Hörmander’s L2-Method, ∂¯-Problem and Polyanalytic Function Theory in One Complex Variable

In this paper we consider the classical ∂ ¯ -problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using sui...

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Bibliographic Details
Published in:Complex analysis and operator theory Vol. 17; no. 3; p. 41
Main Authors: Alpay, Daniel, Colombo, Fabrizio, Diki, Kamal, Sabadini, Irene, Struppa, Daniele C.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.04.2023
Springer Nature B.V
Subjects:
Analysis
Complex variables
Decomposition
Fourier transforms
Hermite polynomials
Mathematics
Mathematics and Statistics
Operator Theory
Operators (mathematics)
Variables
44A15
Functions of exponential type
Fock space
problem
46E22
Polyanalytic functions
spaces
30H20
Hörmander’s estimate
ISSN:1661-8254, 1661-8262
Online Access:Get full text
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Summary:In this paper we consider the classical ∂ ¯ -problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the ∂ ¯ -problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.
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ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-023-01340-0