Rectifiability via a square function and Preiss' theorem

Xavier Tolsa, Tatiana Toro

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)

Abstract

© The Author(s) 2014. Published by Oxford University Press. All rights reserved. Let E be a set in ℝ<sup>d</sup> with finite n-dimensional Hausdorff measure H<sup>n</sup> such that lim inf<inf>r→0</inf> r<sup>-n</sup>H<sup>n</sup>(B(x, r) ∩ E) >0 for H<sup>n</sup>-a.e. x ∈ E. In this paper, it is shown that E is n-rectifiable if and only if ∫<inf>0</inf><sup>1</sup>|H<sup>n</sup>(B(x, r) ∩ E)/r<sup>n</sup>-H<sup>n</sup>(B(x, 2r) ∩ E)/(2r)n|<sup>2</sup> dr/r <∞ for H<sup>n</sup>-a.e. x ∈ E, and also if and only if lim/r→0(H<sup>n</sup>(B(x, r) ∩ E)/r<sup>n</sup>-H<sup>n</sup>(B(x, 2r) ∩ E)/(2r)<sup>n</sup> =0 for H<sup>n</sup>-a.e. x ∈ E. Other more general results involving Radon measures are also proved.
Original languageEnglish
Pages (from-to)4638-4662
JournalInternational Mathematics Research Notices
Volume2015
Issue number13
DOIs
Publication statusPublished - 1 Jan 2015

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