-{\sf
Re}\,\mu)}{|\Gamma(1-\mu)|\,|\Gamma(\mu)|\,m!}\,.$

An application of Stirling’s formula

(6.4)
$\lim\limits_{n\to\infty}\frac{\Gamma(\alpha+n)}{\Gamma(n+1)}e^{-(\alpha-1)\log(n+1)}\to
1\,,\quad\alpha\in\mathbb{C}\,,$

to each of the first three quotients in (6.3) shows that there is a constant
$\gamma>0$ depending only on $\mu$, $m$ and $K$ such that

$|\rho_{m+1}(n,\omega)|\leq\gamma\cdot\left(n+1\right)^{{\sf
Re}\,\mu-1}\left(n+1\right)^{1-{\sf
Re}\,\mu}\left(n+1\right)^{-m-1}|\omega|^{m+1}=\gamma\cdot\left(\frac{|\omega|}{n+1}\right)^{m+1}\,.$

In particular, $\rho_{m+1}(n,\omega)\to 0$ as $n\to\infty$ uniformly for
$\omega\in K$. This completes the proof of (6.1). ∎

We are now in a position to prove Theorem 2.2.

###### Proof Theorem 2.2.

Let $f\in X_{\lambda}(\mathbb{D})$, and write $\lambda=4\mu(\mu-1)$ for some
complex number $\mu\in\mathbb{C}$ with ${\sf Re}\,\mu\geq 1/2$. By [8, Theorem
16.18] there are uniquely determined coefficients $c_{n}\in\mathbb{C}$ such
that

$f(z)=\sum\limits_{n=-\infty}^{\infty}c_{n}P^{\mu}_{n}(z,\overline{z})\,;$

the series converges absolutely and pointwise for each $z\in\mathbb{D}$. As it
is shown in the proof of [8, Theorem 1.6.18], the coefficients $c_{n}$ do have
the additional property that

(6.5) $\sum\limits_{n=-\infty}^{\infty}|c_{n}|r^{|n|}<\infty\quad\text{ for
all }0<r<1\,.$

We proceed to show that (6.5) and Lemma 6.1 together guarantee that the series

(6.6) $F(z,w):=\sum\limits_{n=-\infty}^{\infty}c_{n}P^{\mu}_{n}(z,w)$

converges locally uniformly for $(z,w)\in\mathbb{D}^{2}$, and hence defines a
function ${F\in\mathcal{H}(\mathbb{D}^{2})}$ with the required properties. The
identity principle shows further that $F$ is then uniquely determined.

It remains to prove the local uniform convergence of the series (6.6) in
$\mathbb{D}^{2}$. Fix $r\in(0,1)$. We begin by noting that (4.11) and (4.7)
lead to

(6.7)
$\begin{split}\left|P^{\mu}_{n}(z,w)\right|&=\left|\binom{\mu}{|n|}\right|\,\left|F^{\mu}_{-n}(z,w)\right|\leq\frac{|\Gamma(\mu+|n|)|}{|\Gamma(\mu)|\,\Gamma(|n|+1)}\left|{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right)\right|\,r^{|n|}\end{split}$

for all $|z|,|w|\leq r$. Since the Möbius transformation

$\xi\mapsto\frac{\xi}{\xi-1}$

maps the unit disk $\mathbb{D}$ conformally onto the half–plane
$\\{\zeta\in\mathbb{C}\,:\,{\sf Re}\,\zeta<1/2\\}$, the set

$K:=\left\\{\frac{zw}{zw-1}\,:\,|z|\leq r,\,|w|\leq r\right\\}$

is a compact subset of $\mathbb{C}\setminus[1,\infty)$. Thus, Lemma 6.1
implies

${}_{2}F_{1}\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right)\to 1\quad\text{
uniformly for }|z|\leq r,\,|w|\leq r\,$

as $|n|\to\infty$. Combining this with Stirling’s formula (6.4) we see from
inequality (6.7) that there is a constant $\gamma>0$ such that

(6.8) $\left|P^{\mu}_{n}(z,w)\right|\leq\gamma\cdot\left(|n|+1\right)^{{\sf
Re}\,\mu-1}\,r^{|n|}$

for all $|z|\leq r$, $|w|\leq r$ and every $n\in\mathbb{Z}$. This estimate
together with (6.5) implies that the series (6.6) converges uniformly for
$|z|,|w|\leq r$, as required. In particular, we have shown that the
restriction map

$\mathcal{R}_{h}:X_{\lambda}(\mathbb{D}^{2})\to
X_{\lambda}(\mathbb{D})\,,\qquad\mathcal{R}_{h}(F)(z):=F(z,\overline{z})\quad(z\in\mathbb{D})\,$

is bijective. Since $X_{\lambda}(\mathbb{D}^{2})$ and
$X_{\lambda}(\mathbb{D})$ are both Fréchet spaces with respect to the topology
of locally uniform convergence on $\mathbb{D}^{2}$ resp. $\mathbb{D}$ (see
[38, Corollary 1 to Theorem 4.2.4] for the fact that $X_{\lambda}(\mathbb{D})$
is a Fréchet space) and the restriction map $\mathcal{R}_{h}$ is obviously
continuous, its inverse is continuous as well by the Open Mapping Theorem.
This completes the proof of Theorem 2.2. ∎

######  Theorem 6.2 (Smooth eigenfunctions of $\Delta_{\widehat{\mathbb{C}}}$
on $\widehat{\mathbb{C}}$ vs. holomorphic eigenfunctions of $\Delta_{zw}$ on
$\Omega$)

Let $\lambda\in\mathbb{C}$. Then the following are equivalent:

  * (i)

There is a function $f\in C^{2}(\widehat{\mathbb{C}})$ such that
$\Delta_{\widehat{\mathbb{C}}}f=\lambda f$ on $\widehat{\mathbb{C}}$.

  * (ii)

There is a function $F\in\mathcal{H}(\Omega)$ such that $\Delta_{zw}F=\lambda
F$ on $\Omega$.

  * (iii)

$\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$.

Theorem 6.2 is a special case of Theorem 2.5.

###### Proof.

The implication (ii) $\Rightarrow$ (iii) is Corollary 5.2, and (iii)
$\Rightarrow$ (ii) is Theorem 4.2. Clearly, (ii) implies (i), so we only need
to prove that (i) implies (ii). Accordingly, we write $\lambda=4\mu(\mu-1)$
with $\mu\in\mathbb{C}$ and ${\sf Re}\,\mu\geq 1/2$. For $n\in\mathbb{Z}$
consider

$f_{n}(z):=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}}f(\eta
z)\,\eta^{-(n+1)}\,d\eta=\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(e^{it}z)e^{-int}\,dt\,.$

Then $f_{n}\in C^{2}(\widehat{\mathbb{C}})$ is $n$–homogeneous and
$\Delta_{\widehat{\mathbb{C}}}f_{n}=4\mu(\mu-1)f_{n}$ on
$\widehat{\mathbb{C}}$. Arguing in a similar way as in the proof of Theorem
4.2 we see that there is a constant $c_{n}\in\mathbb{C}$ such that

$f_{n}(z)=c_{n}\binom{-\mu}{|n|}{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;\frac{|z|^{2}}{1+|z|^{2}}\right)\cdot\begin{cases}\overline{z}^{|n|}&\text{
if }n\geq 0\,,\\\ (-z)^{|n|}&\text{ if }n\leq 0\,.\end{cases}$

Therefore, the behavior of $f_{n}$ as $z\to\infty$ depends essentially on the
value $\prescript{}{2}{F_{1}}\left(\mu,1-\mu;|n|+1;1\right)$. It is
well–known, see [3, Theorem 2.1.3], that

