The John-Nirenberg inequality and a Sobolev inequality in general domains
type: Item
instance of: scholarly article
title: The John-Nirenberg inequality and a Sobolev inequality in general domains
zbMATH Open document ID: 0779.30018
author: Ritva Hurri-Syrjänen
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-09-05T00:00:00Z
review text: The classical John-Nirenberg lemma [\textit{F. John} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 14, 415-426 (1961; Zbl 0102.043)] states that if the mean oscillation of a function \(u\) in a cube \(D\) is uniformly bounded in each subcube, parallel to \(D\), then the oscillation of \(u\) is exponentially integrable in \(D\); more precisely, \[ \int_ D\exp(c| u(x)-a_ D|)dx\leq\text{const}.| D|,\tag{1} \] where \(a_ D\) is the integral average of \(u\) in \(D\). In the paper under review the author is looking for a class of domains \(D\) for which it holds that (1) is true as soon as the mean oscillation of \(u\) is uniformly bounded in each, say dyadic, cube \(Q\) such that, for a fixed constant \(\tau>1\), the enlarged cube \(\tau Q\) is still a subcube of \(D\). It appears that these domains are exactly the ones that satisfy a quasihyperbolic boundary condition. Hence e.g. the John domains belong to the class in question.
Mathematics Subject Classification ID: 30C99
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 399044
zbMATH Keywords: John-Nirenberg inequality
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: mean oscillation
zbMATH Keywords: John domains
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1191
OpenAlex ID: W2045501233
DOI: 10.1006/JMAA.1993.1191
description: scientific article; zbMATH DE number 399044