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 [Reference Rudin37]. 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,

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

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

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

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 $\psi \in \mathcal {M}({\mathbb D})$ .Footnote ² It is called nontrivial if $Y\not =\{0\}$ and $Y\not =X$ .

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 analog of Theorem 2.1 with $\Delta _{{\mathbb D}}$ replaced by the differential operator $\Delta _{zw}$ (see Theorem 2.6). 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) $$ \begin{align} \Delta_{zw} F=\lambda F \, \end{align} $$

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 2.5) 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 to 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

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

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 $\Delta _{{\mathbb D}} f=\lambda f$ on ${\mathbb D}$ . 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

$$ \begin{align*}X_{\lambda}({\mathbb D}) \to X_{\lambda}({\mathbb D}^2)\end{align*} $$

is continuous.

Theorem 2.2 gives rise to the following definition.

Definition 2.4 Let $\lambda \in {\mathbb D}$ . Then the continuous bijective linear mapping

$$ \begin{align*}\mathcal{R}_h : X_{\lambda}({\mathbb D}^2) \to X_{\lambda}({\mathbb D}) , \qquad \mathcal{R}_h(F)(z):=F(z,\overline{z}) \quad (z \in {\mathbb D}) \,\end{align*} $$

is called the hyperbolic restriction map. Its continuous inverse

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

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. [Reference Fritzsche and Grauert14, p. 97]).

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 [Reference Heins, Moucha and Roth18]). In particular, the largest possible maximal domain of existence of any eigenfunction of $\Delta _{zw}$ in $\mathcal {H}({\mathbb D}^2)$ is $\Omega $ . Note, this also means that the maximal domain of existence for the constant eigenfunctions is $\Omega $ .

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 nontrivial 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 $\Delta _{\widehat {{\mathbb C}}}$ .

With Theorem 2.6, we have reached two of our goals, a conceptual characterization of exceptional eigenvalues and the finite-dimensional nontrivial Möbius invariant subspaces of the eigenspaces $X_{\lambda }({\mathbb D})$ of the hyperbolic Laplacian $\Delta _{\mathbb D}$ . Next, we address the infinite-dimensional nontrivial 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 2.6, 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) $$ \begin{align} \Delta_{zw} \left( F \circ T \right)=\left(\Delta_{zw} F\right) \circ T \qquad \text{ for all } F \in \mathcal{H}(\Omega) \, \end{align} $$

holds for all $T \in {\textsf {Aut}}(\Omega )$ . The reason is simply that the automorphism group ${\textsf {Aut}}(\Omega )$ is much too large (see [Reference Heins, Moucha and Roth18]). However, $\Delta _{zw}$ is invariant under the subgroup $\mathcal {M}$ of ${\textsf {Aut}}(\Omega )$ defined by

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

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 $T \in {\textsf {Aut}}(\Omega )$ for which the invariance property (2.3) holds does necessarily belong to the subgroup $\mathcal {M}$ , see [Reference Heins, Moucha and Roth18, 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 $\mathcal {M}({\mathbb D})$ ,Footnote ³ 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

$$ \begin{align*}\mathcal{M}({\mathbb D}^2):={\textsf{Aut}}({\mathbb D}^2) \cap \mathcal{M} ,\end{align*} $$

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

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.

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 2.8, the Möbius invariant subspaces of $X_{\lambda }({\mathbb D})$ . The following subdomains of $\Omega $ play the essential role for this purpose:

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

$$ \begin{align*}\Omega=\Omega_+ \cup \Omega_- \cup \{ (\infty,\infty)\}\end{align*} $$

and

$$ \begin{align*}\mathcal{H}(\Omega)=\mathcal{H}(\Omega_+) \cap \mathcal{H}(\Omega_-) .\end{align*} $$

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 [Reference Heins, Moucha and Roth18]) and also for studying invariant differential operators of Peschl–Minda type acting on $\mathcal {H}(\Omega )$ (see [Reference Heins, Moucha, Roth and Sugawa19]). 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.

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

Remark 2.11 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 nontrivial infinite-dimensional Möbius invariant subspace Y of $X_{\lambda }({\mathbb D}^2)$ exactly one of the triples

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

is a Runge triple. Theorem 2.10 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 Sections 4 and 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 [Reference Berenstein and Gay8, 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 [Reference Berenstein and Gay8], 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, Theorems 2.6 and 2.10. By and large, we follow Rudin’s [Reference Rudin37] 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 8.5. We close the paper with Section 9, which connects the Poisson–Fourier modes to the invariant differential operators of Peschl–Minda type studied in [Reference Heins, Moucha, Roth and Sugawa19].

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 [Reference Bauer4–Reference Bauer and Ruscheweyh6] 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 [Reference Rudin37] 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 $n=1$ . Third, 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 [Reference Helgason20–Reference Helgason22] 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 [Reference Kashiwara, Kowata, Minemura, Okamoto, Oshima and Tanaka23, Reference Koryani26], who generalized the theory to higher-dimensional symmetric spaces, and Maass [Reference Maass29], 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 ${\mathbb D}\times {\mathbb D}$ by ${\mathbb D}^2$ 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 for the set of positive integers, 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

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

and the flip chart

$$ \begin{align*}\Omega \cap \big( \widehat{{\mathbb C}} \setminus \{0\} \times \widehat{{\mathbb C}} \setminus \{0\} \big) \to {\mathbb C}^2 , \qquad (u,v) \mapsto (1/v,1/u) .\end{align*} $$

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

$$ \begin{align*}\Delta_{zw} f(z,w)=4 (1-zw)^2 \partial_z \partial_w f(z,w) ,\end{align*} $$

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 [Reference Range35]))

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

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

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 [Reference Rudin37] 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) $$ \begin{align} f_n(\eta z,w/\eta)=\eta^n f_n(z,w) \qquad \text{for all } \eta \in \partial {\mathbb D} \end{align} $$

