A concept of absolute continuity and its characterization in terms of convergence in variation (Q2836141)

Using the new concept of multidimensional \(\varphi\)-variation, \(V^{\varphi}\), in the sense of Tonelli and of multidimensional \(\varphi\)-absolute continuity, introduced by the authors in [Commun. Appl. Nonlinear Anal. 20, No. 1, 1--20 (2013; Zbl 1279.26028)], the main result can be stated as follows.NEWLINENEWLINETheorem. A function \(f:\mathbb{R}^{n}_{+}\to \mathbb{R}\) is \(\varphi\)-absolute continuous, if and only if there exists a \(\lambda>0\), such that NEWLINE\[NEWLINE\lim_{w\to +\infty}V^{\varphi}[\lambda(T_{w}(f)-f)]=0,NEWLINE\]NEWLINE where \(T_{w}(f)(s)=\int_{\mathbb{R}_{+}^{n}}K_{w}(t)f(s t)\frac{d t}{(t)}\) (with \(s t=(s_{1} t_{1},\dots, s_{n} t_{n})\) and \((t)=t_{1}\cdot t_{2}\cdot \dots\cdot t_{n}\)) denote the multidimensional Mellin operators.NEWLINENEWLINEThe result is new even in the one-dimensional case.