$\lim\limits_{x\to
1-}\frac{{}_{2}F_{1}\left(\mu,1-\mu;|n|+1;1\right)}{-\log(1-x)}=\frac{1}{\Gamma(\mu)\Gamma(1-\mu)}\,\quad\text{
if }n=0\;,$

and, see [3, Theorem 2.2.2],

${}_{2}F_{1}\left(\mu,1-\mu;|n|+1;1\right)=\frac{\Gamma(|n|+1)\Gamma(|n|)}{\Gamma(|n|+1-\mu)\Gamma(|n|+\mu)}\,\quad\text{
if }n\not=0\,.$

Hence, if $n=0$, then $f_{n}$ is defined at the point $\infty$ only if
$c_{0}=0$ or $\mu\in\mathbb{N}$. If $n\not=0$, then
$\prescript{}{2}{F_{1}}\left(\mu,1-\mu,|n|+1,1\right)=0$ if and only if
$\mu=|n|+1+k$ for some $k\in\mathbb{N}_{0}$. Thus,
$f_{n}:\widehat{\mathbb{C}}\to\mathbb{C}$ is defined at the point $\infty$
only if $c_{n}=0$ or $\mu\in\mathbb{N}$, $\mu>|n|$. We conclude that either
$c_{n}=0$ for every $n\in\mathbb{Z}$ and then $f\equiv 0$, or $c_{n}\not=0$
for at least one $n\in\mathbb{Z}$ and then $\lambda=4m(m+1)$ for some
$m\in\mathbb{N}_{0}$. In the latter case, we see that $c_{n}\not=0$ forces
$m+1=\mu=|n|+1+k$ for some $k\in\mathbb{N}_{0}$, so $m=|n|+k$. In particular,
$c_{n}=0$ for all $|n|>m$, so $f(z)=F(z,-\overline{z})$ for

$F(z,w)=\sum\limits_{n=-m}^{m}c_{n}P^{\mu}_{n}(z,w)\in\mathcal{H}(\Omega)\,.\qed$

######  Remark 8.

We see that the spectrum of the hyperbolic Laplacian $\Delta_{\mathbb{D}}$ is
$\mathbb{C}$ whereas the spectrum of the spherical Laplacian
$\Delta_{\widehat{\mathbb{C}}}$ is notably smaller as it only consists of the
scalars $4m(m+1)$ for $m\in\mathbb{N}_{0}$. Furthermore, by Theorem 6.2 every
eigenfunction of the spherical Laplacian can be extended to the whole domain
$\Omega$. This is different to the hyperbolic case where the extension to
$\mathbb{D}^{2}$ provided by Theorem 2.2 is maximal at least for the category
of rotationally invariant domains as the following example shows.

###### Example 6.3

Fix $\mu\in\mathbb{C}$ with ${\sf Re}\,\mu\geq\frac{1}{2}$, let
$(z_{0},w_{0})\in\Omega\setminus\mathbb{D}^{2}$ with
$z_{0}w_{0}\not\in[1,\infty)$, and define

$b_{n}:=\left(\frac{\Gamma(\mu+|n|)}{\Gamma(\mu)|n|!}\prescript{}{2}{F_{1}}\left(\mu,1-\mu;|n|+1;\frac{z_{0}w_{0}}{z_{0}w_{0}-1}\right)\right)^{-1}\,.$

Note that asymptotically

$|b_{n}|\sim\left(|n|+1\right)^{1-{\sf Re}\,\mu}\qquad(|n|\to\infty)\,.$

This follows from Lemma 6.1 and Stirling’s formula (6.4). In view of (6.8)
this shows that the series

$F(z,w)=\sum\limits_{n=-\infty}^{\infty}b_{n}P_{n}^{\mu}(z,w)$

converges locally uniformly in $\mathbb{D}^{2}$. Hence
$F\in\mathcal{H}(\mathbb{D}^{2})$ and, since every function $P^{\mu}_{n}$ is
an eigenfunction of $\Delta_{zw}$, we have $\Delta_{zw}F=4\mu(\mu-1)F$ in
$\mathbb{D}^{2}$. In particular,

$f(z):=F(z,\overline{z})=\sum\limits_{n=-\infty}^{\infty}b_{n}P^{\mu}_{n}(z,\overline{z})$

is of class $C^{2}(\mathbb{D})$ and an eigenfunction of $\Delta_{\mathbb{D}}$.
By Theorem 2.2, $F\in\mathcal{H}(\mathbb{D}^{2})$ is the uniquely determined
function in $\mathcal{H}(\mathbb{D}^{2})$ such that $F(z,\overline{z})=f(z)$
in $\mathbb{D}$. Now assume $F\in\mathcal{H}(D)$, where $D$ is a rotationally
invariant domain such that $\mathbb{D}^{2}\subsetneq D\subseteq\Omega$ and
$(z_{0},w_{0})\in D\setminus\mathbb{D}^{2}$. By Theorem 5.1 there are
coefficients $(\tilde{b}_{n})_{n\in\mathbb{Z}}\subseteq\mathbb{C}$ such that

$F(z,w)=\sum\limits_{n=-\infty}^{\infty}\tilde{b}_{n}P_{n}^{\mu}(z,w)$

for all $(z,w)\in D$. Since the coefficients are uniquely determined, we
conclude $\tilde{b}_{n}=b_{n}$. This is a contradiction because

$F(z_{0},w_{0})=\sum\limits_{n=0}^{\infty}w_{0}^{n}+\sum\limits_{n=1}^{\infty}z_{0}^{n}$

is a divergent series since $|z_{0}|\geq 1$ or $|w_{0}|\geq 1$.

###### Corollary 6.4

Let $\lambda\in\mathbb{C}$ and $D$ a rotationally invariant subdomain of
$\Omega$ such that $\mathbb{D}^{2}\subsetneq D$. Then there exists a function
$f\in X_{\lambda}(\mathbb{D}^{2})$ that cannot be analytically continued to
$D$.

##  7\. Poisson Fourier modes and the Möbius group

Recall the Möbius group $\mathcal{M}(\mathbb{D}^{2})$ of the bidisk from
(2.5). In order to give a characterization of the Möbius invariant eigenspaces
of $\Delta_{zw}$ (e.g. proving Theorem 2.8), it will turn out that all we need
to understand are precompositions of PFM with automorphisms in
$\mathcal{M}(\mathbb{D}^{2})$, that is

$P^{\mu}_{n}\circ T\text{ with }{\sf Re}\,\mu\geq 1/2\text{ and
}T\in\mathcal{M}(\mathbb{D}^{2})\,.$

Moreover, since $\mathcal{M}(\mathbb{D}^{2})$ is closely related to the Möbius
group $\mathcal{M}$ of $\Omega$ from (2.4), we can use the following
description of $\mathcal{M}$.

###### Lemma 7.1

The group $\mathcal{M}$ is generated by the flip map
$\mathcal{F}(u,v)\coloneqq(1/v,1/u)$ and the mappings

(7.1)
$T_{z,w}(u,v)\coloneqq\left(\frac{z-u}{1-wu},\frac{w-v}{1-zv}\right)\quad\textrm{and}\quad\varrho_{\gamma}(u,v)\coloneqq\left(\gamma
u,\frac{1}{\gamma}v\right)\,,\qquad(u,v)\in\Omega$

with $(z,w)\in\Omega\cap\mathbb{C}^{2}$ and $\gamma\in\mathbb{C}^{*}$. More
precisely, for every $T\in\mathcal{M}$ there exist $z,w\in\mathbb{C}$ and
$\gamma\in\mathbb{C}^{*}$ such that

$T=\varrho_{\gamma}\circ T_{z,w}\quad\text{or}\quad T=\varrho_{\gamma}\circ
T_{z,w}\circ\mathcal{F}\,.$

######  Remark 9.

Lemma 7.1 is exactly Lemma 2.2 in [19] where we have replaced the
automorphisms $\Phi_{z,w}$ by the $T_{z,w}=\Phi_{z,w}\circ\varrho_{-1}$
automorphisms in (7.1). The reason will become apparent in (7.3) and (7.4).
Essentially, the automorphisms $T_{z,w}$ in (7.1) are precisely the
automorphisms of $\Omega$ interchanging a given point
$(z,w)\in\Omega\cap(\mathbb{C}\times\mathbb{C})$ with $(0,0)$ – instead of
only sending $(0,0)$ to $(z,w)$. See [32, Sec. 2.3] for more details on this.

