Vol. 14, No. 6, 2021

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Quantitative comparisons of multiscale geometric properties

Jonas Azzam and Michele Villa

Vol. 14 (2021), No. 6, 1873–1904
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 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 satisfies a Carleson packing condition, that is, for any surface cube R,

QR Q(diamQ)d (diamR)d.

We show that, for lower content regular sets that aren’t necessarily Ahlfors regular, if βE(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

d(R) + QR Q(diamQ)d 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