On the Cauchy transform of weighted Bergman spaces

TL;DR

This paper characterizes the Cauchy transform of weighted Bergman spaces on Jordan domains, extending previous results to include weights constant on Green function level lines, with explicit results for the unit disk.

Contribution

It provides a new description of the Cauchy transform of weighted Bergman spaces with specific weights, generalizing known results for unweighted spaces.

Findings

K(B_2(G,ω)) is characterized for weights constant on Green function level lines.

For the unit disk, the Cauchy transform equals a weighted Dirichlet space under certain conditions.

The results extend previous descriptions of the Cauchy transform for unweighted and integrable Jordan domains.

Abstract

The problem of describing the range of a Bergman space B_2(G) under the Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of \bar G. The description of K(B_2(G)) for an integrable Jordan domain is given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127]. In the present paper we give a description of K(B_2(G,\omega)) analogous to the one given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127] for a weighted Bergman space B_2(G,\omega) with a weight \omega\ which is constant on level lines of the Green function of G. In the case G=D, the unit disk, and under some additional conditions on the weight \omega, K(B_2(D,\omega))=B_2^1(C\bar{D}, \omega^{-1}), a weighted analogue of a Dirichlet space.