Quantitative comparisons of multiscale geometric properties

Azzam, Jonas; Villa, Michele

doi:10.2140/apde.2021.14.1873

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Quantitative comparisons of multiscale geometric properties

Jonas Azzam and Michele Villa

Vol. 14 (2021), No. 6, 1873–1904

DOI: 10.2140/apde.2021.14.1873

Abstract

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and, in particular, is not necessarily Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$ -regular set $E$ , if we consider the set $ℬ$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $ℬ$ satisfies a Carleson packing condition, that is, for any surface cube $R$ ,

\sum_{\binom{Q \subseteq R}{Q \in ℬ}} {(diam Q)}^{d} ≲ {(diam R)}^{d} .

We show that, for lower content regular sets that aren’t necessarily Ahlfors regular, if $β_{E} (R)$ denotes the square sum of $β$ -numbers over subcubes of $R$ as in the traveling salesman theorem for higher-dimensional sets, presented by Azzam and Schul, then

ℋ^{d} (R) + \sum_{\binom{Q \subseteq R}{Q \in ℬ}} {(diam Q)}^{d} \sim β_{E} (R) .

We prove similar results for other uniform rectifiability criteria, such as the local symmetry, local convexity, and generalized weak exterior convexity conditions.

En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.

Keywords

rectifiability, traveling salesman, beta numbers, coronizations, corona decomposition, uniform rectifiability, quantitative rectifiability

Mathematical Subject Classification 2010

Primary: 28A12, 28A75, 28A78

Milestones

Received: 23 July 2019

Revised: 30 October 2019

Accepted: 3 March 2020

Published: 7 September 2021

Authors

Jonas Azzam
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Michele Villa
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Vol. 14, No. 6, 2021