Solid Cauchy transform on weighted poly-Bergman spaces

TL;DR

This paper investigates the properties of the weighted solid Cauchy transform on weighted poly-Bergman spaces, focusing on boundedness, explicit action on disc polynomials, and the description of its range.

Contribution

It provides a detailed analysis of the boundedness and range of the weighted solid Cauchy transform on weighted poly-Bergman spaces, including explicit computations involving disc polynomials.

Findings

Boundedness of the weighted solid Cauchy transform established.

Explicit action on disc polynomials derived.

Range of the transform on weighted true poly-Bergman spaces characterized.

Abstract

The so-called weighted solid Cauchy transform, from inside the unit disc into the complement of its closure, is considered and their basic properties such as boundedness is studied for appropriate probability measures. The action the disc polynomials is explicitly computed and used to describe the range of its restriction on the weighted true poly-Bergman spaces.