The Bergman kernel for the Vekua equation

TL;DR

This paper extends classical results about the Bergman kernel to solutions of the Vekua equation, providing new insights into their existence and construction.

Contribution

It introduces the construction of the Bergman kernel and projection operator for solutions of the Vekua equation, expanding the theory beyond analytic functions.

Findings

Existence of the Bergman kernel for Vekua solutions

Explicit construction methods for the Bergman kernel

Extension of classical analytic function results

Abstract

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial _{\overline{z}}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.