On Hyperholomorphic Bergman Type Spaces in Domains of C2

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Abstract

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.

Translated title of the contributionSobre espacios tipo Bergman hiperholomórficos en dominios de C 2
Original languageEnglish
Article number30
JournalComplex Analysis and Operator Theory
Volume17
Issue number2
DOIs
StatePublished - Mar 2023

Keywords

  • Covariant and invariant conformal properties
  • Holomorphic function theory in two complex variables
  • Quaternionic weighted Bergman spaces
  • Reproducing kernel

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