TY - JOUR
T1 - On Hyperholomorphic Bergman Type Spaces in Domains of $$\mathbb C^2$$
AU - González-Cervantes, José Oscar
AU - Bory-Reyes, Juan
N1 - Funding Information:
Secretaría de Investigación y Posgrado, Instituto Politécnico Nacional pls confirm the name Instituto Politécnico Nacional (Grant Number SIP20211188, SIP20221274) and CONACYT.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/3
Y1 - 2023/3
N2 - 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≅ C
2, 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 C
1(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 C
2, in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.
AB - 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≅ C
2, 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 C
1(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 C
2, in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.
KW - Covariant and invariant conformal properties
KW - Holomorphic function theory in two complex variables
KW - Quaternionic weighted Bergman spaces
KW - Reproducing kernel
UR - https://doi.org/10.1007/s11785-023-01336-w
U2 - 10.1007/s11785-023-01336-w
DO - 10.1007/s11785-023-01336-w
M3 - Artículo
SN - 1661-8254
VL - 17
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 2
M1 - 30
ER -