Therefore, understanding precompositions of a PFM $P_{n}^{\mu}$ with elements
${T\in\mathcal{M}(\mathbb{D}^{2})}$ breaks down to understanding
precompositions with the above generators. Note that the mappings
$\mathcal{F}$ and $T_{z,w}$ for $w\neq\overline{{z}}$ are not elements of
$\mathcal{M}(\mathbb{D}^{2})$. However, the precompositions with these
mappings still make sense for those PFM defined on all of $\Omega$, that is,
in view of Theorem 4.2 the precompositions

$P^{-m}_{n}\circ T\text{ with }m\in\mathbb{N}_{0}\text{ and }T\in\mathcal{M}$

are well–defined. In the case that $T=\varrho_{\gamma}$ we have
${P_{n}^{\mu}\circ T=\gamma^{-n}P_{n}^{\mu}}$ by the $(-n)$–homogeneity of the
PFM for all $\mu\in\mathbb{C}$. Next, if $T=\mathcal{F}$, it is easily seen by
direct verification based on (4.17) that

$(P_{n}^{-m}\circ\mathcal{F})(z,w)=(-1)^{m}P_{n}^{-m}(z,w)\,$

for $m\in\mathbb{N}_{0}$ and $|n|\leq m$. In order to treat the case that
$T=T_{z,w}$ resp. $T=T_{z,\overline{{z}}}$, some preliminary observations are
useful. First, every automorphism $T_{z,w}$ is induced by a Möbius
transformation $\psi_{z,w}$ of the form

(7.2) $\psi_{z,w}(u):=\frac{z-u}{1-wu}\,,\qquad$

in the sense that $T_{z,w}(u,v)=(\psi_{z,w}(u),1/\psi_{z,w}(1/v))$.

######  Remark 10.

Choosing $w=\overline{z}$ in (7.2) yields all self–inverse automorphisms of
$\mathbb{D}$ except for the identity, and, similarly, the choice
$w=-\overline{z}\in\mathbb{D}$ yields all self–inverse rigid motions of
$\widehat{\mathbb{C}}$.

Further, we will employ the definition of the PFM via integrals from (4.10).
By Remark 5, when considering points $(z,\overline{{z}})\in\mathbb{D}^{2}$,
this definition coincides with the $n$–th Fourier mode of the $\mu$–power of
the Poisson kernel on the unit disk. The Poisson kernel on $\mathbb{D}$
satisfies the well–known properties

(7.3)
$P\left(z,\overline{{z}};\xi\right)=\frac{\psi_{z,\overline{{z}}}^{\prime}(\xi)}{\psi_{z,\overline{{z}}}(\xi)}\xi$

and

(7.4)
$P\left(\psi_{z,\overline{{z}}}(u),\overline{{\psi_{z,\overline{{z}}}(u)}};\psi_{z,\overline{{z}}}(\xi)\right)=\frac{P(u,\overline{{u}};\xi)}{P(z,\overline{{z}};\xi)}$

where $z,u\in\mathbb{D}$, $\xi\in\mathop{}\\!\partial\mathbb{D}$ and
$\psi_{z,\overline{{z}}}\in\mathcal{M}(\mathbb{D})$, see [38, Th. 3.3.5].

######  Remark 11.

One can define $P$ on $\Omega\times\partial\mathbb{D}$. Then $P$ is never
zero, and for each $\xi\in\partial\mathbb{D}$ the function $P(\cdot;\xi)$ is
meromorphic in the sense of [13, Chapter VI.2]. The identities (7.3) and (7.4)
then take the form

$\displaystyle P\left(\psi_{z,w}^{-1}(0),1/\psi_{z,w}^{-1}(\infty);\xi\right)$
$\displaystyle=\frac{\psi_{z,w}^{\prime}(\xi)}{\psi_{z,w}(\xi)}\xi=P\left(z,w;\xi\right)\,,$
$\displaystyle
P\left(\psi_{z,w}(u),\frac{1}{\psi_{z,w}\left(1/v\right)};\psi_{z,w}(\xi)\right)$
$\displaystyle=\frac{P(u,v;\xi)}{P\left(\psi_{z,w}^{-1}(0),1/\psi_{z,w}^{-1}(\infty);\xi\right)}=\frac{P(u,v;\xi)}{P\left(z,w;\xi\right)}\,,$

where $(z,w)\in\Omega\cap\mathbb{C}^{2}$, $(u,v)\in\Omega$ and
$\xi\in\partial\mathbb{D}$.

With these preparations we are now in a position to analyze the
precompositions of $P^{\mu}_{n}$ with elements of
$\mathcal{M}(\mathbb{D}^{2})$ resp. $\mathcal{M}$. We start with the simplest
case, $P^{m+1}_{0}\circ T_{u,v}$:

###### Proposition 7.2

Let $m\in\mathbb{N}_{0}$, $(z,w)\in\Omega$ and
$(u,v)\in\Omega\cap\mathbb{C}^{2}$. Then

(7.5) $\big{(}P_{0}^{m+1}\circ T_{u,v}\big{)}(z,w)=\big{(}P_{0}^{-m}\circ
T_{u,v}\big{)}(z,w)=\sum_{j=-m}^{m}P^{m+1}_{-j}\left(u,v\right)P^{-m}_{j}(z,w)\,.$

###### Proof.

The way of reasoning is as follows: first, fix $z,u\in\mathbb{D}$ and let
$v:=\overline{u}$. Then $\psi_{u,\overline{{u}}}\in\mathcal{M}(\mathbb{D})$
which means, in particular, that
$T_{u,\overline{{u}}}(z,\overline{{z}})\in\\{(t,\overline{{t}})\,:\,t\in\mathbb{D}\\}$.
By the two variable identity principle, Lemma 3.1, if we can show (7.5) in
this special case, then, keeping $u\in\mathbb{D}$ fixed, (7.5) also holds for
points $(z,w)\in\Omega$ since both sides of the equation are holomorphic
functions in $\Omega$. Next, fix $(z,w)\in\Omega$ and note that both sides of
(7.5) are holomorphic as functions of $(u,v)\in\Omega\cap\mathbb{C}^{2}$.
Assuming (7.5) for $v=\overline{{u}}\in\mathbb{D}$ then implies the claim,
again by Lemma 3.1.

