Hausdorff measure on $n$-dimensional manifolds in~$\mathbb R^m$ and $n$-dimensional variations

A. V. Potepun

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 143-150

Résumé

If $f\colon[a;b]\to\mathbb R^m$ is an injective continuous mapping and $f_1,\dots,f_m$ are coordinate functions of $f$, then the curve $f([a;b])$ is rectifiable if and only if the variations of all $f_k$ are finite. By Jordan's theorem for the length of the curve we have $$ V_{f_i}([a;b])\le l(f([a;b]))\le\sum_{k=1}^mV_{f_k}([a;b]),\quad i=1,\dots,m. $$ The length $l(f([a;b]))$ is $H_1(f([a;b]))$, where $H_1$ is one-dimensional Hausdorff measure in $\mathbb R^m$. In this article, the notion of the variation $V_f[a;b]$ of a function $$ f\colon[a;b]\to\mathbb R $$ is generalized to the variation $V_f(A)$ of a continuous mapping $f\colon G\to\mathbb R^n$, where $G$ is an open subset of $\mathbb R^n$, on a set $A\subset G$, $A=\bigcup_{i\in I}K_i$, where $I$ is countable, all $K_i$ are compact. Suppose $f\colon G\to\mathbb R^m$, $G\subset\mathbb R^n$, $n\le m$, $f_1,\dots,f_m$ are the coordinate functions of $f$. If $1\le i_1$, $\alpha=\{i_1,\dots,i_n\}$, then $f_\alpha$ is the mapping with the coordinate functions $f_{i_1},\dots,f_{i_n}$: $$ f_\alpha\colon \begin{cases} x_{i_1}=f_{i_1}(t_1,\dots,t_n)\\ \dots\dots\dots\dots\dots\dots\\ x_{i_n}=f_{i_n}(t_1,\dots,t_n) \end{cases} \quad(t_1,\dots,t_n)\in G. $$ The main result states that if $f$ is a continuous injective mapping, $f\colon G\to\mathbb R^m$, $n\le m$, $G$ is an open subset of $\mathbb R^n$, $A\subset G$, $A=\bigcup_{i\in I}K_i$, $I$ is countable, all $K_i$ are compact, then $$ V_{f_\alpha}(A)\le H_n(f(A)), $$ where $V_{f_\alpha}(A)$ is the variation of $f_\alpha$ on $A$, $H_n$ is $n$-dimensional Hausdorff measure in $\mathbb R^m$.