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) $$ \begin{align} f_n(z,w)=y_n(zw) z^n \quad \text{ if } n \ge 0 \quad \text{ and } \quad f_n(z,w)=y_n(zw) w^{|n|} \quad \text{ if } n \le 0 \, \end{align} $$

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) $$ \begin{align} 4 \left(1-t\right)^2 \left[ t y_n"(t)+(|n|+1) y_n'(t) \right]=\lambda y_n(t) \end{align} $$

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) $$ \begin{align} t (1-t) \hat{y}_n"+\big[c-(a+b+1) t\big] \hat{y}_n'-a b \hat{y}_n=0 \end{align} $$

with

$$ \begin{align*}a=\mu, \, b=\mu+|n|, \, c=|n|+1 .\end{align*} $$

It is well-known (see [Reference Olver, Olde Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain33, Section 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

$$ \begin{align*} {}_2F_1\left(a,b;c;t\right):=\sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_k}\frac{t^k}{k!},\quad |t|<1, \end{align*} $$

where

(4.5) $$ \begin{align} (\alpha)_k:=\prod\limits_{j=0}^{k-1}(\alpha+j), \, \qquad \alpha\in{\mathbb C}, \, k\in{\mathbb N}_0 \end{align} $$

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

(4.6) $$ \begin{align} y_{n}(t)=(1-t)^\mu{}_2F_1\left(\mu,\mu+|n|;|n|+1;t\right) \end{align} $$

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 4.1 Note that if $\mu \in {\mathbb C}$ 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 [Reference Abramowitz, Stegun, Danos and Rafelski1, Equation (15.3.3)])

$$ \begin{align*} {}_2F_1\left(a,b;c;t\right) &= (1-t)^{c-a-b} {}_2F_1\left(c-a,c-b;c;t\right) . \end{align*} $$

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

By a standard fact about hypergeometric functions (see [Reference Abramowitz, Stegun, Danos and Rafelski1, 15.3.1]), the function $y_n$ has a holomorphic extension at least to the slit plane

$$ \begin{align*}{\mathbb C} \setminus\, [1,\infty).\end{align*} $$

Returning to (4.2) with this choice of $y_n$ , we therefore see that

(4.7) $$ \begin{align} F^{\mu}_n(z,w):= (1-zw)^\mu {}_2F_1\left(\mu,\mu+|n|;|n|+1;zw\right) \cdot \begin{cases} z^n, & \text{ if } n \ge 0 , \\ w^{|n|}, & \text{ if } n \le 0 ,\end{cases} \end{align} $$

is holomorphic at least on the domain

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

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 ${}_2F_1\left (\mu ,\mu +|n|;|n|+1;zw\right )$ (see [Reference Erdelyi11, Section 2.5.1, Formula (10), p. 81]) shows that for every $n \in {\mathbb Z}$ ,

(4.8) $$ \begin{align} \binom{\mu+|n|-1}{|n|} F^\mu_n(z,w)=\frac{(1-z w)^{\mu}}{2\pi} \int \limits_{0}^{2\pi} \frac{e^{in t}}{\left( 1-z e^{-it} \right)^{\mu} \left(1-w e^{it} \right)^{\mu}} \, dt , \end{align} $$

with the generalized binomial coefficients $ \binom {\mu }{n} := \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 function

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

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

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

(4.11) $$ \begin{align} P^\mu_n=(-1)^n \binom{-\mu}{|n|} F^\mu_{-n} =\binom{\mu+|n|-1}{|n|} F^\mu_{-n}. \end{align} $$

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 domain $D \subseteq \Omega $ rotationally invariant if ${(\eta z, w/\eta ) \in D}$ for all $(z,w) \in D$ and all $\eta \in \partial {\mathbb D}$ . Summarizing our considerations leads to the following complete description of the n-homogeneous eigenfunctions of $\Delta _{zw}$ .

Theorem 4.4 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 $\textsf {Re}\, \mu \ge 1/2$ . Then there is a constant $c \in {\mathbb C}$ such that

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

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

1. (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.

2. (E) (Exceptional case)

   If $\lambda =4m(m+1)$ for some $m \in {\mathbb N}_0$ , then the maximal domain of existence of f is

   $$ \begin{align*}\begin{array}{lcl} \Omega_+ & & n < -m\,; \\[2mm] \Omega_- & \text{ if }\quad & n>m \,; \\[2mm] \Omega & & \! |n|\le m. \end{array}\end{align*} $$

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 2.5). Then $F^\mu _n$ has a holomorphic extension to some point $(z_0,w_0) \in \Omega $ such that $z_0w_0 \in {\mathbb R} \cup \{\infty \}$ and $z_0w_0> 1$ . We first consider the case $n \ge 0$ . By definition of $F^\mu _n$ , see (4.7), and in view of [Reference Abramowitz, Stegun, Danos and Rafelski1, Eq. 15.3.4], we have

(4.12) $$ \begin{align} \nonumber F^{\mu}_n(z,w) &= (1-zw)^{\mu}{}_2F_1\left(\mu,\mu+n;|n|+1;zw\right)z^n \\ &= {}_2F_1\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right) z^n, \end{align} $$

from which we infer that

$$ \begin{align*}t \mapsto {}_2F_1\left(\mu,1-\mu;|n|+1;t\right)\end{align*} $$

is holomorphic in a neighborhood of the point $x_0:=z_0w_0/(z_0w_0-1) \ge 1 $ . However, see [Reference Olver, Olde Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain33, 15.2.3],

(4.13) $$ \begin{align} \nonumber\lim \limits_{y \to 0^+} & \big[ {}_2F_1\left(\mu,1-\mu;|n|+1;x+i y\right) -{}_2F_1\left(\mu,1-\mu;|n|+1;x-i y\right)\big]\\ & \quad =\frac{2\pi i}{\Gamma(\mu) \Gamma(1-\mu)} \left(x-1 \right)^{|n|} {}_2F_1\left(|n|+1-\mu,|n|+\mu;|n|+1;1-x\right) \end{align} $$

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 $n \le 0$ follows from the case $n \ge 0$ and $F^\mu _{-n}(z,w)=F^{\mu }_n(w,z)$ .