It remains to show (7.5) for $w=\overline{{z}}$ and $v=\overline{{u}}$ with
$z,u\in\mathbb{D}$. For this purpose we compute

$\displaystyle\big{(}P_{0}^{m+1}\circ
T_{u,\overline{{u}}}\big{)}(z,\overline{{z}})$
$\displaystyle\overset{\eqref{eq:Poissonplusminus}}{=}\big{(}P_{0}^{-m}\circ
T_{u,\overline{{u}}}\big{)}(z,\overline{{z}})=\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\left(\psi_{u,\overline{{u}}}(z),\frac{1}{\psi_{u,\overline{{u}}}\left(1/\overline{{z}}\right)};e^{it}\right)^{-m}\,dt$
$\displaystyle\overset{\eqref{eq:PoissonkernelasMoebiustrafo}}{=}\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};\psi_{u,\overline{{u}}}^{-1}(e^{it})\big{)}^{-m}P\big{(}u,\overline{{u}};\psi_{u,\overline{{u}}}^{-1}(e^{it})\big{)}^{m}\,dt$
$\displaystyle\overset{\psi_{u,\overline{{u}}}(e^{is})=e^{it}}{=}\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};e^{is}\big{)}^{-m}P\big{(}u,\overline{{u}};e^{is}\big{)}^{m+1}\,ds\,.$

In the last step we used (7.3). Now we can interpret the resulting integral as
the inner product on $L^{2}([0,2\pi],\mathbb{C})$, the space of square
integrable functions $f:[0,2\pi]\to\mathbb{C}$. Using Parseval’s identity in
(P) we obtain

$\displaystyle\big{(}P_{0}^{-m}\circ
T_{u,\overline{{u}}}\big{)}(z,\overline{{z}})$ $\displaystyle=\left\langle
P\left(u,\overline{{u}};e^{is}\right)^{m+1},\overline{{P(z,\overline{{z}};e^{is})^{-m}}}\right\rangle_{L^{2}([0,2\pi],\mathbb{C})}$
$\displaystyle\overset{\mathrm{(P)}}{=}\sum_{j=-\infty}^{\infty}\left\langle
P\left(u,\overline{{u}};e^{it}\right)^{m+1},e^{-ijt}\right\rangle_{L^{2}([0,2\pi],\mathbb{C})}\left\langle
e^{-ijs},\overline{{P(z,\overline{{z}};e^{is})^{-m}}}\right\rangle_{L^{2}([0,2\pi],\mathbb{C})}$
$\displaystyle=\sum_{j=-\infty}^{\infty}\frac{1}{(2\pi)^{2}}\int\limits_{0}^{2\pi}P\left(u,\overline{{u}};e^{it}\right)^{m+1}e^{ijt}\,dt\int\limits_{0}^{2\pi}P(z,\overline{{z}};e^{is})^{-m}e^{-ijs}\,ds$
$\displaystyle=\sum_{j=-m}^{m}P_{-j}^{m+1}(u,\overline{{u}})P_{j}^{-m}(z,\overline{{z}})\,,$

where the series reduces to a finite sum because $P_{j}^{-m}$ vanishes for
$|j|>m$, see Remark 6(b). ∎

The previous proof can be modified to establish the following result.

###### Theorem 7.3

Let $\mu\in\mathbb{C}$, ${\sf Re}\,\mu\geq 1/2$, $n\in\mathbb{N}_{0}$ and
$u\in\mathbb{D}$. Then

(7.6a) $\displaystyle\big{(}P_{n}^{\mu}\circ T_{u,\overline{{u}}}\big{)}(z,w)$
$\displaystyle=\sum\limits_{j=-\infty}^{\infty}\left(\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}(-1)^{k-n-\ell}\binom{n}{k}\binom{-n}{\ell}u^{\ell}\overline{{u}}^{k}P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right)P^{\mu}_{-j}(z,w)$
(7.6b) $\displaystyle\big{(}P_{-n}^{\mu}\circ
T_{u,\overline{{u}}}\big{)}(z,w)$
$\displaystyle=\sum\limits_{j=-\infty}^{\infty}\left(\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}(-1)^{n-k+\ell}\binom{n}{k}\binom{-n}{\ell}u^{k}\overline{{u}}^{\ell}P^{1-\mu}_{j+k-n-\ell}(u,\overline{{u}})\right)P^{\mu}_{-j}(z,w)\,$

for all $(z,w)\in\mathbb{D}^{2}$. Moreover, both series converge locally
uniformly and absolutely w.r.t. $(z,w)$ in $\mathbb{D}^{2}$.

###### Proof.

Using (7.3) and (7.4) we compute

$\displaystyle\big{(}P_{n}^{\mu}\circ
T_{u,\overline{{u}}}\big{)}(z,\overline{{z}})$
$\displaystyle=\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};e^{is}\big{)}^{\mu}P\big{(}u,\overline{{u}};e^{is}\big{)}^{1-\mu}\psi_{u,\overline{{u}}}(e^{is})^{-n}\,ds$
$\displaystyle=\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};e^{is}\big{)}^{\mu}P\big{(}u,\overline{{u}};e^{is}\big{)}^{1-\mu}(1-\overline{{u}}e^{is})^{n}(u-e^{is})^{-n}\,ds\,.$

Note that taking complex powers is unproblematic since the appearing Poisson
kernels are positive real quantities. Applying Lemma 3.1 and, additionally,
the generalized binomial theorem leads to

$\displaystyle\big{(}P_{n}^{\mu}$ $\displaystyle\circ
T_{u,\overline{{u}}}\big{)}(z,w)$
$\displaystyle=\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}\binom{n}{k}\binom{-n}{\ell}(-1)^{k-n+\ell}\overline{{u}}^{k}u^{\ell}\frac{1}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,w;e^{is}\big{)}^{\mu}P\big{(}u,\overline{{u}};e^{is}\big{)}^{1-\mu}e^{is(k-n-\ell)}\,ds\,.$

Interpreting the above integral as a $L^{2}([0,2\pi],\mathbb{C})$ inner
product as it was done in the proof of Proposition 7.2 shows (7.6a). A similar
computation yields (7.6b).

Let $K\subseteq\mathbb{D}^{2}$ compact. It remains to prove that

$\displaystyle\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}\left|\binom{n}{k}\binom{-n}{\ell}\overline{u}^{k}u^{\ell}\right|\sum_{j=-\infty}^{\infty}\left|P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right|\sup_{(z,w)\in
K}\left|P^{\mu}_{-j}(z,w)\right|<\infty\,.$

We have

$\displaystyle\left|P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right|\leq\frac{1}{2\pi}\int\limits_{0}^{2\pi}\left|P(u,\overline{{u}};e^{it})^{1-\mu}\right|\,dt:=M_{u}<\infty$

as $u\in\mathbb{D}$ is fixed. Since $K$ is compact, we find a non–negative
real number $r<1$ such that $|z|,|w|\leq r$ for all $(z,w)\in K$. Hence, using
(6.8) we can ensure that

$\displaystyle\sum_{j=-\infty}^{\infty}\sup_{(z,w)\in
K}\left|P^{\mu}_{-j}(z,w)\right|$
$\displaystyle\leq\sum_{j=-\infty}^{\infty}\gamma_{r}\cdot\left(|j|+1\right)^{{\sf
Re}\,\mu-1}\,r^{|j|}:=N_{r,\mu}<\infty\,,$

where $\gamma_{r}>0$ depends on $r$ (and $\mu$). Thus,

$\displaystyle\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}$
$\displaystyle\left|\binom{n}{k}\binom{-n}{\ell}\overline{u}^{k}u^{\ell}\right|\sum_{j=-\infty}^{\infty}\left|P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right|\sup_{(z,w)\in
K}\left|P^{\mu}_{-j}(z,w)\right|$
$\displaystyle\leq\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}\left|\binom{n}{k}\binom{-n}{\ell}\overline{u}^{k}u^{\ell}\right|\cdot
M_{u}\cdot N_{r,\mu}<\infty$

since the binomial series over $k$ and $\ell$ converge absolutely. ∎

Let $m,n\in\mathbb{N}_{0}$, $n\leq m$. For $\mu=-m$ the $j$–series in (7.6)
terminate. Moreover in this case, when replacing $(u,\overline{{u}})$ by
$(u,v)\in\mathbb{D}^{2}$ in (7.6), a similar argument as in the proof of
Theorem 7.3 shows that the series in (7.6) are absolutely and locally
uniformly convergent with respect to $(u,v)$ in $\mathbb{D}^{2}$, too.
Further, we know that $P_{\pm n}^{-m}\circ T_{u,v}\in\mathcal{H}(\Omega)$.
Thus, both sides of (7.6a) resp. (7.6b) (with $(u,\overline{u})$ replaced by
$(u,v)$) define holomorphic functions w.r.t. $(u,v)\in\mathbb{D}^{2}$ which
allows us to apply the two variable identity principle, Lemma 3.1. We obtain:

