Solid Cauchy transform on weighted poly-Bergman spaces

Abstract

In the present paper, we are concerned with the concrete description of the range and the null space of the restriction of the weighted solid Cauchy transform $\mathcal{C}_{c}^{\mu}$, from inside the unit disc into the complement of its closure, to the weighted true poly-Bergman spaces in the unit disc.
The main tool is an explicit expression of its action on the so-called disc polynomials.
To this end, we begin by studying the basic properties of $\mathcal{C}_{c}^{\mu}$ such as boundedness for appropriate probability measures, and proving that the disc polynomials form an orthogonal basis of the considered weighted true poly-Bergman spaces. Such spaces are reintroduced here \`a la Ramazanov and \`a la Vasilevski for which we give closed expression of their reproducing kernel.