Absolutely continuous functions of Rado, Reichelderfer, and Malý
type: Item
instance of: scholarly article
title: Absolutely continuous functions of Rado, Reichelderfer, and Malý
zbMATH Open document ID: 0981.26009
author: Marianna Csörnyei
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-07-24T00:00:00Z
review text: The aim of the paper is to compare different definitions of absolute continuity and bounded variation for functions of \(n\)-variables. Given a symmetric convex set \(K\) with non-empty interior, let \({\mathcal K}\) be the class of all dilated and translated copies of \(K\) and let us say that a continuous function with compact support is \(K\)-absolutely continuous (\(K\)-AC in short) if for any \(\varepsilon>0\) there exists \(\delta>0\) such that for any disjoint family \((K_i)\subset{\mathcal K}\) we have \[ \sum_i{\mathcal L}^n(K_i)<\delta\qquad\Longrightarrow\qquad \sum_i\omega^n (f,K_i)<\varepsilon. \] Here \(\omega(f,A):=\sup_A f-\inf_A f\) is the oscillation of \(f\) in \(A\). Analogously, we say that \(f\) is of bounded variation with respect to \(K\) (\(K\)-BV in short) if for any disjoint family \((K_i)\subset{\mathcal K}\) we have \[ \sum_i\omega^n (f,K_i)<\infty. \] When \(K=Q\) is a cube these concepts have been studied by Rado and Reichelderfer in connection with the validity of the area formula and the differentiability almost everywhere. More recently, Malý introduced and studied this class of functions when \(K=B\) is a ball. The author shows a remarkable sensitivity of \(K\)-absolute continuity to the shape of \(K\), providing an example of a continuous function with compact support in \({\mathbf R}^2\) which is \(B\)-AC but not \(Q\)-AC, and not even \(Q\)-BV.
Mathematics Subject Classification ID: 26A46
Mathematics Subject Classification ID: 26A45
zbMATH DE Number: 1563453
zbMATH Keywords: absolutely continuous functions
zbMATH Keywords: BV functions
zbMATH Keywords: bounded variation
zbMATH Keywords: area formula
zbMATH Keywords: differentiability almost everywhere
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6962
OpenAlex ID: W1987106457
cites work: Absolutely continuous functions of several variables
cites work: Q5807746
cites work: Q3227685
DOI: 10.1006/JMAA.2000.6962
description: scientific article; zbMATH DE number 1563453