A new function space and applications
type: Item
instance of: scholarly article
title: A new function space and applications
zbMATH Open document ID: 1339.46028
DOI: 10.4171/JEMS/551
author: Jean Bourgain
author: Haïm Brezis
author: Petru Mironescu
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2015-10-16T00:00:00Z
review text: This article is concerned with showing that certain functions with values in \(\mathbb{Z}\) are constant. The starting point are the known results that if a \(\mathbb{Z}\)-valued function is contained in any of the function spaces \(\mathrm {VMO}(\Omega)\), \(W^{1,1}(\Omega)\), or \(W^{1/p,p}(\Omega)\), then it is constant. The authors introduce a space \(B\) that contains the above three spaces and they further introduce a subspace \(B_0\) of \(B\). They show that if \(f\) is \(\mathbb{Z}\)-valued and in \(B_0\), then \(f\) is constant. Actually, for \(f\) being a characteristic function (and \(n\geq 2\)), they show a quantitative improvement of the result. This quantitative result is extended to \(\mathbb{Z}\)-valued functions in \(B\) (under some smallness condition on the function, which is also shown to be necessary).
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6495629
zbMATH Keywords: \(\mathrm {BMO}\)
zbMATH Keywords: \(\mathrm {VMO}\)
zbMATH Keywords: \(\mathrm {BV}\)
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: integer-valued functions
zbMATH Keywords: constant functions
zbMATH Keywords: isoperimetric inequality
reviewed by: Thomas Zürcher
MaRDI profile type: Publication
cites work: Perimeter of sets and BMO-type norms
cites work: Lifting in Sobolev spaces
cites work: Degree theory of BMO. I: Compact manifolds without boundaries
cites work: On functions of bounded mean oscillation
cites work: Functions of Vanishing Mean Oscillation
full work available at URL: https://doi.org/10.4171/jems/551
OpenAlex ID: W2134269164
description: scientific article; zbMATH DE number 6495629