Complete weighted Bergman spaces have bounded point evaluations

Abstract

Let $\hat{I}\text{c◯}\subset \mathbb{C}$ be an arbitrary domain in the one-dimensional complex plane equipped with a positive Radon measure $\hat{I}¼$. For any $1\le p<\infty$, it is shown that the weighted Bergman space $A^p(\hat{I}\text{c◯},\hat{I}¼)$ of holomorphic functions is a Banach space if and only if $A^p(\hat{I}\text{c◯},\hat{I}¼)$ has locally uniformly bounded point evaluations. In particular, in the case $p=2$, any complete Bergman space $A^2(\hat{I}\text{c◯},\hat{I}¼)$ is automatically a reproducing kernel Hilbert space.

6 pages