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

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 ${}_u^{\theta }\mathcal{D}[f]={}^{\theta }\mathcal{D}[f]+uf$ with $u\in \mathbb{H}$ defined in $C^1(\Omega ,\mathbb{H})$, where \[ {}^θ\mathcal D:= \frac{\partial}{\partial \bar z_1} + ie^{iθ}\frac{\partial}{\partial z_2}j = \frac{\partial}{\partial \bar z_1} + ie^{iθ}j\frac{\partial}{\partial \bar z_2},\hspace{0.5cm} θ\in[0,2π). \] 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.