Global integrability of the Jacobian and quasiconformal maps
type: Item
instance of: scholarly article
title: Global integrability of the Jacobian and quasiconformal maps
zbMATH Open document ID: 0816.30014
published in: Michigan Mathematical Journal
publication date: 1994-08-17T00:00:00Z
review text: Let \(f: D\to D'\) be a \(K\)-quasiconformal map, where \(D\) and \(D'\) are domains in \(\mathbb{R}^ n\). In 1973 \textit{F. W. Gehring} [Acta Math. 130, 265-277 (1973; Zbl 0258.30021)] proved that for some \(\varepsilon> 0\), \(J^{1+ \varepsilon}_ f\), where \(J_ f\) if the Jacobian of \(f\), is locally integrable in \(D\). More recently, \textit{K. Astala} and \textit{P. Koskela} [J. Anal. Math. 57, 203-220 (1991; Zbl 0772.30022)] answered the question, ``When is \(J^{1+ \varepsilon}_ f\) integrable over \(D\)?''. A crucial step in their argument involves establishing the equivalence of this question with the local Lipschitz question studied by \textit{F. W. Gehring} and \textit{O. Martio} [Ann. Acad. Sci. Fenn., Ser. A I 10, 203-219 (1985; Zbl 0584.30018)]. In the paper under review, the author presents direct proofs of the global integrability of the Jacobian results. These proceed using quasihyperbolic metric and BMO estimates. Furthermore, these proofs lead to ideas and techniques useful in the study of the domains involved.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 614380
zbMATH Keywords: quasihyperbolic metric
zbMATH Keywords: BMO
author: Susan G. Staples
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004829
OpenAlex ID: W1989231039
DOI: 10.1307/MMJ/1029004829
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description: scientific article; zbMATH DE number 614380