Definitions of quasiconformality
type: Item
instance of: scholarly article
title: Definitions of quasiconformality
zbMATH Open document ID: 0832.30013
DOI: 10.1007/BF01241122
author: Juha Heinonen
author: Pekka Koskela
published in: Inventiones Mathematicae
publication date: 1996-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/144269
review text: Let \(f : D \to \mathbb{R}^n\), \(D \subset \mathbb{R}^n\) a domain, be a homeomorphism. The mapping \(f\) is quasiconformal iff \((*)\) \(\limsup_{n \to 0} H(x,f,r) \leq C < \infty\) for all \(x \in D\); here \(H(x,f,r) = L(x,f,r)/l(x,f,r)\) and \(L(x,f,r) = \max \{|f(y) - f(x) |: |y - x |= r\}\); for \(l(x,f,r) \max\) is replaced by min. No simple proof for \(\Leftarrow\) is known. The \(H\)-dilatation can be used to study homeomorphisms between arbitrary metric spaces \(X\) and \(Y\). The authors look for global estimates which can be derived from the boundedness of \(H\). Such an estimate is the quasisymmetry condition [\textit{P. Tukia} \textit{J. Väisälä}: Ann. Acad. Sci. Fenn., Ser. A I 5, 97-114 (1980; Zbl 0443.54011)], which holds in the case \(X = \mathbb{R}^n = Y\). The authors show that this conclusion remains true if \(X = Y\) is a Garnot group [A. Korányi, H. M. Riemann: Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. (to appear)]. As a byproduct they show that the lim sup in \((*)\) can be replaced by the lim inf. The proofs employ elementary covering arguments instead of the usual capacity estimates.
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 746069
zbMATH Keywords: quasiconformality in metric spaces
zbMATH Keywords: Garnot group
Wikidata QID: Q112013286
reviewed by: O. Martio
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 746069