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

$$ \begin{align*}F^\mu_n(z,w)=\frac{G^\mu_n(zw)}{(1-zw)^{\mu-1}} z^{n} ,\end{align*} $$

where

(4.14) $$ \begin{align} \nonumber G^{\mu}_n(zw)&:= (1-zw)^{2 \mu-1} {}_2F_1\left(\mu,\mu+|n|;|n|+1;zw\right)\\ &\hspace{0.11cm}={}_2F_1\left(1-\mu,|n|+1-\mu;|n|+1;zw\right) \end{align} $$

is a polynomial in $zw$ of degree $\mu -n-1 \ge 0$ (see [Reference Abramowitz, Stegun, Danos and Rafelski1, 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 $1-\mu \le n \le 0$ .

(iii) Let $\mu \in {\mathbb N}$ and $|n| \ge \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 \ge \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)

$$ \begin{align*} \lim \limits_{z \to \infty} \frac{G^\mu_n(zw_0)}{(1-zw_0)^{\mu-1}} & =\lim \limits_{z \to \infty} {}_2F_1\left(\mu,1-\mu;|n|+1;\frac{zw_0}{z w_0-1}\right)={}_2F_1\left(\mu,1-\mu;|n|+1;1\right)\\ &= \frac{\Gamma(n+1) \Gamma(n)}{\Gamma(n+1-\mu) \Gamma(n+\mu)} \in (0,\infty), \end{align*} $$

by [Reference Abramowitz, Stegun, Danos and Rafelski1, Eq. 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 $F^\mu _n$ .

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

Remark 4.7 (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 [Reference Helgason21]. The group $\mathcal {M}({\mathbb D})$ is isomorphic to the projective split unitary group

$$ \begin{align*}\operatorname{PSU}(1,1) = \bigg\{ {\scriptsize \begin{pmatrix} a & b \\ \overline{{b}} & \overline{{a}} \end{pmatrix}} \colon |a|^2-|b|^2=1 \bigg\} \Big/ \Big\{ \pm {\scriptsize \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} \Big\}\end{align*} $$

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)

which are known as representative functions or matrix elements. For a systematic discussion, we refer to the textbook [Reference Bröcker and Dieck9, Chapter III]. The corresponding eigenvalues may be computed from representation-theoretic data (see, e.g., [Reference Hall16, Proposition 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 [Reference Helgason20, Theorem 11]). Adapting [Reference Hall16, Example 4.10], one may parameterize the irreducible representations of $\operatorname {PSU}(1,1)$ as follows: let $m \in {\mathbb N}_0$ and

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

$$ \begin{align*} \pi_m \colon \operatorname{SU}(1,1) \longrightarrow \operatorname{GL}(V_m), \quad \pi_m {\scriptsize \begin{pmatrix} a & b \\ \overline{{b}} & \overline{{a}} \end{pmatrix}} p (z,w) := p \big( \overline{{a}}z - bw, -\overline{{b}}z + aw \big) \end{align*} $$

for $m \in {\mathbb N}$ constitutes a complete list of the irreducible finite-dimensional representations of $\operatorname {SU}(1,1)$ (see also [Reference Hall16, Proposition 4.11 and Section 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, parameterizing 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 $1 \le d \le 2m+1$ 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 these considerations only recover the finite-dimensional invariant eigenspace, i.e., the case $(E_0)$ in Theorem 2.10. It would be interesting to study whether this approach generalizes to the other invariant subspaces by incorporating suitable 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 $\textsf {Re}\, \mu \ge 1/2$ . Then there are uniquely determined coefficients $c_n \in {\mathbb C}$ such that

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

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

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

Since D is rotationally invariant, $f_n$ is well-defined. Clearly, $f_n$ is holomorphic on D and n-homogeneous. By Theorem 4.4, there are complex numbers $c_n \in {\mathbb C}$ such that

$$ \begin{align*}f_n(z,w)=c_n P^{\mu}_{-n}(z,w) .\end{align*} $$

(ii) We fix $(z,w) \in D$ and 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

$$ \begin{align*}f\left(\eta z,\frac{w}{\eta} \right)=\sum \limits_{n=-\infty}^{\infty} f_n(z,w) \eta^n ,\end{align*} $$

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

$$ \begin{align*}f\left(z,w \right)=\sum \limits_{n=-\infty}^{\infty} f_n(z,w) , \qquad (z,w) \in D .\end{align*} $$

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 ${K_1:=\{(\eta z,w/\eta ) \, : \, (z,w) \in K, \, r_1 \le |\eta | \le R_1\}}$ is a compact subset of D, and we let ${M_1:=\max \{ |f(z,w)| \, : (z,w) \in K_1\}}$ . Note that

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

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

$$ \begin{align*}|f_n(z,w)| \le M_1 \cdot \rho^{-n} \, \quad \text{ for all } (z,w) \in K \text{ and all } \rho \in [r_1,R_1] .\end{align*} $$

In particular, $|f_n(z,w)| \le M_1 r_1^{-n}$ for all $n < 0$ and $|f_n(z,w)| \le M_1 R_1^{-n}$ for all $n \ge 0$ , and this ensures the absolute and uniform convergence of the two series

$$ \begin{align*}\sum \limits_{n=-\infty}^{-1} f_n(z,w) \quad \text{ and } \quad \sum \limits_{n=0}^{\infty} f_n(z,w)\end{align*} $$

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.

