Hyperholomorphic Bergman Spaces and Bergman Operators Associated with Domains in $${\mathbb{C}}^2$$

Abstract

The work deals with a version of quaternionic analysis adapted in such a way that the set of arising hyperholomorphic functions includes, as a proper subset, all holomorphic mappings. There are considered, for the theory of the corresponding Bergman spaces and Bergman operators, both classic, commonly treated aspects and more specific ones, in particular, conformally invariant or covariant character of certain objects; a description of an algebra generated by the Bergman projection together with a criterion ensuring the Fredholmness of elements of the algebra; relations with the usual holomorphic mappings theory in two complex variables are demonstrated as well. Mathematics Subject Classification (2000). 30G35, 32A25, 47B32, 47B37, 47L30.

On hyperholomorphic Bergman type spaces in domains of $\mathbb C^2$

“…is a quaternionic right-Hilbert space and from deeply similar computations to those presented in [14], [15] and [32] we can see that the valuation functional f → f (z), for z ∈ Ω, is bounded on θ u A(Ω). Riesz representation theorem for quaternionic right Hilbert space, see [7], shows that there exists…”

“…Let Ω ⊂ C 2 be a bounded domain with its boundary ∂Ω a compact 3−dimensional sufficiently smooth hypersurface (co-dimension 1 manifold). The following formulas can be found in many sources (see, for example [14,24]).…”

“…There has been a great deal of work done over the recent years on weighted Bergman spaces for ψ−hyperholomorphic functions in R 4 , see [14][15][16][17]32]. For a natural development of Bergman kernel function in Clifford analysis we refer the reader to [7,33].…”

On Some Categories and Functors in the Theory of Quaternionic Bergman Spaces

“…When the specific case ψ = {1, i, ie iθ j, e iθ j} with 0 ≤ θ < 2π is considered, it coincides with the θ-hyperholomorphy, see [2] and [6]; although the case of an arbitrary ψ is more general, at the same time the proofs of all results are quite similar to their antecedents for θ-hyperholomorphy and they are mostly omitted here. The interested reader may consult [2].…”

“…In [2] there are more results about the θ-hyperholomorphy that have their ψ-hyperholomorphic versions; here we expose those facts in the ψ-hyperholomorphic setting only which are necessary for our purposes.…”