The C-property for slice regular functions and applications to the Bergman space

TL;DR

This paper explores the C-property of slice regular functions, demonstrating their decomposition into components satisfying this property, and applies these findings to establish a reproducing property of Bergman kernels in quaternionic analysis.

Contribution

It introduces a novel decomposition of slice regular functions into C-property components and applies this to Bergman kernel analysis in quaternionic spaces.

Findings

Decomposition of slice regular functions into four C-property components

Establishment of a reproducing property for Bergman kernels of the second kind

Enhanced understanding of quaternionic function behavior with respect to C-properties

Abstract

This paper has a twofold purpose: on one hand we deepen the study of slice regular functions by studying their behavior with respect to the so-called C-property and anti-C-property. We show that, for any fixed basis of the algebra of quaternions $\mathbb{H}$ any slice regular function decomposes into the sum of four slice regular components each of them satisfying the C-property. Then, we will use these results to show a reproducing property of the Bergman kernels of the second kind.