###### Corollary 7.4

Let $m,n\in\mathbb{N}_{0}$, $n\leq m$, $(z,w)\in\Omega$ and
$(u,v)\in\mathbb{D}^{2}$. Then

$\displaystyle\big{(}P_{n}^{-m}$ $\displaystyle\circ
T_{u,v}\big{)}(z,w)=\sum_{j=-m}^{m}\left(\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}(-1)^{k-n-\ell}\binom{n}{k}\binom{-n}{\ell}u^{\ell}v^{k}P^{m+1}_{j-k+n+\ell}(u,v)\right)P^{-m}_{-j}(z,w)$
$\displaystyle\big{(}P_{-n}^{-m}$ $\displaystyle\circ
T_{u,v}\big{)}(z,w)=\sum\limits_{j=-m}^{m}\left(\sum_{k=0}^{n}\sum_{\ell=0}^{\infty}(-1)^{n-k+\ell}\binom{n}{k}\binom{-n}{\ell}u^{k}v^{\ell}P^{m+1}_{j+k-n-\ell}(u,\overline{{u}})\right)P^{-m}_{-j}(z,w)\,.$

In particular, $P^{-m}_{n}\circ
T_{u,v}\in\mathrm{span}\\{P^{-m}_{j}\,:\,j=-m,\ldots,m\\}\subseteq\mathcal{H}(\Omega)$.

##  8\. Proof of Theorems 2.5 and 2.8: invariant eigenspaces

Using the theory we have developed so far, we are now able to give an explicit
description of the eigenspaces of the Laplace operator $\Delta_{zw}$ on
$\mathbb{D}^{2}$ resp. on $\Omega$. This will allow us to conclude Theorems
2.5 and 2.8 afterwards.

######  Theorem 8.1 (Möbius invariant subspaces of
$X_{\lambda}(\mathbb{D}^{2})$)

Let $\lambda\in\mathbb{C}$.

  * (NE)

If $\lambda\neq 4m(m+1)$ for all $m\in\mathbb{N}_{0}$, then
$X_{\lambda}(\mathbb{D}^{2})$ has no non–trivial Möbius invariant subspaces.

  * (E)

If $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$, then
$X_{\lambda}(\mathbb{D}^{2})$ has precisely three non–trivial Möbius invariant
subspaces
$X_{\lambda}^{+}(\mathbb{D}^{2}),X_{\lambda}^{-}(\mathbb{D}^{2}),X_{\lambda}^{0}(\mathbb{D}^{2})$
of which exactly one, say $X_{\lambda}^{0}(\mathbb{D}^{2})$, is finite
dimensional of dimension $2m+1$. Explicitly, these spaces are given by

(8.1a) $\displaystyle X_{\lambda}^{+}(\mathbb{D}^{2})$
$\displaystyle=\operatorname{clos}_{\mathbb{D}^{2}}\left(\mathrm{span}\\{P_{n}^{m+1}\,:\,-m\leq
n<\infty\\}\right)\,,$ (8.1b) $\displaystyle X_{\lambda}^{-}(\mathbb{D}^{2})$
$\displaystyle=\operatorname{clos}_{\mathbb{D}^{2}}\left(\mathrm{span}\\{P_{n}^{m+1}\,:\,-\infty<n\leq
m\\}\right)\,,$ (8.1c) $\displaystyle X_{\lambda}^{0}(\mathbb{D}^{2})$
$\displaystyle=\mathrm{span}\\{P_{n}^{m+1}\,:\,-m\leq n\leq m\\}\,,$

where $\operatorname{clos}_{\mathbb{D}^{2}}$ denotes the closure with respect
to the topology of locally uniform convergence on $\mathbb{D}^{2}$.

Note that Rudin gives an explicit characterization of the three non–trivial
subspaces in the second case of his theorem (see Theorem 2.1), too. We will
discuss the similarities and the differences between Rudin’s and our approach
in Remark 13.

Three crucial ingredients that we will use for the proof of Theorem 8.1 are
the representation of eigenfunctions in terms of PFM from Theorem 5.1, the
pullback formula for PFM from Theorem 7.3 and the differential operators
$D^{+}$ and $D^{-}$ defined by

(8.2) $D^{+}f(z,w)\coloneqq\partial_{z}f(z,w)-w^{2}\partial_{w}f(z,w)\,,\qquad
D^{-}f(z,w)\coloneqq\partial_{w}f(z,w)-z^{2}\partial_{z}f(z,w)$

for $f\in\mathcal{H}(D)$ and any subdomain $D\subseteq\mathbb{C}^{2}$. We note
an alternative description of theses operators based on the automorphisms from
(7.1).

###### Lemma 8.2

Let $D\subseteq\mathbb{C}^{2}$ be a subdomain, $f\in\mathcal{H}(D)$. For all
$(z,w)\in D$ it then holds that

(8.3) $\begin{split}D^{+}f(z,w)=\partial_{u}\left(f\circ
T_{u,0}\circ{\varrho}_{-1}\right)\Big{|}_{u=0}(z,w)\,,\\\
D^{-}f(z,w)=\partial_{v}\left(f\circ
T_{0,v}\circ\varrho_{-1}\right)\Big{|}_{v=0}(z,w)\,.\end{split}$

What makes these operators useful is that they act by shifts of the Fourier
index on the PFM.

###### Lemma 8.3

The operators $D^{+}$ and $D^{-}$ commute with $\Delta_{zw}$. Furthermore, for
$\mu\in\mathbb{C}$ and $n\in\mathbb{Z}$ it holds that

$D^{+}P_{n}^{\mu}=(\mu-n-1)P_{n+1}^{\mu}\qquad\text{and}\qquad
D^{-}P_{n}^{\mu}=(\mu+n+1)P_{n-1}^{\mu}\,.$

###### Proof.

The operator $D^{+}$ commutes with $\Delta_{zw}$ in view of the latter’s
$\mathcal{M}$–invariance, (8.3) and commutativity of partial derivatives.
Moreover, $D^{+}$ lowers homogeneity in the sense of (4.1) by one degree.
Since $P_{n}^{\mu}$ is a $(-n)$–homogeneous eigenfunction of $\Delta_{zw}$ to
the eigenvalue ${4\mu(\mu-1)}$, this implies that $D^{+}P_{n}^{\mu}$ is a
${-(n+1)}$–homogeneous eigenfunction of $\Delta_{zw}$ to the same eigenvalue.
Hence, Theorem 4.2 yields ${D^{+}P_{n}^{\mu}=cP_{n+1}^{\mu}}$ for some
${c\in\mathbb{C}}$. Applying (8.2) and evaluating (4.16) in $(z,w)=(0,1)$
yields $c=\mu-n-1$. Note that one has to distinguish the cases $n\geq 0$ and
$n<0$. The second equality in (8.3) may be derived analogously. ∎

######  Remark 12.

One may thus regard the operators $D^{+}$ and $D^{-}$ as ladder operators and
together with their commutator

$\frac{1}{2}\big{(}D^{-}\circ D^{+}-D^{+}\circ
D^{-}\big{)}=z\partial_{z}-w\partial_{w}\,,$

which is the Euler vector field corresponding to the homogeneity (4.1), they
generate the Lie algebra of $\mathcal{M}$. By (8.3), all three operators may
moreover be regarded as fundamental vector fields of the natural action of
$\mathcal{M}$ as automorphisms of $\Omega$.

