On summability of partial derivatives of Lipschitz functions
type: Item
instance of: scholarly article
title: On summability of partial derivatives of Lipschitz functions
zbMATH Open document ID: 1150.26009
published in: Ricerche di Matematica
publication date: 2007-11-13T00:00:00Z
review text: Let \(X\) be a bounded measurable subset in \(R^k\), a function \(f\:X\to R^k\) and \(\gamma\in(0,1]\). In [Real Anal. Exch. 29, No. 2, 713--728 (2003--2004; Zbl 1094.26009)] the authors showed the importance of the function \(M_{(f,\gamma)}(x)=\limsup_{y\to x}| f(x)-f(y)|/| x-y|^\gamma\), \(x\in X\), in the study of \(\gamma\)-Hölder functions. Here, they show that the summability of \(M_{(f,1)}\) in \(X\) is equivalent with the summability of \(f\) and differentiability of its partial derivatives in \(X\). As a consequence they prove that \(f\) is differentiable almost everywhere if and only if \(f\) can be in a certain way approximated by Lipschitz functions.
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 28A20
zbMATH DE Number: 5211211
zbMATH Keywords: Parameter of regularity
zbMATH Keywords: Lipschitz functions
author: Emma D'Aniello
author: Paolo De Lucia
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
description: scientific article