Sparse domination of weighted composition operators on weighted Bergman spaces in the upper half-plane

TL;DR

This paper applies harmonic analysis techniques to establish sparse domination estimates for weighted composition operators on weighted Bergman spaces in the upper half-plane, leading to new characterizations of their boundedness and compactness.

Contribution

It introduces a novel approach using sparse domination to analyze weighted composition operators, extending results to new weights and the unit ball in complex space.

Findings

New characterizations of boundedness and compactness of operators

Weighted type estimates for Bergman-class functions

Extension of results to the unit ball in complex space

Abstract

The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the $A_2$ theorem for Calder\'on-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer--testing conditions. We also extend our results to the unit ball $\mathbb B$ in $\mathbb C^n$.