###### Proof of Theorem 8.1.

Write $\lambda=4\mu(\mu-1)$ for $\mu\in\mathbb{C}$, ${\sf Re}\,\mu\geq 1/2$,
and let $Y$ be a non–trivial Möbius invariant subspace of
$X_{\lambda}(\mathbb{D}^{2})$. The way of reasoning is as follows: we first
show that $Y$ contains a PFM $P_{n}^{\mu}$ for some $n\in\mathbb{Z}$. If the
assumption of (NE) holds, then this will imply that $P_{n}^{\mu}\in Y$ for all
$n\in\mathbb{Z}$. Conversely, if the assumption of (E) holds for
$m\in\mathbb{N}_{0}$, the existence of $P_{n}^{-m}$ for some $n\in\mathbb{Z}$
will imply that certain other PFM need to be contained in $Y$, too. Here, we
will need to distinguish three cases which will lead to the three specific
spaces described on the right–hand side of (8.1). This shows that there are at
most three possible choices for $Y$. Then, in a second step, we show that the
three spaces found in the first step are in fact Möbius invariant, which then
proves the existence of exactly three non–trivial proper Möbius invariant
subspaces.

  * Step 1:

    * (i)

Using (8.2) it follows from $Y$ being closed and Möbius invariant that the
differential quotients $D^{+}f,D^{-}f\in Y$. Therefore, $Y$ is invariant with
respect to $D^{+}$ and $D^{-}$, and, by iteration,
$(D^{+})^{k}f,(D^{-})^{k}f\in Y$ for every $k\in\mathbb{N}$.

    * (ii)

Since $Y$ is Möbius invariant, $Y$ is, in particular, rotationally invariant.
Thus, for every $n\in\mathbb{Z}$, we have $f_{n}\in Y$ where $f_{n}$ is
defined by

$f_{n}(z,w):=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}}f\left(\eta
z,\frac{w}{\eta}\right)\eta^{-(n+1)}\,d\eta\,.$

By Theorem 4.2, $f_{n}$ is a multiple of the PFM $P_{n}^{\mu}$. Moreover, by
Theorem 5.1, there are $c_{n}\in\mathbb{C}$ for all $n\in\mathbb{Z}$ depending
on $f$ such that $f$ has the representation

$f=\sum_{n=-\infty}^{\infty}c_{n}P_{n}^{\mu}\,.$

Since $Y\neq\\{0\\}$, we may assume $f\not\equiv 0$, i.e. $c_{n}\neq 0$ for
some $n\in\mathbb{Z}$. Thus, the above observations imply $P_{n}^{\mu}\in Y$
for some $n\in\mathbb{Z}$.

    * (iii)

By Part (ii) there exists $P_{n}^{\mu}\in Y$ for some $n\in\mathbb{Z}$, and by
Part (i) we conclude ${(D^{+})^{k}P_{n}^{\mu},(D^{-})^{k}P_{n}^{\mu}\in Y}$
for every $k\in\mathbb{N}$. Lemma 8.3 shows that these functions are multiples
of PFM again, so ${(\mu-n-k)_{k}P_{n+k}^{\mu}}$,
$(\mu+n-k)_{k}P_{n-k}^{\mu}\in Y$ for every $k\in\mathbb{N}$. Recall that the
prefactors denote (rising) Pochhammer symbols, see (4.5).

For $\mu\not\in\mathbb{Z}$ the factors $(\mu\pm n-k)_{k}$ never vanish. In
this case, $P_{n}^{\mu}\in Y$ for every $n\in\mathbb{Z}$. Using Theorem 5.1
again, this implies $Y=X_{\lambda}(\mathbb{D}^{2})$ which proves (NE).

For $\mu\in\mathbb{N}$ the factors $(\mu\pm n-k)_{k}$ vanish for appropriate
$k\in\mathbb{N}$. This leads to the dichotomy in case (E): assume
$\mu=m+1\in\mathbb{N}$, and $P_{n}^{m+1}\in Y$. Then necessarily
$P_{k}^{m+1}\in Y$ for $-m\leq k\leq m$. Since shifting $P_{k}^{m+1}$ with
$D^{\pm}$ eventually produces the zero function, $P_{q}^{m+1}$, $q<-m$ or
$q>m$, does not necessarily need to be contained in $Y$. However, if
$P_{q}^{m+1}\in Y$ for some $q<-m$, then necessarily all $P_{p}^{m+1}$,
$-\infty<p\leq m$, need to be contained in $Y$ since $P_{q}^{m+1}$ can be
shifted to each of these function without producing the zero function. The
same argument works for $q>m$. In summary, these considerations show that
every non–trivial closed Möbius invariant subspace of
$X_{\lambda}(\mathbb{D}^{2})$ contains one of the three spaces

$\displaystyle Y_{+}$
$\displaystyle:=\operatorname{clos}_{\mathbb{D}^{2}}\left(\mathrm{span}\\{P_{n}^{m+1}\,:\,-m\leq
n<\infty\\}\right)\,,$ $\displaystyle Y_{-}$
$\displaystyle:=\operatorname{clos}_{\mathbb{D}^{2}}\left(\mathrm{span}\\{P_{n}^{m+1}\,:\,-\infty<n\leq
m\\}\right)\,\text{ and}$ $\displaystyle Y_{0}\,$
$\displaystyle:=\mathrm{span}\\{P_{n}^{m+1}\,:\,-m\leq n\leq m\\}\,.$

  * Step 2:

It remains to show that $Y_{+},Y_{-}$ and $Y_{0}$ are in fact Möbius
invariant. Let $\mu-1=m\in\mathbb{N}_{0}$, $n\in\mathbb{Z}$ and
$z,w,u\in\mathbb{D}$.

    * (iv)

Assume $n\geq 0$. Theorem 7.3 shows that $(P_{n}^{m+1}\circ
T_{u,\overline{{u}}})(z,w)$ is a series of scalar multiples of
$P_{k}^{m+1}(z,w)$ with $-m\leq k<\infty$. To see this recall that by Remark 6
we have $P_{n}^{-m}\equiv 0$ if $|n|>m$. Moreover, Theorem 7.3 also implies
that this series converges locally uniformly with respect to $(z,w)$ on
$\mathbb{D}^{2}$. Thus, $P_{n}^{m+1}\circ T_{u,\overline{{u}}}\in Y_{+}$. The
same argument applies to $P_{-n}^{m+1}\circ T_{u,\overline{{u}}}$.

    * (v)

Part (iv) implies that every linear combination

$\sum_{n=-m}^{N}c_{n}P_{n}^{m+1}\circ T_{u,\overline{{u}}}\,$

belongs to $Y_{+}$. Since, by definition, every element in $Y_{+}$ is a series
of the form $\sum_{n=-m}^{\infty}c_{n}P^{m+1}_{n}$, Möbius invariance of
$Y_{+}$ follows. The same argument applies to $Y_{-}$.

    * (vi)

Now, consider $-m\leq n\leq m$. In this case $P_{n}^{m+1}$ is a multiple of
$P_{n}^{-m}$ by (4.18), and Corollary 7.4 shows that $(P_{n}^{m+1}\circ
T_{u,\overline{{u}}})(z,w)$ can be expressed as linear combination of
$P_{k}^{-m}(z,w)$ with $-m\leq k\leq m$. This shows Möbius invariance of
$Y_{0}$.∎

######  Remark 13.

  * (a)

In his proof of Theorem 2.1 Rudin employs exclusively a specific differential
operator $A$, see [37, Formula (5), p. 143], which in our terminology can be
defined by

$A:=D^{+}+D^{-}\,.$

The operator ${A^{\sharp}:=i\left(D^{+}-D^{-}\right)}$ is “conjugate” to $A$
in the sense that

$A-iA^{\sharp}=2D^{+}\quad\text{ and }\quad A+iA^{\sharp}=2D^{-}\,.$

Note the analogy with the Wirtinger derivatives $\partial_{z}$ and
$\partial_{\bar{z}}$ which satisfy

$\partial_{x}-i\partial_{y}=2\partial_{z}\quad\text{ and
}\quad\partial_{x}+i\partial_{y}=2\partial_{\bar{z}}\,.$

We note that in a related, but different context, namely in [36], Rudin also
considers the operators $D^{+}$ and $D^{-}$. He denotes them by $Q$ and
$\overline{Q}$, see [36, Section 4.1, Formula (1)].

