The Hausdorff measure on \(n\)-dimensional manifolds in \(\mathbb{R}^m\) and \(n\)-dimensional variations (Q2307895)

In the paper under review, the author introduces the notion of variation \(V_f(A)\) for every continuous function \(f\colon G\to\mathbb{R}^n\), where \(G\subseteq\mathbb{R}^n\) is an open subset and the subset \(A\subseteq G\) is a countable union of compact sets. In special case when \(A=[a_1,b_1)\times\cdots\times[a_n,b_n)\) with \(a_j<b_j\) in \(\mathbb R\) for \(j=1,\dots,n\), the author recovers the notion introduced in the author's earlier paper [Zap. Nauchn. Semin. POMI 327, 168--206 (2005; Zbl 1083.58012)]. With the above notation, the author proves the formula \(V_f(A)=\int_{\mathbb{R}^n}N_f(A,y)\mathrm{d}y\), where \(N_f(A,y)\) is the cardinality of the set \(f^{-1}(y)\cap A\) for \(y\in\mathbb{R}^n\). The main result of the present paper concerns an arbitrary continuous injective function \(\varphi=(\varphi_1,\dots,\varphi_m)\colon G\to\mathbb{R}^m\), where \(G\subseteq \mathbb{R}^n\) is an open subset as above, while \(n\le m\). For an arbitrary set \(\alpha=\{i_1,\dots,i_n\}\) with \(1\le i_1<\cdots<i_n\le m\), the author denotes \(\varphi_\alpha:=(\varphi_{i_1},\dots,\varphi_{i_n})\colon G\to\mathbb{R}^n\). With this notation, the author proves that if a subset \(A\subseteq G\) can be written as a countable union of compact sets, then \(V_{\varphi_\alpha}(A)\le H_n(\varphi(A))\), where \(H_n\) stands for the \(n\)-dimensional Hausdorff measure in \(\mathbb{R}^m\).