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1. Problems on Holomorphic Function Spaces


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      Problem 1.05.

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


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          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 ).


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              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.


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                  Problem 1.2.

                  What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?


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                      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?


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                          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.


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                              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?


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                                  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}?


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                                      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.


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                                          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.


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                                              Problem 1.55.

                                              Is there a Fatou–Bieberbach domain with real analytic boundary?


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                                                  Problem 1.6.

                                                  Can region of attractions for non-polynomial automorphisms of \mathbb{C}^2 have C^1 boundary?


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                                                      Problem 1.65.

                                                      Is there a Fatou–Bieberbach domain contained in \mathbb{C}^* \times \mathbb{C}^*?


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                                                          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?


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                                                              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?)


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                                                                  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.