Vanishing Carleson measures and power compact weighted composition operators

TL;DR

This paper characterizes Carleson measures and vanishing Carleson measures on weighted Bergman spaces using t-Berezin transform and averaging functions, and applies these to characterize power bounded and power compact weighted composition operators.

Contribution

It introduces new characterizations of Carleson measures and applies these to analyze power boundedness and compactness of weighted composition operators.

Findings

Characterization of Carleson measures via t-Berezin transform and averaging functions.

Criteria for power boundedness of weighted composition operators.

Criteria for power compactness of weighted composition operators.

Abstract

In this paper, we characterize Carleson measure and vanishing Carleson measure on Bergman spaces with admissible weights in terms of {\it t-Berezin transform} and {\it averaging function} as key tools. Moreover, power bounded and power compact weighted composition operators are characterized as application of Carleson measure and vanishing Carleson measure respectively on Bergman spaces with admissible weights.