Complete weighted Bergman spaces have bounded point evaluations

TL;DR

This paper establishes that for weighted Bergman spaces on any domain in the complex plane, being a Banach space is equivalent to having locally uniformly bounded point evaluations, ensuring the existence of reproducing kernels.

Contribution

It proves that complete weighted Bergman spaces are Banach spaces if and only if they have locally uniformly bounded point evaluations, generalizing known results.

Findings

Weighted Bergman spaces are Banach spaces iff they have bounded point evaluations.

Complete weighted Bergman spaces are automatically reproducing kernel Hilbert spaces for p=2.

The results hold for arbitrary domains in the complex plane.

Abstract

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