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LIPSCHITZ SPACES AND BOUNDED MEAN OSCILLATION OF HARMONIC MAPPINGS

Part of: Geometric function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  18 January 2013

SH. CHEN ,
S. PONNUSAMY ,
M. VUORINEN  and
X. WANG
SH. CHEN
Affiliation:
Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008,PR China email [email protected]
S. PONNUSAMY*
Affiliation:
Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India
M. VUORINEN
Affiliation:
Department of Mathematics, University of Turku, Turku 20014, Finland email [email protected]
X. WANG
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081,PR China email [email protected]
*
✉[email protected][email protected]
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Abstract

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We first study the bounded mean oscillation of planar harmonic mappings. Then we establish a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings. Finally, we obtain sharp estimates on the Lipschitz number of planar harmonic mappings in terms of the bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BM{O}_{2} $ as a Banach space.

Keywords

harmonic mappingmajorantLipschitz spaceBMOpequivalent modulusharmonic Bloch spaceGreen’s theorem

MSC classification

Secondary: 30H10: Hardy spaces 30H30: Bloch spaces 30C20: Conformal mappings of special domains 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 143 - 157
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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