On Hyperholomorphic Bergman Type Spaces in Domains of $$\mathbb C^2$$

Quaternionic analysis is a branch of classical referring to different generalizations the Cauchy-Riemann equations quaternion skew field \(\mathbb H\) context. In this work we deals with H-\)valued \((\theta , u)-\)hyperholomorphic functions related elements kernel Helmholtz operator parameter \(u \in \mathbb H\), just in same way as usual quaternionic set harmonic functions. Given domain \(\Omega \subset H\cong C^2\), our main goal us discuss Bergman spaces theory for class \({}^{\theta }_{u}\mathcal D [f]= {}^{\theta } \mathcal [f] + u f\) \(u\in defined \(C^1(\Omega H)\), where $$\begin{aligned} {}^\theta D:= \frac{\partial }{\partial \bar{z}_1} ie^{i\theta }\frac{\partial z_2}j = }j\frac{\partial \bar{z}_2}, \theta [0,2\pi ). \end{aligned}$$Using guiding fact that includes, proper subset, all complex valued holomorphic two variables obtain some assertions and operators domains particular, existence reproducing kernel, its projection their covariant invariant properties certain objects.