Zero sequences, factorization and sampling measures for weighted Bergman spaces

TL;DR

This paper characterizes zero sets in weighted Bergman spaces with various weights, generalizes factorization results, and explores applications to operators and sampling measures, advancing understanding of these function spaces.

Contribution

It provides a comprehensive characterization of zero sets, extends factorization theorems, and studies sampling measures in weighted Bergman spaces with new techniques.

Findings

Characterization of zero sets for weighted Bergman spaces with radial and non-radial weights

Generalization of Horowitz's factorization result for these spaces

Analysis of sampling measures and dominating sets for doubling weights

Abstract

The zero sets of the Bergman space $A^p_\omega$ induced by either a radial weight $\omega$ admitting a certain doubling property or a non-radial Bekoll\'e-Bonami type weight are characterized in the spirit of Luecking's results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz's factorization result from 1977 for functions in $A^p_\omega$. The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space $A^p_\omega$ induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.