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Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

Bo-Yong Chen, John Erik Fornæss, Jujie Wu

Abstract

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overlineΩ$ of a bounded non-smooth pseudoconvex domain $Ω$, in the Bergman space $ H^2(Ω,\varphi)$ with a plurisubharmonic weight $\varphi$. As an application, we show that the Hartogs domain $$ Ω_α: = \{(z,w) \in D\times \C: |w|< δ^α_D(z) \}, \ \ \ α>0, $$ where $D\subset \subset \C$ and $δ_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated.

Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

Abstract

We study the density of functions which are holomorphic in a neighbourhood of the closure of a bounded non-smooth pseudoconvex domain , in the Bergman space with a plurisubharmonic weight . As an application, we show that the Hartogs domain where and denotes the boundary distance, is Bergman complete if and only if every boundary point of is non-isolated.
Paper Structure (6 sections, 20 theorems, 128 equations)

This paper contains 6 sections, 20 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Theorem \ref{['th:moregeneral']} and Corollary \ref{['co:co1']}
  4. Proofs of Theorem \ref{['cor:Main22']} and Theorem \ref{['th:deltabijin1']}
  5. Proofs of Theorem \ref{['th:Bergmancomplete1']} and Theorem \ref{['th:approxidelta1']}
  6. Appendix: Examples for Theorem \ref{['cor:Main22']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\varphi$ a psh function in a neighbourhood of $\overline{\Omega}$. Suppose there exists a Stein neighbourhood basis $\{\Omega^t\}_{0<t\le t_0}$ of $\overline{\Omega}$ such that where $\eta (t)$ is a non-negative continuous increasing function about $t$ satisfying for some $r_0\ll 1$. Then $\mathcal{O}(\overline{\Omega}) \cap H^

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1: Kobayashi criterion
  • ...and 28 more