Higher dimensional Ahlfors regular sets and chordarc curves in \(R^n\)
For \(n=2\) and rectifiable curves, the topics mentioned in the title are well studied (a brief summary of that work is included in the paper). The authors generalize this to higher values of \(n\) and to curves of arbitrary Hausdorff dimension \(\alpha\in [1, n)\). A set \(E\subset {\mathbb R}^n\) is said to be \(\alpha\)-dimensional Ahlfors regular if the \({\mathcal H}^{\alpha}\)-measure (i.e., the \(\alpha\)-Hausdorff measure) of the part of \(E\) inside an arbirary \(r\)-ball centered at a point in \(E\) is comparable to \(r^{\alpha}\). A Jordan curve \(C\subset {\mathbb R}^n\) is said to be \(\alpha\)-dimensional chordarc (respectively, of bounded turning) if a component \(A\) of \(C\setminus\{ x, y\}\) satisfies \({\mathcal H}^{\alpha}(A)\asymp |x-y|^{\alpha}\) (respectively, \(\text{diam } A\leq C|x-y|\)). Here is a selection of results. First, \(\alpha\)-dimensional chordarc Jordan curves do exist in \({\mathbb R}^n\) for every \(\alpha\in [1, n)\). Next, for a Jordan curve \(C\subset {\mathbb R}^n\) the following are equivalent: (a) \(C\) is \(\alpha\)-dimensional chordarc; (b) \(C\) is of bounded turning and is \(\alpha\)-dimensional Ahlfors regular; (c) \(C\) is of bounded turning and \({\mathcal H}^{\alpha}(A)\asymp (\text{ diam} (A))^{\alpha}\) for all arcs \(A\subset C\). Further, for arbitrary \(\alpha\in (0, n)\) and arbitrary sets, \(\alpha\)-dimensional Ahlfors regularity is stable under the Möbius transformations. Finally, \(E\subset {\mathbb R}^n\) is \(\alpha\)-dimensional Ahlfors regular if and only if \(E\) is porous and \({\mathcal H}^{\alpha}(\varphi (E))\approx 1\) for all Möbius \(\varphi\) with \(\text{ diam} (\varphi (E))=1\). Some examples and counterexamples are presented.
scientific article; zbMATH DE number 1286534