The restricted content and the \(d\)-dimensional Analyst's travelling salesman theorem for general sets
A multiscale geometric criterion for a set \(E \subset \mathbb{R}^2\) to be contained in a rectifiable curve has been proved by \textit{P. W. Jones} [Invent. Math. 102, No. 1, 1--15 (1990; Zbl 0731.30018)] who introduced \(\beta\)-numbers. A characterization of subsets of higher dimensional surfaces was studied by \textit{M. Villa} [``Higher dimensional Jordan curves'', Preprint, \url{arXiv:1908.10289}] with the use of a variant of Jones' \(\beta\)-numbers introduced by \textit{J. Azzam} and \textit{R. Schul} [Math. Ann. 370, No. 3--4, 1389--1476 (2018; Zbl 1388.28005)]. Villa considered lower regular subsets (with respect to a \(d\)-dimensional Hausdorff content) of a certain class of surfaces. In the first section of the paper the author has defined a new \(d\)-dimensional variant of Jones' \(\beta\)-number for any set in \(\mathbb{R}^n\) and \(1 \leqslant d < n\) (def. 1.11, def. 3.13 and remark 3.16): Definition 1.11. Let \(1 \leq p < \infty\), \(E \subseteq \mathbb{R}^n\), \(B\) be a ball centred on \(E\) and \(L\) a \(d\)-plane. Define \begin{align*} \beta^{d,p}_E(B,L)^p&=\frac{1}{r^d_B}\int\left(\frac{\operatorname{dis} (x,L)}{r_B}\right)^p d\mathscr{R}^d_{E\cap B}\\ &=\frac{p}{r^d_B} \int^1_0 \mathscr{R}^d_{E\cap B}(\{x\in E :\operatorname{dis} (x,L)>tr_B\})t^{p-1}dt, \end{align*} and \[ \beta^{d,p}_E(B)=\inf\{ \beta^{d,p}_E (B,L): \text{ is a \(d\)-plane} \}. \] The main results of the paper are Theorems 1.13 and 1.14 Theorem 1.13. Let \(1 \leq d < n\), \(C_0 > 1\) and \(1 \leq p < p(d)\). There exists a constant \(c_1 > 0\) such that the following holds. Suppose \(E \subseteq F \subseteq \mathbb{R}^n\), where \(F\) is \((c,d)\)-lower content regular for some \(c\geq c_1\). Let \(\mathscr{D}^E\) be a system of Christ-David cubes for \(E\), and let \(\mathscr{D}^F\) be an extension of \(\mathscr{D}^E\) for \(F\), as above. Let \(Q^E_0 \in \mathscr{D}^E\) and let \(Q^F_0\) be the cube in \(\mathscr{D}^F\) with the same centre and side length as \(Q^E_0\). Then \[ \begin{multlined} \operatorname{diam} (Q^E_0)^d + \sum_{{^{Q \in \mathscr{D}^E}_{Q \subseteq Q^E_0}}} \beta^{d,p}_E (C_0B_Q)^2 l(Q)^d\\ \lesssim_{n,d,c,p,C_0}\operatorname{diam} (Q^F_0)^d + \sum_{{^{Q \in \mathscr{D}^F}_{Q \subseteq Q^F_0}}} \breve{\beta}^{d,p}_F (C_0 B_Q)^2 l(Q)^d. \end{multlined}\tag{1.8} \] Theorem 1.14. Let \(1 \leq d < n\), \(C_0 > 1\) and \(1 \leq p < p(d)\). Let \(E \subseteq \mathbb{R}^n\), \(\mathscr{D}^E\) denote a system of Christ-David cubes for \(E\) and let \(Q^E_0 \in \mathscr{D}^E\) be such that \(\operatorname{diam} (Q^E_0) \geq \lambda l (Q^E_0)\) for some \(0 < \lambda \leq 1\). Then there exists a \((c_1, d)\)-lower content regular set \(F\) (with \(c_1\) as in the previous theorem) such that \(E \subseteq F\) and the following holds. Let \(\mathscr{D}^F\) be an extension of \(\mathscr{D}^E\) for \(F\), and let \(Q^F_0\) denote the cube in \(\mathscr{D}^F\) with the some centre and side length as \(Q^E_0\). Then \[ \begin{multlined} \operatorname{diam} (Q^F_0)^d + \sum_{{^{Q \in \mathscr{D}^F}_{Q \subseteq Q^F_0}}} \breve{\beta}^{d,p}_F (C_0B_Q)^2 l(Q)^d\\ \lesssim_{n,d,c_1,p,C_0,\lambda} \operatorname{diam} (Q^E_0)^d + \sum_{{^{Q \in \mathscr{D}^E}_{Q \subseteq Q^E_0}}} \beta^{d,p}_E (C_0 B_Q)^2 l(Q)^d. \end{multlined}\tag{1.9} \] Furthermore, if the right-hand side of (1.9) is finite then \(F\) is \(d\)-rectifiable. In the inequality (1.18) it should be understand that the inequality holds with dependency on \(n\), \(d\), \(c\), \(p\), \(C_0\) (see page 50). Sections 2 and 3 are devoted to study the properties of \(\beta^{d,p}_E\). The proof of Th.1.13 is included in Section 4 and the proof of Th.1.14 in Section 5.
scientific article; zbMATH DE number 7472318
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