On Hyperholomorphic Bergman Type Spaces in Domains of C²

Quaternionic analysis is a branch of classical analysis referring to different generalizations of the Cauchy-Riemann equations to the quaternion skew field H context. In this work we deals with H- valued (θ, u) - hyperholomorphic functions 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≅ C², our main goal us to discuss the Bergman spaces theory for this class of functions as elements of the kernel of uθD[f]=θD[f]+uf with u∈ H defined in C¹(Ω , H) , where θD:=∂∂z¯1+ieiθ∂∂z2j=∂∂z¯1+ieiθj∂∂z¯2,θ∈[0,2π).Using as a guiding fact that (θ, 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 C², in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.