On hyperholomorphic Bergman type spaces in domains of $\mathbb C^2$

TL;DR

This paper develops a Bergman spaces theory for a class of quaternionic hyperholomorphic functions related to the Helmholtz operator, extending complex analysis concepts to quaternionic domains in a7^2.

Contribution

It introduces and studies a new class of quaternionic hyperholomorphic functions and establishes foundational properties of their Bergman spaces, including reproducing kernels and invariance.

Findings

Existence of a reproducing kernel for the Bergman space.

Development of Bergman projection and operator properties.

Extension of complex analysis results to quaternionic hyperholomorphic functions.

Abstract

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb H$. In this work we deals with a well-known $(\theta, u)-$hyperholomorphic $\mathbb H-$valued functions class related to elements of the kernel of the Helmholtz operator with a parameter $u \in\mathbb H$, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Given a domain $\Omega\subset\mathbb H\cong \mathbb C^2$, we define and study a Bergman spaces theory for $(\theta, u)-$ hyperholomorphic quaternion-valued functions introduced as elements of the kernel of ${}^{\theta}_{u}\mathcal D [f]= {}^{\theta} \mathcal D [f] + u f$ with $u\in \mathbb H$ defined in $C^1(\Omega, \mathbb H)$, where \[ {}^\theta\mathcal D:= \frac{\partial}{\partial \bar z_1} + ie^{i\theta}\frac{\partial}{\partial z_2}j = \frac{\partial}{\partial \bar z_1} + ie^{i\theta}j\frac{\partial}{\partial \bar z_2},\hspace{0.5cm} \theta\in[0,2\pi). \] Using as a guiding fact that $(\theta, u)-$hyperholomorphic functions includes, as a proper subset, all complex valued holomorphic functions of two complex variables we obtain some assertions for the theory of Bergman spaces and Bergman operators in domains of $\mathbb C^2$, in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.