Toward the theory of ring \(Q\)-homeomorphisms
type: Item
instance of: scholarly article
title: Toward the theory of ring \(Q\)-homeomorphisms
zbMATH Open document ID: 1161.30009
published in: Israel Journal of Mathematics
publication date: 2009-02-17T00:00:00Z
review text: The authors investigate so-called ring \(Q\)-homeomorphisms, including, in particular, \(Q\)-homeo-morphisms, where a homeomorphism \(f : D \rightarrow {\mathbb R}^N \cup {\infty}\) is said to be a \(Q\)-homeomorphism if the modulus \(M\) of \(f \Gamma\) satisfies the following inequality \[ M(f \Gamma) \leq \int\int_{D} Q(x) \rho^n(x) d m(x) \] for every path \(\Gamma\) in \(D\) and any admissible function \(\rho\) with \(Q(x)\) being a measurable function. In terms of the majorant \(Q(x)\), a series of criteria for normality is given based on estimates of the distortion of the spherical distance under ring \(Q\)-homeomorphisms. In particular, it is shown that the class \(\Re_{Q, \triangle}\) of all ring \(Q\)-homeomorphisms \(f\) of a domain \(D \subset {\mathbb R}^N\) into \(\overline{{\mathbb R}^N}\), \(N\geq 2\), with \(h\big(\overline{{\mathbb R}^N}\setminus f(D)\big) \geq \triangle > 0\) forms a normal family if \(Q(x)\) has finite mean oscillation in \(D\).
reviewed by: Sergei V. Rogosin
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5508717
zbMATH Keywords: \(Q\)-homeomorphism
zbMATH Keywords: ring \(Q\)-homeomorphism
zbMATH Keywords: capacity
zbMATH Keywords: uniform equicontinuity
zbMATH Keywords: BMO
zbMATH Keywords: FMO
zbMATH Keywords: Sobolev classes
author: Vladimir I. Ryazanov
author: Evgeny Sevost'yanov
MaRDI profile type: Publication
full work available at URL: s11856-008-1058-2
OpenAlex ID: W2053037304
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DOI: 10.1007/S11856-008-1058-2
description: scientific article; zbMATH DE number 5508717