Representation Theorems for the Cauchy Transform in Weighted Bergman–Dirichlet Spaces

Abstract

In this paper we analyse the integrability of Cauchy transforms of functions and measures. As an application of general Paley–Wiener theorems, we show that the integrability of the Cauchy transform of a function or measure is deeply related to the integrability of the Fourier transform of the corresponding function (measure). Our main results are representation theorems for the Cauchy transform in weighted spaces of Bergman–Dirichlet type.