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

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

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

$$ \begin{align*}f_n(z,w)=c_n P^{\mu}_{-n}(z,w) \,\end{align*} $$

with $\mu \in {\mathbb C}$ such that $\lambda =4\mu (\mu -1)$ and $\textsf {Re}\,\mu \ge 1/2$ . Hence, $\lambda = 4 m (m+1)$ for some $m \in {\mathbb N}_0$ since otherwise $P^{\mu }_{-n}$ is not holomorphic on $\Omega $ by Theorem 4.4. If $\lambda =4m(m+1)$ , then for each $|n|> m$ , the function $P^{m+1}_{-n}$ is not holomorphic on $\Omega $ again by Theorem 4.4, 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 [Reference Berenstein and Gay8, 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 [Reference Berenstein and Gay8]. We begin with the following lemma which is crucial for our approach.

Lemma 6.1 Let $\mu \in {\mathbb C}$ with $\textsf {Re}\, \mu>0$ . Then

(6.1) $$ \begin{align} \lim \limits_{n \to \infty} {}_2F_1\left(\mu,1-\mu;n+1;\omega\right) = 1 \end{align} $$

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

Proof Fix $m \in {\mathbb N}_0$ such that $m> \textsf {Re}\, \mu -2$ , and let $n \in {\mathbb N}_0$ . Then by [Reference Erdelyi11, Formula (11), p. 76] or [Reference MacRobert30, p. 84] there are complex numbers $A_k$ , $k=1, \ldots , m$ , such that

$$ \begin{align*} {}_2F_1\left(\mu,1-\mu;n+1;\omega\right) = 1+\sum \limits_{k=1}^m \frac{A_k}{\left( n+1 \right) \ldots \left(n+k \right)} \frac{\omega^k}{k!}+\rho_{m+1}(n,\omega) , \end{align*} $$

where

(6.2) $$ \begin{align} \rho_{m+1}(n,\omega):=\tilde{\gamma}_n \int \limits_{0}^1 \int \limits_0^1 t^{1-\mu+m} (1-t)^{n+\mu-1} \left( 1- s t \omega\right)^{-\mu-1-m} (1-s)^m \, ds \, dt \, \cdot \, \omega^{m+1} \end{align} $$

and

$$ \begin{align*} \tilde{\gamma}_n:=\frac{\Gamma(n+1) \Gamma(\mu+m)}{\Gamma(1-\mu) \Gamma(\mu+n) \Gamma(\mu) \, m!} . \end{align*} $$

Note that our choice of the nonnegative 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,

$$ \begin{align*}M_K:=\min \limits_{\omega \in K,0 \le s,t \le 1} |1-st \omega|>0 ,\end{align*} $$

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

(6.3) $$ \begin{align} \nonumber |\rho_{m+1}(n,\omega)| & \le |\tilde{\gamma}_{n}| \int \limits_{0}^1 t^{1-\textsf{Re}\,\mu+m} (1-t)^{n+\textsf{Re}\,\mu-1} \, dt \frac{|\omega|^{m+1}}{M_K^{m+1+\textsf{Re}\,\mu}} \\ & \nonumber = |\tilde{\gamma}_n| \frac{\Gamma(2+m-\textsf{Re}\,\mu) \, \Gamma(n+\textsf{Re}\,\mu)}{\Gamma(2+m+n)} \frac{|\omega|^{m+1}}{M_K^{m+1+\textsf{Re}\,\mu}} \\ &= \gamma^* \frac{\Gamma(n+\textsf{Re}\,\mu)}{\Gamma(n+1)} \frac{\Gamma(n+1)}{|\Gamma(n+\mu)|} \frac{\Gamma(n+1)}{\Gamma(2+m+n)} \frac{|\omega|^{m+1}}{M_K^{m+1+\textsf{Re}\,\mu}}, \end{align} $$

where

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

An application of Stirling’s formula

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

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

$$ \begin{align*}|\rho_{m+1}(n,\omega)| \le \gamma \cdot \left(n+1 \right)^{\textsf{Re}\,\mu-1} \left(n+1 \right)^{1-\textsf{Re}\,\mu} \left(n+1 \right)^{-m-1} |\omega|^{m+1} =\gamma \cdot \left( \frac{|\omega|}{n+1} \right)^{m+1} .\end{align*} $$

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 of 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 $\textsf {Re}\,\mu \ge 1/2$ . By [Reference Berenstein and Gay8, Theorem 16.18], there are uniquely determined coefficients $c_n \in {\mathbb C}$ such that

$$ \begin{align*}f(z)=\sum \limits_{n=-\infty}^{\infty} c_n P^{\mu}_n(z,\overline{z}) \, ;\end{align*} $$

the series converges absolutely and pointwise for each $z \in {\mathbb D}$ . As it is shown in the proof of [Reference Berenstein and Gay8, Theorem 1.6.18], the coefficients $c_n$ do have the additional property that

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

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

(6.6) $$ \begin{align} F(z,w):=\sum \limits_{n=-\infty}^{\infty} c_n P^{\mu}_n(z,w) \end{align} $$

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) $$ \begin{align} \nonumber \left| P^{\mu}_n(z,w) \right| & = \left| \binom{\mu}{|n|} \right| \, \left|F^{\mu}_{-n}(z,w) \right| \\ &\le \frac{|\Gamma(\mu +|n|)|}{|\Gamma(\mu)| \, \Gamma(|n|+1)} \left| {}_2F_1\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right) \right|\, r^{|n|} \end{align} $$

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

$$ \begin{align*}\xi \mapsto \frac{\xi}{\xi-1} \end{align*} $$

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

$$ \begin{align*}K:=\left\{ \frac{zw}{zw-1} \, : \, |z| \le r, \, |w| \le r \right\}\end{align*} $$

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

$$ \begin{align*}{}_2F_1\left(\mu,1-\mu;|n|+1;\frac{zw}{zw-1}\right) \to 1 \quad \text{ uniformly for } |z|\le r, \, |w| \le r\,\end{align*} $$

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) $$ \begin{align} \left| P^{\mu}_n(z,w)\right| \le \gamma \cdot \left(|n|+1 \right)^{\textsf{Re}\,\mu-1} \, r^{|n|} \end{align} $$

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

$$ \begin{align*}\mathcal{R}_h : X_{\lambda}({\mathbb D}^2) \to X_{\lambda}({\mathbb D}) , \qquad \mathcal{R}_h(F)(z):=F(z,\overline{z}) \quad (z \in {\mathbb D}) \,\end{align*} $$

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 [Reference Rudin38, 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 is a special case of Theorem 2.6.

