A remark on differentiable functions with partial derivatives in \(L^p\)
type: Item
instance of: scholarly article
title: A remark on differentiable functions with partial derivatives in \(L^p\)
zbMATH Open document ID: 1124.26009
author: Calogero Vetro
author: Cristina Di Bari
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-03-08T00:00:00Z
review text: Let \(\Omega\) be an open set of \(\mathbb{R}^n\), \(E\) be a measurable subset of \(\Omega\) and \(p\) be a positive real number. It is introduced a definition of \(p,\delta\)-variation on \(E\) of a function \(f:\Omega\to \mathbb{R}\) and it is proved Theorem. For a function \(f:\Omega\to \mathbb{R}\) the following conditions are equivalent: (i) the function \(f\) is differentiable almost everywhere on \(E\) with partial derivatives belonging to \(L^p(E)\); (ii) \(f\) is a function with bounded \(p,\delta\)-variation on \(E\). The theorem generates corresponding results of \textit{D. Bongiorno} [Ill. J. Math. 46, No. 2, 557--570 (2002; Zbl 1036.26010)] and \textit{J. Malý} J. Math. Anal. Appl. 231, 492--508 (1999; Zbl 0924.26008)]. For functions with bounded \(p,\delta\)-variation change of variables formula is also proved.
reviewed by: Giorgi Oniani
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 2141779
zbMATH Keywords: differentiability
zbMATH Keywords: variation
zbMATH Keywords: functions of several variables
zbMATH Keywords: gage
zbMATH Keywords: partition
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.06.044
OpenAlex ID: W2008193366
cites work: A regularity condition in Sobolev spaces \(W_{\operatorname{loc}}^{1,p}(\mathbb{R}^n)\) with \(1\leq p<n\).
cites work: Sulla differenziabilita delle funzioni a valori in uno spazio reale di Banach riflessivo
cites work: Q5565773
cites work: Absolutely continuous functions of several variables
DOI: 10.1016/J.JMAA.2004.06.044
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description: scientific article; zbMATH DE number 2141779