On a left-α-ψ-hyperholomorphic Bergman space.

Given a structural set ψ ⊂ H , some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is ψ D = ∑ i = 0 3 ψ i ∂ ∂ x i . The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: α ψ D [ f ] = α f + ψ D [ f ] , where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to α ψ D and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties. [ABSTRACT FROM AUTHOR]