Solid Cauchy transform on weighted poly-Bergman spaces

Rachid El Harti, Abdelatif Elkachkouri, Allal Ghanmi


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.


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