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 $\overline{\Omega }$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $H^2(\Omega ,\phi )$ with a plurisubharmonic weight $\phi$. 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 ${\delta }_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated.

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