Absolutely continuous functions of Rado, Reichelderfer, and Malý
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.
scientific article; zbMATH DE number 1563453