Hölder continuous Sobolev mappings and the Lusin N property
Let \(|\cdot |\) and \({\mathcal H}^n\) denote, respectively, the \(n\)-dimensional Lebesgue measure and the \(n\)-dimensional Hausdorff measure. The author gives a new proof of the following theorem: Let \(\Omega\) be a domain in \({\mathbb{R}}^n\) and \(f\in W^{1,n}({\mathbb{R}}^m)\), \(m\geq n\geq 2\), a Hölder continuous mapping. Then, \({\mathcal H}^n (f(E))=0\) for each \(E\subset \Omega\), such that \(|E|=0\). This result was obtained by \textit{J. Malý} and \textit{O. Martio} [J. Reine Angew. Math. 458, 19--36 (1995; Zbl 0812.30007)] for \(n=m\). Then, she generalizes this theorem to a metric setting.
scientific article; zbMATH DE number 6475315