Uniform domains and the ubiquitous quasidisk
type: Item
instance of: scholarly article
title: Uniform domains and the ubiquitous quasidisk
zbMATH Open document ID: 0615.30016
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1987-01-01T00:00:00Z
review text: A domain \(D\subset {\bar {\mathbb{R}}}^ 2\) is a K-quasidisk if it is the image of an euclidean disk under a K-quasiconformal selfmapping of \({\bar {\mathbb{R}}}^ 2\). A simple metric characterization of quasidisks is due to L. Ahlfors. The article summarizes several equivalent characterizations for quasidisks. Many of these have been initiated by the author. Equivalent analytic characterizations for bounded plane simply connected domains D: (i) A BMO extension domain (Jones), (ii) a Sobolev extension domain (Jones) and (iii) a quasiextremal distance domain (Gehring- Martio). Equivalent metric conditions: (i) uniform (Martio-Sarvas), (ii) linearly locally connected (Gehring), (iii) decomposable (Martin) and (iv) \(k_ D\leq cj_ D\) where \(k_ D\) is the quasihyperbolic metric and \[ j_ D(z_ 1,z_ 2)=\log (1+| z_ 1-z_ 2| /d(z_ 1,\partial D))(1+| z_ 1+z_ 2| /d(z_ 2,\partial D)) \] (Jones, Gehring-Osgood). There are also a number of function theoretic characterizations based on the quasiconformal extension property, the Schwarzian derivative and its counterpart \(| f''/f'|\). The article completes in a nice way the lecture notes [\textit{F. W. Gehring}, Characteristic properties of quasidisks (1982; Zbl 0495.30018)] of the author and contains a comprehensive list of references.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 3996059
zbMATH Keywords: uniform domains
zbMATH Keywords: quasidisks
zbMATH Keywords: Schwarzian derivative
author: F. W. Gehring
reviewed by: O. Martio
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3996059