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## 1. Problems on Holomorphic Function Spaces

1. #### Problem 1.05.

Is there a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ with $0 < {\rm dim} \, A^2(\Omega) < \infty$?
• #### Problem 1.1.

What is the dimension of the Bergman space of a short $\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)

As an example of short $\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \searrow 0$ rapidly. Then let $\Omega = \bigcup_{n \geq 1} ( \, \mbox{basin of attraction of} \, f_n \, {\rm at}\, 0 )$.
• #### Problem 1.15.

Is there a pseudoconvex domain $\Omega$ with ${\rm dim} \, A^2(\Omega) \neq 0$ such that there is no bounded plurisubharmonic function $\phi$ on $\Omega$.
• #### Problem 1.2.

What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?
• #### Problem 1.25.

Let $\Delta$ be the unit disk in $\mathbb{C}$ and let $\omega > 0$ be an upper semicontinuous function on $\Delta$. Consider the weightd Bergman space $A^2 (\Delta, \omega)$. Is it possible that $0 < {\rm dim} \, A^2(\Delta, \omega) < \infty$?
• #### Problem 1.3.

Let $b \in H^\infty (\Delta)$ with $|| b ||_{H^\infty} \leq 1$. Define $H(b):=(I-T_b T_{\overline{b}})^{1/2} H^2(\Delta)$, where $T_b$ denotes the multiplication operator by $b$.

1. When are polynomials dense in $H(b)$? (known in affirmative for non-extremal $b$ or inner $b$)

2. How to generate the $H(b)$s as reproducing kernel Hilbert spaces to polydisk or the ball in $\mathbb{C}^n$?

3. Find integral representations of these spaces in setting beyond the unit disk $\Delta$.
• #### Problem 1.35.

Given a $\overline{\partial}$-closed $(0,1)$-form $f \in L^\infty$ on a polydisk or bounded symmetric domain, is there a $u$ solving $\overline{\partial}u = f$ with the estimate $||u||_\infty \leq c ||f||_\infty$?
• #### Problem 1.4.

Define $\mathcal{H}_{p, \alpha}:=\left\{ f \,\, \mbox{entire}\, : \left(\frac{\alpha p}{2\pi}\right)^n \left(\int_{\mathbb{C}^n} |f|^p e^{-\alpha p ||z||^2/2} dm \right)^{1/p} < \infty \right\}$. What is the best constant for the Hölder inequality between $\mathcal{H}_{p, \alpha}$ and $\mathcal{H}_{q, \alpha}$?
• #### Problem 1.45.

Given a Fatou–Bieberbach domain $\Omega$, must there exist a pseudoconvex domain $\tilde{\Omega}$ such that $\tilde{\Omega}$ with $\Omega \subsetneq \tilde{\Omega} \subsetneq \mathbb{C}^n$.
• #### Problem 1.5.

Given a Kähler manifold $(M, \omega)$. Define $H^2_\omega:=\{f: \Delta \rightarrow M : \int_\Delta \log |z|^2 f^* \omega > -\infty \}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.
• #### Problem 1.55.

Is there a Fatou–Bieberbach domain with real analytic boundary?
• #### Problem 1.6.

Can region of attractions for non-polynomial automorphisms of $\mathbb{C}^2$ have $C^1$ boundary?
• #### Problem 1.65.

Is there a Fatou–Bieberbach domain contained in $\mathbb{C}^* \times \mathbb{C}^*$?
• #### Problem 1.7.

For a domain $\Omega \subset \mathbb{C}$, define \begin{equation*} H^2(\Omega):={ f \,\, \mbox{holomorphic} \, \, \mbox{on} \, \, \Omega : |f|^2 \, \, \mbox{has a harmonic majorant} }. \end{equation*} Does log-capacity$(\mathbb{C}\setminus \Omega) > 0$ imply ${\rm dim}\, H^2 (\Omega) > 1$?
• #### Problem 1.75.

Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szegö projection (defined using surface area/modified measure?)
• #### Problem 1.8.

Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?

Cite this as: AimPL: Problems on holomorphic function spaces and complex dynamics, available at http://aimpl.org/scvproblems.