The Riesz transform and quantitative rectifiability for general Radon measures

Daniel Girela-Sarrión, Xavier Tolsa

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

© 2017, Springer-Verlag GmbH Germany, part of Springer Nature. In this paper we show that if μ is a Borel measure in Rn+1 with growth of order n, such that the n-dimensional Riesz transform Rμ is bounded in L2(μ) , and B⊂ Rn+1 is a ball with μ(B) ≈ r(B) n such that:(a)there is some n-plane L passing through the center of B such that for some δ> 0 small enough, it holds (Formula presented.)(b)for some constant ε> 0 small enough, (Formula presented.) where mμ,B(Rμ1) stands for the mean of Rμ1 on B with respect to μ, then there exists a uniformly n-rectifiable set Γ , with μ(Γ ∩ B) ≳ μ(B) , and such that μ| Γ is absolutely continuous with respect to Hn| Γ. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.
Original languageEnglish
Article number16
JournalCalculus of Variations and Partial Differential Equations
Volume57
Issue number1
DOIs
Publication statusPublished - 1 Feb 2018

Keywords

  • 28A75
  • 28A78
  • 42B20
  • 49Q20

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