On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects

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Abstract

This is a continuation of our work (González-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil-Théodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms. © 2009 Birkhäuser Verlag Basel/Switzerland.
Original languageAmerican English
Pages (from-to)237-251
Number of pages211
JournalComplex Analysis and Operator Theory
DOIs
StatePublished - 1 Mar 2011

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Invariance
Vector Field
Quaternionic Analysis
Bergman Kernel
Conformal Map
Operator Space
Operator Theory
Bergman Space
Reproducing Kernel
Functor
Field Theory
Continuation
Invariant
Series
Term
Object

Cite this

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title = "On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects",
abstract = "This is a continuation of our work (Gonz{\'a}lez-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil-Th{\'e}odoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms. {\circledC} 2009 Birkh{\"a}user Verlag Basel/Switzerland.",
author = "Gonz{\'a}lez-Cervantes, {J. Oscar} and Luna-Elizarrar{\'a}s, {M. E.} and M. Shapiro",
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AU - Shapiro, M.

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N2 - This is a continuation of our work (González-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil-Théodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms. © 2009 Birkhäuser Verlag Basel/Switzerland.

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