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 -regular set , if we consider the set of surface cubes (in the sense of Christ and David) near which does not look approximately like a union of planes, then is UR if and only if satisfies a Carleson packing condition, that is, for any surface cube ,
We show that, for lower content regular sets that aren’t necessarily Ahlfors regular, if denotes the square sum of -numbers over subcubes of as in the traveling salesman theorem for higher-dimensional sets, presented by Azzam and Schul, then
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.
Citation
Jonas Azzam. Michele Villa. "Quantitative comparisons of multiscale geometric properties." Anal. PDE 14 (6) 1873 - 1904, 2021. https://doi.org/10.2140/apde.2021.14.1873
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