A remark on differentiable functions with partial derivatives in \(L^p\)
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.
scientific article; zbMATH DE number 2141779
scientific article