Higher dimensional Ahlfors regular sets and chordarc curves in \(R^n\)
type: Item
instance of: scholarly article
title: Higher dimensional Ahlfors regular sets and chordarc curves in \(R^n\)
zbMATH Open document ID: 0932.28004
author: Manouchehr Ghamsari
author: David A. Herron
published in: Rocky Mountain Journal of Mathematics
publication date: 2000-02-16T00:00:00Z
full work available at URL: CONT281.html
review text: 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.
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 1286534
zbMATH Keywords: Jordan curve
zbMATH Keywords: boundary turning
zbMATH Keywords: von Koch snowflake
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: rectifiable curve
zbMATH Keywords: Hausdorff dimension
reviewed by: S. V. Kislyakov
MaRDI profile type: Publication
OpenAlex ID: W2090456021
cites work: Q3310101
cites work: Harmonic measure and arclength
cites work: Opérateurs intégraux singuliers sur certaines courbes du plan complexe
cites work: Digital sundials, paradoxical sets, and Vitushkin's conjecture
cites work: Q3995301
cites work: Classification of a quasi-circles by Hausdorff dimension
cites work: Q3731958
cites work: Hardy spaces, $A_\infty$, and singular integrals on chord-arc domains
cites work: Extension of quasisymmetric and Lipschitz embeddings of the real line into the plane
cites work: Bilipschitz extensions of maps having quasiconformal extensions
DOI: 10.1216/RMJM/1181071829
description: scientific article; zbMATH DE number 1286534