Real-variable characterizations of Bergman spaces

TL;DR

This paper surveys recent real-variable characterizations of Bergman spaces, emphasizing maximal and area integral functions, and introduces a new proof using vector-valued Calderón-Zygmund operators for area integral characterizations.

Contribution

It provides a new proof for area integral characterizations of Bergman spaces using Calderón-Zygmund operator techniques, with sharp estimates of Bergman kernels and metrics.

Findings

New proof of area integral characterizations

Application of Calderón-Zygmund operators to Bergman spaces

Sharp estimates of Bergman kernel and metric

Abstract

In this paper, we give a survey of results obtained recently by the present authors on real-variable characterizations of Bergman spaces, which are closely related to maximal and area integral functions in terms of the Bergman metric. In particular, we give a new proof of those results concerning area integral characterizations through using the method of vector-valued Calder\'{o}n-Zygmund operators to handle Bergman singular integral operators on the complex ball. The proofs involve some sharp estimates of the Bergman kernel function and Bergman metric.