On the Cauchy transform of the Bergman space

TL;DR

This paper characterizes the Cauchy transform of the Bergman space for various domains, establishing a key relation for quasidisks that links the transform's range to another Bergman space.

Contribution

It proves a fundamental relation between the Cauchy transform of the Bergman space and a Bergman space on the complement for quasidisks, extending understanding of these transforms.

Findings

For quasidisks, K(B_2^*(G)) equals B_2^1(ℂ\bar{G})

Describes the range of the Cauchy transform for a large class of domains

Provides a new characterization of the Cauchy transform's image in Bergman spaces

Abstract

The range of the Bergman space B_2(G) under the Cauchy transform K is described for a large class of domains. For a quasidisk G the relation K(B_2^*(G))=B_2^1(\mathbb C\setminus\bar{G}) is proved.