Equivalent Bergman Spaces with Inequivalent Weights

TL;DR

This paper demonstrates that for any weighted space of holomorphic functions, there exists an equivalent weight with a Bergman kernel that has zeros, and constructs examples of weights with kernels having infinitely many zeros.

Contribution

It proves the existence of equivalent weights with zeroes in their Bergman kernels and provides explicit examples of weights with infinitely many kernel zeros.

Findings

Existence of equivalent weights with zeroes in Bergman kernels

Construction of radial weights with infinitely many kernel zeros

Bergman kernels can have complex zero structures despite equivalent weights

Abstract

We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(\mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.