Quasiextremal distance domains and extension of quasiconformal mappings
type: Item
instance of: scholarly article
title: Quasiextremal distance domains and extension of quasiconformal mappings
zbMATH Open document ID: 0596.30031
DOI: 10.1007/BF02792549
published in: Journal d'Analyse Mathématique
publication date: 1985-01-01T00:00:00Z
review text: A domain D in \(\bar R^ 2\) is said to be a K-quasidisc if it is the image of an open disc under a K-quasiconformal self-mapping of \(\bar R^ 2\). The paper under review is concerned with the extension to more general classes of domains D in \(\bar R^ n\) of the following two basic properties of quasidiscs. 1.Extremal distance property. If D is a quasidisc and \(F_ 1\) and \(F_ 2\) are disjoint continua in D, then \(mod(\Gamma)\leq M mod(\Gamma_ D)\) where \(\Gamma\) and \(\Gamma_ D\) are the families of curves which join \(F_ 1\) and \(F_ 2\) in \(\bar R^ 2\) and D, respectively, and where M is a constant which depends only on D and mod(\(\Gamma)\) is the modulus of \(\Gamma\). 2. Extension property. If D is a quasidisc and if f is a quasiconformal mapping of D onto a domain D' in \(\bar R^ 2\), then f has a quasiconformal extension to \(\bar R^ 2\) if and only if D' is a quasidisc. The authors prove that properties 1 and 2 are related in the following sense. If D and D' are as in 2 and if D has the extremal distance property, then D and D' have the extension property iff D' has the extremal distance property. The authors call domains satisfying property 1 QED-domains (quasi-extremal distance domains) and study their geometric properties. They also show that a simply-connected plane domain with a non-degenerated boundary is a QED-domain if it is a quasidisc.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 3959880
zbMATH Keywords: quasidiscs
zbMATH Keywords: quasiconformal extension
zbMATH Keywords: quasi-extremal distance domains
author: F. W. Gehring
author: O. Martio
reviewed by: M. Vuorinen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02792549
OpenAlex ID: W2074145476
description: scientific article; zbMATH DE number 3959880