Proof The implication (ii) $\Rightarrow $ (iii) is Corollary 5.2, and (iii) $\Rightarrow $ (ii) is Theorem 4.4. 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 $\textsf {Re}\,\mu \ge 1/2$ . For $n \in {\mathbb Z}$ , consider

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

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 4.4, we see that there is a constant $c_n \in {\mathbb C}$ such that

$$ \begin{align*}f_n(z)=c_n \binom{-\mu}{|n|} {}_2F_1\left(\mu,1-\mu;|n|+1;\frac{|z|^2}{1+|z|^2}\right) \cdot \begin{cases} \overline{z}^{|n|}, & \text{ if } n \ge 0, \\ (-z)^{|n|}, & \text{ if } n \le 0.\end{cases}\end{align*} $$

Therefore, the behavior of $f_n$ as $z \to \infty $ depends essentially on the value ${}_2F_1\left (\mu ;1-\mu ;|n|+1\right ){1}$ . It is well-known, see [Reference Andrews, Askey and Roy3, Theorem 2.1.3], that

$$ \begin{align*}\lim \limits_{x \to 1-} \frac{{}_2F_1\left(\mu,1-\mu;|n|+1;1\right)}{-\log (1-x)}=\frac{1}{\Gamma(\mu) \Gamma(1-\mu)} \, \quad \text{ if } n=0 \; ,\end{align*} $$

and, see [Reference Andrews, Askey and Roy3, Theorem 2.2.2],

$$ \begin{align*}{}_2F_1\left(\mu,1-\mu;|n|+1;1\right) =\frac{\Gamma(|n|+1) \Gamma(|n|)}{\Gamma(|n|+1-\mu) \Gamma(|n|+\mu)} \, \quad \text{ if } n\not=0 .\end{align*} $$

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 ${}_2F_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 $\lambda =4 m(m+1)$ 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

$$ \begin{align*} F(z,w)=\sum \limits_{n=-m}^m c_n P^{\mu}_n(z,w) \in \mathcal{H}(\Omega).\\[-41pt] \end{align*} $$

Example 6.4 Fix $\mu \in {\mathbb C}$ with $\textsf {Re}\,\mu \geq \frac {1}{2}$ , let $(z_0,w_0)\in \Omega \setminus {\mathbb D}^2$ with $z_0 w_0 \not \in [1,\infty )$ , and define

$$\begin{align*}b_n :=\left(\frac{\Gamma(\mu+|n|)}{\Gamma(\mu)|n|!}{}_2F_1\left(\mu;1-\mu;|n|+1\right){\frac{z_0w_0}{z_0w_0-1}}\right)^{-1}.\end{align*}$$

Note that asymptotically

$$ \begin{align*}|b_n| \sim \left( |n|+1 \right)^{1-\textsf{Re}\,\mu} \qquad (|n| \to \infty) .\end{align*} $$

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

$$ \begin{align*}F(z,w)=\sum\limits_{n=-\infty}^\infty b_n P_n^{\mu}(z,w)\end{align*} $$

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,

$$ \begin{align*}f(z):=F(z,\overline{z})=\sum \limits_{n=-\infty}^{\infty} b_n P^\mu_n(z,\overline{z})\end{align*} $$

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

$$ \begin{align*}F(z,w)=\sum\limits_{n=-\infty}^\infty \tilde{b}_n P_n^{\mu}(z,w)\end{align*} $$

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

$$\begin{align*}F(z_0,w_0)=\sum\limits_{n=0}^\infty w_0^n+\sum\limits_{n=1}^\infty z_0^n\end{align*}$$

is a divergent series since $|z_0|\geq 1$ or $|w_0|\geq 1$ .

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 2.10), it will turn out that all we need to understand are precompositions of PFM with automorphisms in $\mathcal {M}({\mathbb D}^2)$ , that is,

$$\begin{align*}P^{\mu}_n \circ T \text{ with }\textsf{Re}\, \mu \ge 1/2 \text{ and }T \in \mathcal{M}({\mathbb D}^2).\end{align*}$$

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 and the mappings

(7.1)

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

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

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 4.4, the precompositions

$$\begin{align*}P^{-m}_n \circ T \text{ with }m\in{\mathbb N}_0\text{ and }T \in \mathcal{M}\end{align*}$$

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

$$ \begin{align*} (P_{ n}^{-m}\circ\mathcal{F})(z,w)=(-1)^mP_{n}^{-m}(z,w)\, \end{align*} $$

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

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

Further, we will employ the definition of the PFM via integrals from (4.10). By Remark 4.3, when considering points $(z,\overline {{z}})\in {\mathbb D}^2$ , this definition coincides with the nth 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) $$ \begin{align} P\left(z,\overline{{z}};\xi\right)=\frac{\psi_{z,\overline{{z}}}'(\xi)}{\psi_{z,\overline{{z}}}(\xi)}\xi \end{align} $$

and

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

where $z,u\in {\mathbb D}$ , $\xi \in \partial {\mathbb D}$ , and $\psi _{z,\overline {{z}}}\in \mathcal {M}({\mathbb D})$ (see [Reference Rudin38, Theorem 3.3.5]).

