On functions with derivatives in a Lorentz space
type: Item
instance of: scholarly article
title: On functions with derivatives in a Lorentz space
zbMATH Open document ID: 0976.26004
author: Janne Kauhanen
author: Pekka Koskela
author: Jan Malý
published in: Manuscripta Mathematica
publication date: 2001-12-27T00:00:00Z
review text: The authors of this deep and interesting paper pursue the following question: what are ``minimal'' assumptions on a function \(f\) of several variables which guarantee that \(f\) has properties such as continuity, differentiability, absolute continuity or the Lusin N-property (\(f(E)\) is of \(n\)-dimensional Hausdorff measure zero whenever the same is true for \(E\)). The main result is that one such sharp sufficient condition guaranteeing the N-condition is that \(f\) belongs to the Sobolev space \(W^{1,1}_{\text{loc}}\) and its gradient \(\nabla f\) belongs to the Lorentz space \(L^{n,1}\). Interestingly, the same condition is also sufficient for other important properties such as continuity, \(n\)-absolute continuity or differentiability a.e. In an earlier work of \textit{A. Cianchi} and the reviewer [Ark. Mat. 36, No. 2, 317-340 (1998)] it was shown that the above-mentioned condition is sufficient (and, in a~certain sense, also necessary) for \(f\) to belong to \(L^\infty\). It is therefore interesting and even somewhat surprising that the same condition is equivalent to the \(n\)-absolute continuity of \(f\). The paper contains plenty of nice and original thoughts. A~useful by-product of the argument is a~new characterization of the Lorentz space \(L^{p,q}\) which can turn out to be quite handy.
reviewed by: Luboš Pick
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 1355508
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: oscillation of functions
zbMATH Keywords: Lusin N-property
zbMATH Keywords: differentiability
zbMATH Keywords: absolute continuity
MaRDI profile type: Publication
full work available at URL: s002290050197
OpenAlex ID: W2049060790
Wikidata QID: Q109967582
DOI: 10.1007/S002290050197
description: scientific article; zbMATH DE number 1355508