  * (b)

Step 1 of our proof closely follows the argumentation of Rudin’s proof (Steps
1–3 in his labelling). However, because of the usage of his $A$–operator (see
Part (a)), which does not respect homogeneity when applied to a PFM, Rudin
additionally needs to consider the projections on $n$–homogeneous components
in his proof. In our approach based on the $D^{\pm}$–operators this issue does
not arise, since these operators lower resp. raise the degree of homogeneity
by exactly one.

  * (c)

Step 2 of our proof, where we made heavy use of Proposition 7.2 resp. Theorem
7.3, is different to Rudin’s approach (Step 4 in his labeling). In short,
Rudin makes a series expansion in the disk automorphism parameter and uses
again that Möbius invariant subspaces are invariant under repeated application
of $A$ (see Part (a)).

  * (d)

The proof of Theorem 8.1 heavily relies on the PFM. In fact, the usage of
$n$–homogeneous functions is also crucial in the original proof of Theorem 2.1
in [37], although Rudin does not explicitly define these functions but works
with projections on the subspaces of $n$–homogeneous eigenfunctions.

We are now in a position to prove Theorems 2.8 and 2.5.

###### Proof of Theorem 2.8.

Let $\lambda\in\mathbb{C}$ and $Y\subseteq\mathcal{H}(\mathbb{D}^{2})$ be a
non–trivial Möbius invariant subspace of $X_{\lambda}(\mathbb{D}^{2})$. Let
first $Y=X_{\lambda}(\mathbb{D}^{2})$ be the full eigenspace. Every PFM is
holomorphic at least on $\Omega_{*}$. By rotational invariance of
$\mathbb{D}^{2}$ and Theorem 5.1 this implies that
$Y\cap\mathcal{H}(\Omega_{*})$ is dense in $Y$, i.e. (NE) holds. Assume now
that $Y$ is a proper subspace of $X_{\lambda}(\mathbb{D}^{2})$. By Theorem
8.1, this is only possible if $\lambda$ is an exceptional eigenvalue.
Furthermore,

$Y\in\big{\\{}X_{\lambda}^{+}(\mathbb{D}^{2}),X_{\lambda}^{-}(\mathbb{D}^{2}),X_{\lambda}^{0}(\mathbb{D}^{2})\big{\\}}\,.$

Combining (8.1a), (8.1b), resp. (8.1c) with Corollary 5.2 yields
($\text{E}_{+}$ ), ($\text{E}_{-}$ ) resp. ($\text{E}_{0}$ ). We have already
discussed the additional statements in the latter case in Theorem 8.1.

Finally, Corollary 6.4 shows that none of the density statements
($\text{E}_{+}$ ), ($\text{E}_{-}$ ) and (NE) may be improved to holomorphic
extensibility of every element. ∎

###### Proof of Theorem 2.5.

The equivalence of (i), (ii) and (iii) was already established in Theorem 6.2.
Let $\lambda=4m(m+1)$ for some $m\in\mathbb{N}_{0}$. By (8.1c) we then have
$\dim(X_{\lambda}(\Omega))=2m+1$. Investing moreover Theorem 4.2 yields (a).
Remark 8 proves the inclusion “$\subseteq$” in (2.2). Conversely, restricting
an eigenfunction $F\in X_{\lambda}(\Omega)$ to the rotated diagonal
$\\{(z,-\overline{{z}})\colon z\in\widehat{\mathbb{C}}\\}$ yields an
eigenfunction of $\Delta_{\widehat{\mathbb{C}}}$. This completes the proof. ∎

The finite–dimensional spaces $X_{4m(m+1)}(\Omega)$ can also be characterized
in terms of the PFM $P_{0}^{-m}$ only.

###### Corollary 8.4

Let $m\in\mathbb{N}_{0}$. The function $P_{0}^{-m}$ is cyclic in
$X_{4m(m+1)}(\Omega)$ with respect to the natural action of $\mathcal{M}$ by
pullbacks. That is,

(8.4) $X_{4m(m+1)}(\Omega)=\operatorname{span}\big{\\{}P_{0}^{-m}\circ
T\;\big{|}\;T\in\mathcal{M}\big{\\}}\,.$

In addition,

(8.5) $X_{4m(m+1)}(\Omega)=\operatorname{span}\big{\\{}P_{0}^{-m}\circ
T_{u,\overline{{u}}}\;\big{|}\;u\in\mathbb{D}\big{\\}}=\operatorname{span}\big{\\{}P_{0}^{-m}\circ
T_{u,-\overline{{u}}}\;\big{|}\;u\in\mathbb{C}\big{\\}}\,.$

###### Proof.

We begin by showing the first equality in (8.5), which implies (8.4). Recall
that, by (8.1c), the set $\\{P^{-m}_{j}\colon-m\leq j\leq m\\}$ constitutes a
basis of $X_{4m(m+1)}(\Omega)$. By (7.5),

$P_{0}^{-m}\circ
T_{u,\overline{{u}}}=\sum_{j=-m}^{m}P^{m+1}_{-j}(u,\overline{{u}})P^{-m}_{j}$

for every $u\in\mathbb{D}$. We interpret the claim as a change of basis from
${\\{P^{-m}_{j}\;|\;-m\leq j\leq m\\}}$ to ${\\{P_{0}^{-m}\circ
T_{u_{k},\overline{{u}}_{k}}\;|\;-m\leq k\leq m\\}}$. Therefore, we have to
find suitable $u_{-m},\ldots,u_{m}\in\mathbb{D}$ such that the coefficient
matrix

$\big{(}P_{j}^{m+1}(u_{k},\overline{{u}}_{k})\big{)}_{-m\leq j,k\leq m}$

is invertible. However, it is true in general that given linearly independent
complex valued functions $F_{1},\dots,F_{M}$ for some $M\in\mathbb{N}$ on a
set containing at least $M$ elements, it is possible to find the same number
of points $x_{1},\dots,x_{M}$ in the same set such that the matrix
$(F_{j}(x_{k}))_{1\leq j,k\leq M}$ is invertible, see [15, Proof of Prop.
7.28]. The second equality in (8.5) follows similarly. ∎

##  9\. Poisson Fourier modes and Peschl–Minda operators

In [19] the classical Peschl–Minda differential operators, which were
introduced by Peschl [34] and studied e.g. by [2, 17, 24, 25, 31, 39, 40, 42],
have been extended to differential operators acting on holomorphic functions
defined on subdomains of $\Omega\cap\mathbb{C}^{2}$. It is the purpose of this
section to put these (generalized) Peschl–Minda operators into the context of
the present paper. In particular, we relate the Poisson Fourier modes with the
Peschl–Minda operators.

We first briefly recall the definition from [19]. Let $U$ be an open subset of
$\Omega\cap\mathbb{C}^{2}$ and ${f\in C^{\infty}(U)}$. The Peschl–Minda
derivative $D^{m,n}f$ at the point $(z,w)\in U$ is defined by

$D^{m,n}f(z,w):=\frac{\partial^{m+n}}{\partial u^{m}\partial
v^{n}}\left(f\circ{T}_{z,w}\circ\varrho_{-1}\right)(u,v)\bigg{|}_{(u,v)=(0,0)}\,.$

We write $D_{z}^{n}\coloneqq D^{n,0}$ and $D_{w}^{n}\coloneqq D^{0,n}$ and
refer to these operators as _pure_ Peschl–Minda operators. Comparing with
(8.3), we note that the roles of $(z,w)$ and $(u,v)$ have been swapped, which
yields the operators $D^{1}_{z}$ and $D^{1}_{w}$ instead of $D^{+}$ and
$D^{-}$, respectively.

