Further properties of the Bergman spaces of slice regular functions

TL;DR

This paper advances the understanding of Bergman spaces for slice regular functions by explicitly describing kernels, comparing two theories, and exploring transformations and reflection principles.

Contribution

It provides explicit Bergman kernels for the unit ball and half space, compares two Bergman theories with weights, and relates kernels via Schwarz reflection.

Findings

Explicit Bergman kernel for the unit ball

Comparison of two Bergman theories with weights

Relation of kernels through Schwarz reflection

Abstract

In this paper we continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paper we mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitable weights are taken into account. Finally, we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.