Remark 7.4 One can define the generalized Poisson kernel P even 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 [Reference Fischer and Lieb13, Chapter VI.2]. The identities (7.3) and (7.4) then take the form

$$ \begin{align*} P\left(\psi_{z,w}^{-1}(0),1/\psi_{z,w}^{-1}(\infty);\xi\right)&=\frac{\psi_{z,w}'(\xi)}{\psi_{z,w}(\xi)}\xi =P\left(z,w;\xi\right),\\ P\left(\psi_{z,w}(u),\frac{1}{\psi_{z,w}\left(1/v\right)};\psi_{z,w}(\xi)\right)&=\frac{P(u,v;\xi)}{P\left(\psi_{z,w}^{-1}(0),1/\psi_{z,w}^{-1}(\infty);\xi\right)}=\frac{P(u,v;\xi)}{P\left(z,w;\xi\right)} , \end{align*} $$

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 7.5 Let $m\in {\mathbb N}_0$ , $(z,w)\in \Omega $ , and $(u,v) \in \Omega \cap {\mathbb C}^2$ . Then

(7.5) $$ \begin{align} \big( P_0^{m+1} \circ T_{u,v} \big)(z,w)=\big( P_0^{-m} \circ T_{u,v} \big)(z,w)=\sum_{j=-m}^{m} P^{m+1}_{-j}\left(u,v\right) P^{-m}_{j}(z,w). \end{align} $$

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

$$ \begin{align*} \big( P_0^{m+1} \circ T_{u,\overline{{u}}} \big)(z,\overline{{z}}) &\overset{({4.18})}{=}\big( P_0^{-m} \circ T_{u,\overline{{u}}} \big)(z,\overline{{z}}) \\ &= \frac{1}{2\pi} \int\limits_{0}^{2\pi} P\left(\psi_{u,\overline{{u}}}(z),\frac{1}{\psi_{u,\overline{{u}}}\left(1/\overline{{z}}\right)};e^{it}\right)^{-m}\,dt\\ &\overset{({7.4})}{=} \frac{1}{2\pi} \int\limits_{0}^{2\pi} P\big(z,\overline{{z}}; \psi_{u,\overline{{u}}}^{-1}(e^{it})\big)^{-m} P\big(u,\overline{{u}};\psi_{u,\overline{{u}}}^{-1}(e^{it})\big)^{m}\,dt\\ &\overset{\psi_{u,\overline{{u}}}(e^{is})=e^{it}}{=}\frac{1}{2\pi} \int\limits_{0}^{2\pi} P\big(z,\overline{{z}}; e^{is}\big)^{-m} P\big(u,\overline{{u}};e^{is}\big)^{m+1}\,ds. \end{align*} $$

In the last step, we used (7.3). Now, we can interpret the resulting integral as the inner product $\langle \,\cdot , \,\cdot \,\rangle $ 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

$$ \begin{align*} \big(P_0^{-m} \circ T_{u,\overline{{u}}}\big)(z,\overline{{z}}) &=\left\langle P\left(u,\overline{{u}};e^{is}\right)^{m+1}, \overline{{P(z,\overline{{z}};e^{is})^{-m}}}\right\rangle\\ &\overset{\mathrm{(P)}}{=}\sum_{j=-\infty}^{\infty}\left\langle P\left(u,\overline{{u}};e^{it}\right)^{m+1},e^{-ijt}\right\rangle \left\langle e^{-ijs},\overline{{P(z,\overline{{z}};e^{is})^{-m}}}\right\rangle\\ &=\sum_{j=-\infty}^{\infty}\frac{1}{(2\pi)^2}\int\limits_0^{2\pi} P\left(u,\overline{{u}};e^{it}\right)^{m+1}e^{ijt}\,dt \int\limits_0^{2\pi}P(z,\overline{{z}};e^{is})^{-m}e^{-ijs}\,ds\\ &=\sum_{j=-m}^{m}P_{-j}^{m+1}(u,\overline{{u}})P_j^{-m}(z,\overline{{z}}) , \end{align*} $$

where the series reduces to a finite sum because $P_j^{-m}$ vanishes for $|j|>m$ (see Remark 4.6(b)).

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

Theorem 7.6 Let $\mu \in {\mathbb C}$ , $\textsf {Re}\,\mu \geq 1/2$ , $n\in {\mathbb N}_0$ , and $u\in {\mathbb D}$ . Then

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

and

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

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

$$ \begin{align*} \big( P_n^{\mu} \circ T_{u,\overline{{u}}} \big)(z,\overline{{z}}) &=\frac{1}{2\pi} \int\limits_0^{2\pi} P\big(z,\overline{{z}}; e^{is}\big)^{\mu} P\big(u,\overline{{u}};e^{is}\big)^{1-\mu}\psi_{u,\overline{{u}}}(e^{is})^{-n}\,ds\\ &= \nonumber \frac{1}{2\pi } \int\limits_0^{2\pi} P\big(z,\overline{{z}}; e^{is}\big)^{\mu} P\big(u,\overline{{u}};e^{is}\big)^{1-\mu}(1-\overline{{u}}e^{is})^n(u-e^{is})^{-n}\,ds. \end{align*} $$

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

$$ \begin{align*} &\big( P_n^{\mu} \circ T_{u,\overline{{u}}} \big)(z,w)\nonumber\\ & \quad = \sum_{k=0}^n \sum_{\ell=0}^\infty \binom{n}{k} \binom{-n}{\ell} (-1)^{k-n+\ell}\overline{{u}}^ku^\ell \frac{1}{2\pi } \int\limits_0^{2\pi} P\big(z,w; e^{is}\big)^{\mu} P\big(u,\overline{{u}};e^{is}\big)^{1-\mu}e^{is(k-n-\ell)}\, ds. \end{align*} $$

Interpreting the above integral as a $L^2([0,2\pi ],{\mathbb C})$ inner product as it was done in the proof of Proposition 7.5 shows (7.6a). A similar computation yields (7.6b).