Given a linear mapping $T\colon V\longrightarrow W$ we write
$\ker(T)\coloneqq\\{x\in V\colon Tx=0\\}$ for its kernel. We observe that the
pure Peschl–Minda operators reproduce the generalized Poisson kernel from
(4.9) in the following way:

###### Lemma 9.1

Let $\mu\in\mathbb{C}$, $(z,w)\in\mathbb{D}^{2}$, $\xi\in\partial\mathbb{D}$
and $k\in\mathbb{N}_{0}$. Then

(9.1a) $\displaystyle D^{k}_{z}P(z,w;\xi)^{\mu}$
$\displaystyle=(\mu)_{k}P(z,w;\xi)^{\mu}\left(\psi_{z,w}(\xi)\right)^{-k}$
(9.1b) $\displaystyle D^{k}_{w}P(z,w;\xi)^{\mu}$
$\displaystyle=(\mu)_{k}P(z,w;\xi)^{\mu}\left(\psi_{z,w}(\xi)\right)^{k}$

with $\psi_{z,w}$ from (7.2). In particular, if $\mu=-m\in(-\mathbb{N}_{0})$,
then

$P^{-m}\in\ker(D^{m+1}_{z})\cap\ker(D_{w}^{m+1})\,.$

###### Proof.

If $w=\overline{{z}}$, this is most easily proved by an induction on $k$ and
the help of Proposition 3.8 in [19]. Then, the general result follows from
Lemma 3.1. ∎

We note in passing that Lemma 9.1 for $w=\overline{{z}}$ says that up to
multiplication with a unimodular constant the (classical) Poisson kernel
$z\mapsto P(z,\overline{z};\xi)$ is a joint eigenfunction of the (classical)
Peschl–Minda operators studied e.g. in [24]. Remarkably, the Peschl–Minda
operators also act as weighted shifts when applied to the zeroth PFM.

###### Proposition 9.2

For $\mu\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$ we have

$(-\mu+1)_{n}\cdot
P_{n}^{\mu}=D^{n}_{z}P^{\mu}_{0}\quad\textrm{and}\quad(-\mu+1)_{n}\cdot
P_{-n}^{\mu}=D^{n}_{w}P^{\mu}_{0}\,.$

###### Proof.

Let $z\in\mathbb{D}$. We prove the claim for points $(z,\overline{{z}})$ first
from which then follows the result by the identity principle (Lemma 3.1). We
compute

$\displaystyle D^{n,0}P_{0}^{\mu}(z,\overline{{z}})$
$\displaystyle=\frac{1}{2\pi}\int\limits_{0}^{2\pi}D^{n,0}P\big{(}z,\overline{{z}};e^{it}\big{)}^{\mu}\,dt$
$\displaystyle\overset{\eqref{eq:DzderivativePa}}{=}\frac{1}{2\pi}\int\limits_{0}^{2\pi}(\mu)_{n}P\big{(}z,\overline{{z}};e^{it}\big{)}^{\mu}\left(\psi_{z,\overline{{z}}}(e^{it})\right)^{-n}\,dt$
$\displaystyle\overset{\eqref{eq:PoissonkernelasMoebiustrafo}}{=}\frac{(\mu)_{n}}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};\psi_{z,\overline{{z}}}(e^{it})\big{)}^{-\mu}\left(\psi_{z,\overline{{z}}}(e^{it})\right)^{-n}\,dt$
$\displaystyle\overset{\eqref{eq:MoebiusPoissonID}}{=}\frac{(\mu)_{n}}{2\pi}\int\limits_{0}^{2\pi}P\big{(}z,\overline{{z}};e^{it}\big{)}^{1-\mu}e^{-int}\,dt$
$\displaystyle=(\mu)_{n}P^{1-\mu}_{n}(z,\overline{{z}})\overset{\eqref{eq:Poissonplusminus}}{=}(-\mu+1)_{n}P^{\mu}_{n}(z,\overline{{z}})\,.$

The computation for $D^{0,n}$ and $P_{-n}^{\mu}$ is analogous. ∎

######  Remark 14.

Since Poisson Fourier modes are closely related to hypergeometric functions,
see (4.15), an alternative proof can be given by induction using Gauss’
contiguous identities, more precisely with Eq. 15.2.24 and with a combination
of Eq. 15.2.1 and 15.2.6 in [1].

In view of [19, Corollary 4.4], we find another connection between the PFM and
the pure Peschl–Minda derivatives. Let $\lambda=4m(m+1)$,
$m,n\in\mathbb{N}_{0}$. It is true that

$P_{n}^{-m}\in\ker(D_{w}^{k})\qquad\text{and}\qquad
P_{-n}^{-m}\in\ker(D_{z}^{k})\qquad\text{for all }k>m.$

###### Proposition 9.3

Let $M\in\mathbb{N}_{0}$. Then

(9.2) $\\{P_{n}^{-m}\,:\,m\in\mathbb{N}_{0},\,n\in\mathbb{Z},\,m\leq
M,\,|n|\leq m\\}=\ker(D^{M+1}_{z})\cap\ker(D^{M+1}_{w})\,.$

###### Proof.

Denote the left–hand side of (9.2) by $X_{M}$. Note that
$\ker(D^{M+1}_{z})\cap\ker(D^{M+1}_{w})$ and $X_{M}$ are finite dimensional
spaces. Moreover, $X_{M}$ is the direct sum

$X_{M}=\bigoplus_{m=0}^{M}X_{4m(m+1)}^{0}(\mathbb{D}^{2})$

and $\dim X_{4m(m+1)}^{0}(\mathbb{D}^{2})=2m+1$ by Theorem 8.1. Therefore, we
compute on the one hand $\dim X_{M}=\sum_{m=0}^{M}(2m+1)=(M+1)^{2}$. On the
other hand, by [19, Corollary 4.4], we have

(9.3)
$\ker(D^{M+1}_{z})\cap\ker(D^{M+1}_{w})=\mathrm{span}\bigg{\\{}\frac{z^{j}w^{k}}{(1-zw)^{M}}\colon
0\leq j,k\leq M\bigg{\\}}\,.$

Consequently, $\dim(\ker(D^{M+1}_{z})\cap\ker(D^{M+1}_{w}))=(M+1)^{2}$.
Moreover, combining (4.17) with (9.3) yields
$X_{M}\subseteq(\ker(D^{M}_{z})\cap\ker(D^{M}_{w}))$. This proves (9.2) by
linear algebra. ∎

######  Remark 15.

Note that both the PFM $P_{n}^{-m}$ and the generators from (9.3) may be
understood as holomorphic functions on all of $\Omega$. It thus makes sense to
speak of

$\bigoplus_{M\in\mathbb{N}_{0}}\big{(}\ker(D_{z}^{M+1})\cap\ker(D_{w}^{M+1})\big{)}$

as the subspace of $\mathcal{H}(\Omega)$ consisting of all finite linear
combinations of elements in the kernels of the Peschl–Minda differential
operators. We can even go one step further than Proposition 9.3 and conclude
that as sets

$\displaystyle\operatorname{clos}_{\Omega}\left(\mathrm{span}\\{P_{n}^{-m}\,:\,m\in\mathbb{N}_{0},\,n\in\mathbb{Z},\,|n|\leq
m\\}\right)$
$\displaystyle=\operatorname{clos}_{\Omega}\left(\bigoplus_{M\in\mathbb{N}_{0}}\big{(}\ker(D_{z}^{M+1})\cap\ker(D_{w}^{M+1})\big{)}\right)$
$\displaystyle=\mathcal{H}(\Omega)\,,$

where the last equality follows from a combination of (9.3) with [18, (4.1)
and Cor. 4.8]. This observation leads to the question whether the PFM form a
Schauder basis of $\mathcal{H}(\Omega)$. The second author answers this
question affirmatively in [32]. Perhaps similar results hold for
$\mathcal{H}(\Omega_{\pm})$ and the corresponding Poisson Fourier modes from
Theorem 4.2.

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