Bergman spaces and reproducing kernel for the biquaternionic Vekua equation

TL;DR

This paper studies the properties of Bergman spaces of solutions to a biquaternionic Vekua equation, establishing their structure, continuity, and reproducing kernels, thus advancing quaternionic analysis in bounded domains.

Contribution

It introduces and analyzes the Bergman spaces for the biquaternionic Vekua equation, providing explicit formulas for kernels, projections, and dual spaces, which are novel in quaternionic PDE analysis.

Findings

Bergman spaces are complete, separable, and reflexive.

Solutions are locally Hölder continuous and evaluation maps are bounded.

Explicit reproducing kernels and orthogonal projections are derived for the case p=2.

Abstract

We analyze the main properties of the Bergman spaces of weak $L_p$- solutions for a biquaternionic Vekua equation of the form \[ \mathbf{D}w(x)-\mathbf{Q}_Aw(x)=0 \] on bounded domains of $\mathbb{R}^3$, where the operator $\mathbf{Q}_A$ involves quaternionic conjugation and multiplications, both left and right, by essentially bounded functions. Properties such as completeness, separability, and reflexivity are shown. It is demonstrated that the solutions belonging to the Bergman spaces are locally H\"older continuous and that the evaluation maps are bounded in the $L_p$-norm. Consequently, for the case $p=2$, we obtain a reproducing integral kernel and an explicit formula for the orthogonal projection onto the Bergman space. For $1<p<\infty$, the explicit form for the annihilator of the Bergman space in the dual $L_{p'}$ is presented, along with an orthogonal decomposition for $L_2$.