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

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

We have

$$ \begin{align*} \left\vert P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right\vert\leq\frac{1}{2\pi}\int\limits_0^{2\pi}\left\vert P(u,\overline{{u}};e^{it})^{1-\mu}\right\vert\, dt:=M_u<\infty \end{align*} $$

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

$$ \begin{align*} \sum_{j=-\infty}^{\infty}\sup_{(z,w)\in K}\left\vert P^{\mu}_{-j}(z,w)\right\vert&\leq\sum_{j=-\infty}^{\infty}\gamma_r \cdot \left(|j|+1 \right)^{\textsf{Re}\,\mu-1} \, r^{|j|}:=N_{r,\mu}<\infty, \end{align*} $$

where $\gamma _r>0$ depends on r (and $\mu $ ). Thus,

$$ \begin{align*} \sum_{k=0}^n\sum_{\ell=0}^\infty &\left\vert\binom{n}{k}\binom{-n}{\ell}\overline{u}^ku^\ell\right\vert \sum_{j=-\infty}^{\infty} \left\vert P^{1-\mu}_{j-k+n+\ell}(u,\overline{{u}})\right\vert \sup_{(z,w)\in K}\left\vert P^{\mu}_{-j}(z,w)\right\vert\\ &\leq\sum_{k=0}^n\sum_{\ell=0}^\infty \left\vert\binom{n}{k}\binom{-n}{\ell}\overline{u}^ku^\ell\right\vert \cdot M_u\cdot N_{r,\mu}<\infty \end{align*} $$

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 7.6 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 the following.

Corollary 7.7 Let $m,n\in {\mathbb N}_0$ , $n\leq m$ , $(z,w)\in \Omega $ and $(u,v)\in {\mathbb D}^2$ . Then

$$ \begin{align*} &\big( P_{n}^{-m} \circ T_{u,v} \big)(z,w) \\ & \quad = \sum_{j=-m}^m \left( \sum_{k=0}^n \sum_{\ell=0}^\infty (-1)^{k-n-\ell} \binom{n}{k} \binom{-n}{\ell} u^\ell v^k P^{m+1}_{j-k+n+\ell}(u,v) \right) P^{-m}_{-j}(z,w) \end{align*} $$

and

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

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 2.6 and 2.10: 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 2.6 and 2.10 afterwards.

Note that Rudin [Reference Rudin37] gives an explicit characterization of the three nontrivial 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 8.5.

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 7.6 and the differential operators $D^{+}$ and $D^{-}$ defined by

(8.2)

for $f \in \mathcal {H}(D)$ and any subdomain $D \subseteq {\mathbb C}^2$ . We note an alternative description of these 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) $$ \begin{align} \nonumber D^{+} f(z,w) = \partial_u \left( f \circ T_{u,0} \circ {\varrho}_{-1} \right) \Big|_{u=0} (z,w) , \\ D^{-} f(z,w) = \partial_v \left( f \circ T_{0,v} \circ \varrho_{-1} \right) \Big|_{v=0} (z,w) . \end{align} $$

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

$$ \begin{align*} D^+ P_n^{\mu} = (\mu-n-1) P_{n+1}^{\mu} \qquad\text{and}\qquad D^- P_n^{\mu} = (\mu+n+1) P_{n-1}^{\mu} . \end{align*} $$

Remark 8.4 One may thus regard the operators $D^+$ and $D^-$ as ladder operators and together with their commutator

$$ \begin{align*} \frac{1}{2} \big( D^- \circ D^+ - D^+ \circ D^- \big) = z \partial_z - w \partial_w , \end{align*} $$

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 $ .

We are now in a position to prove Theorems 2.6 and 2.10.

Proof (of Theorem 2.10) Let $\lambda \in {\mathbb C}$ and $Y \subseteq \mathcal {H}({\mathbb D}^2)$ be a nontrivial 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,

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

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

The finite-dimensional spaces $X_{4m(m+1)}(\Omega )$ can also be characterized in terms of the PFM $P_0^{-m}$ only.

Corollary 8.6 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) $$ \begin{align} X_{4m(m+1)}(\Omega) = \operatorname{span} \big\{ P_0^{-m} \circ T \;\big|\; T \in \mathcal{M} \big\} . \end{align} $$

In addition,

(8.5) $$ \begin{align} X_{4m(m+1)}(\Omega) = \operatorname{span} \big\{ P_0^{-m} \circ T_{u,\overline{{u}}} \;\big|\; u \in {\mathbb D} \big\} = \operatorname{span} \big\{ P_0^{-m} \circ T_{u,-\overline{{u}}} \;\big|\; u \in {\mathbb C} \big\} . \end{align} $$

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

$$ \begin{align*} P_0^{-m} \circ T_{u,\overline{{u}}} = \sum_{j=-m}^{m} P^{m+1}_{-j}(u,\overline{{u}}) P^{-m}_{j} \end{align*} $$

for every $u \in {\mathbb D}$ . We interpret the claim as a change of basis from ${\{P^{-m}_{j} \;|\; -m \le j \le m\}}$ to ${\{P_0^{-m} \circ T_{u_k,\overline {{u}}_k} \;|\; -m \le k \le m\}}$ . Therefore, we have to find suitable points ${u_{-m}, \ldots , u_m \in {\mathbb D}}$ such that the coefficient matrix

$$ \begin{align*} \big( P_j^{m+1}(u_k,\overline{{u}}_k) \big)_{-m \le j,k \le m} \end{align*} $$

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 [Reference Gallier and Quaintance15, Proof of Proposition 7.28]). The second equality in (8.5) follows similarly.

