Solid Cauchy transform on the weighted poly-Bergman spaces

Abstract

In the present paper, we deal with the weighted solid Cauchy transform C?s (from inside the unit disc into the complement of its closure) acting on the weighted true poly-Bergman spaces in the unit disc introduced and studied by Ramazanov and Vasilevski. Mainly, we are concerned with the concrete description of its range and its null space. We also give the closed expression of their reproducing kernels. To this end, we begin by studying the basic properties of C?s such as boundedness for appropriate probability measures. The main tool is an explicit expression of its action on the so-called disc polynomials which form an orthogonal basis of the considered weighted true poly-Bergman spaces.