Loewner chains and parametric representation in several complex variables

Let B be the unit ball of C n with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball  B. We show that any Loewner chain f( z, t) and the transition mapping v( z, s, t) associated to f( z, t) satisfy locally Lipschitz conditions...

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Veröffentlicht in:Journal of mathematical analysis and applications Jg. 281; H. 2; S. 425 - 438
Hauptverfasser: Graham, Ian, Kohr, Gabriela, Kohr, Mirela
Format: Journal Article
Sprache:Englisch
Veröffentlicht: San Diego, CA Elsevier Inc 15.05.2003
Elsevier
Schlagworte:
Exact sciences and technology
Loewner chain
Loewner differential equation
Mathematical analysis
Mathematics
Parametric representation
Sciences and techniques of general use
Several complex variables and analytic spaces
Transition mapping
Univalent mapping
Loewner chain
Univalent mapping
Parametric representation
Loewner differential equation
Transition mapping
Parametric method
Lipschitz condition
Mathematical analysis
Loewne chain
Complex variable function
ISSN:0022-247X, 1096-0813
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Zusammenfassung:Let B be the unit ball of C n with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball  B. We show that any Loewner chain f( z, t) and the transition mapping v( z, s, t) associated to f( z, t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈ B. Moreover, we prove that a mapping f∈ H( B) has parametric representation if and only if there exists a Loewner chain f( z, t) such that the family { e − t f( z, t)} t⩾0 is a normal family on B and f( z)= f( z,0) for z∈ B. Also we show that univalent solutions f( z, t) of the generalized Loewner differential equation in higher dimensions are unique when { e − t f( z, t)} t⩾0 is a normal family on  B. Finally we show that the set S 0( B) of mappings which have parametric representation on B is compact.
ISSN:0022-247X
1096-0813
DOI:10.1016/S0022-247X(03)00127-6