9 Poisson–Fourier modes and Peschl–Minda operators

In [Reference Heins, Moucha, Roth and Sugawa19], the classical Peschl–Minda differential operators, which were introduced by Peschl [Reference Peschl34] and studied, e.g., by [Reference Aharonov2, Reference Harmelin17, Reference Kim and Sugawa24, Reference Kim and Sugawa25, Reference Minda31, Reference Schippers39, Reference Schippers40, Reference Sugawa, Begehr, Gilbert and Kajiwara42], 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 [Reference Heins, Moucha, Roth and Sugawa19]. 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

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

We write and 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 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) $$ \begin{align} D^k_zP(z,w;\xi)^{\mu}&=(\mu)_{k}P(z,w;\xi)^{\mu}\left(\psi_{z,w}(\xi)\right)^{-k} \end{align} $$

(9.1b) $$ \begin{align}\kern-5pt D^k_wP(z,w;\xi)^{\mu}&=(\mu)_{k}P(z,w;\xi)^{\mu}\left(\psi_{z,w}(\xi)\right)^{k} \end{align} $$

with $\psi _{z,w}$ from (7.2). In particular, if $\mu =-m\in (-{\mathbb N}_0)$ , then

$$ \begin{align*} P^{-m} \in \ker(D^{m+1}_z) \cap \ker(D_w^{m+1}). \end{align*} $$

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 [Reference Kim and Sugawa24]. 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

$$ \begin{align*} (-\mu+1)_n \cdot P_n^{\mu} = D^n_z P^{\mu}_0 \quad \textrm{and} \quad (-\mu+1)_n \cdot P_{-n}^\mu = D^n_w P^{\mu}_0 . \end{align*} $$

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

$$ \begin{align*} D^{n,0}P_0^{\mu}(z,\overline{{z}})&=\frac{ 1}{2\pi} \int\limits_{0}^{2\pi} D^{n,0}P\big(z,\overline{{z}};e^{it}\big)^{\mu}\, dt \\ &\overset{({9.1a})}{=}\frac{1}{2\pi} \int\limits_{0}^{2\pi}(\mu)_n P\big(z,\overline{{z}};e^{it}\big)^{\mu}\left(\psi_{z,\overline{{z}}}(e^{it})\right)^{-n}\, dt\\ &\overset{({7.4})}{=}\frac{(\mu)_n }{2\pi} \int\limits_{0}^{2\pi}P\big(z,\overline{{z}};\psi_{z,\overline{{z}}}(e^{it})\big)^{-\mu}\left(\psi_{z,\overline{{z}}}(e^{it})\right)^{-n}\, dt\\ &\overset{{7.3}}{=}\frac{(\mu)_n }{2\pi} \int\limits_{0}^{2\pi}P\big(z,\overline{{z}};e^{it}\big)^{1-\mu}e^{-int}\, dt\\ &=(\mu)_nP^{1-\mu}_n(z,\overline{{z}}) \overset{({4.18})}{=}(-\mu+1)_n P^{\mu}_n(z,\overline{{z}}) . \end{align*} $$

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

In view of [Reference Heins, Moucha, Roth and Sugawa19, 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

$$\begin{align*}P_n^{-m}\in\ker(D_w^{k})\qquad\text{and}\qquad P_{-n}^{-m}\in\ker(D_z^{k})\qquad\text{for all }k>m.\end{align*}$$

Proposition 9.4 Let $M\in {\mathbb N}_0$ . Then

(9.2) $$ \begin{align} \{P_n^{-m}\, :\, m\in{\mathbb N}_0,\,n\in{\mathbb Z},\, m\leq M,\, |n|\leq m\}=\ker(D^{M+1}_z)\cap\ker(D^{M+1}_w) . \end{align} $$

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

$$\begin{align*}X_M=\bigoplus_{m=0}^M X_{4m(m+1)}^0({\mathbb D}^2)\end{align*}$$

and $\dim X_{4m(m+1)}^0({\mathbb D}^2)=2m+1$ by Theorem 8.1. Therefore, we compute on the one hand

$$ \begin{align*}\dim X_M = \sum_{m=0}^M (2m+1) = (M+1)^2 .\end{align*} $$

On the other hand, by [Reference Heins, Moucha, Roth and Sugawa19, Corollary 4.4], we have

(9.3) $$ \begin{align} \ker(D^{M+1}_z) \cap \ker(D^{M+1}_w) = \mathrm{span} \bigg\{ \frac{z^j w^k}{(1-zw)^M} \colon 0 \le j,k \le M \bigg\} . \end{align} $$

Consequently, $\dim (\ker (D^{M+1}_z){\kern-1pt}\cap{\kern-1pt} \ker (D^{M+1}_w)) {\kern-1pt}={\kern-1pt} (M{\kern-1pt}+{\kern-1pt}1)^2$ . 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 9.5 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

$$ \begin{align*} \bigoplus_{M\in{\mathbb N}_0}\big(\ker(D_{z}^{M+1})\cap\ker(D_w^{M+1})\big) \end{align*} $$

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 9.4 and conclude that as sets

$$ \begin{align*} &\operatorname{clos}_{\Omega}\left( \mathrm{span}\{P_n^{-m}\,:\, m\in{\mathbb N}_0,\,n\in{\mathbb Z},\, |n|\leq m\}\right) \\ & =\operatorname{clos}_{\Omega}\left(\bigoplus_{M\in{\mathbb N}_0}\big(\ker(D_{z}^{M+1})\cap\ker(D_w^{M+1})\big)\right) \\ & =\mathcal{H}(\Omega) ,\end{align*} $$

where the last equality follows from a combination of (9.3) with [Reference Heins, Moucha and Roth18, Eq. (4.1) and Corollary 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 [Reference Moucha32]. Perhaps similar results hold for $\mathcal {H}(\Omega _\pm )$ and the corresponding Poisson–Fourier modes from Theorem 4.4.