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 H.In this work we deals with a well-known (θ, u)−hyperholomorphic H−valued functions class related to elements of the kernel of the Helmholtz operator with a parameter u ∈ H, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions.Given a domain Ω ⊂ H ∼ = C 2 , we define and study a Bergman spaces theory for (θ, u)− hyperholomorphic quaternion-valued functions introduced as elements of the kernel of θ u D[f ] = θ D[f ] + uf with u ∈ H defined in C 1 (Ω, H), where