The Journal of Analysis

, Volume 24, Issue 1, pp 57–66

Geometric properties of \(\varphi \)-uniform domains

  • Peter Hästö
  • Riku Klén
  • Swadesh Kumar Sahoo
  • Matti Vuorinen
Original Research Paper
  • 19 Downloads

Abstract

We consider proper subdomains G of \({\mathbb R}^n\) and their images \(G'=f(G)\) under quasiconformal mappings f of \({\mathbb R}^n\). We compare the distance ratio metrics of G and \(G'\); as an application we show that \(\varphi \)-uniform domains are preserved under quasiconformal mappings of \({\mathbb R}^n\). A sufficient condition for \(\varphi \)-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If \(G\subset {\mathbb R}^n\) is \(\varphi \)-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar \(\varphi \)-uniform domain whose complement is not \(\psi \)-uniform for any \(\psi \).

Keywords

The distance ratio metric The quasihyperbolic metric Uniform and \(\varphi \)-uniform domains John domains Quasiconformal and quasisymmetric mappings 

Mathematics Subject Classification

Primary 30F45; Secondary 30C65 30L05 30L10 

Copyright information

© Forum D'Analystes, Chennai 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of OuluOuluFinland
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  3. 3.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

Personalised recommendations