Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa

Research output: Contribution to journalArticleResearchpeer-review

10 Citations (Scopus)

Abstract

© European Mathematical Society 2014. For 1 ≤ n < d integers and ρ > 2, we prove that an n-dimensional Ahlfors-David regular measure μ in double-struck Rd is uniformly n-rectifiable if and only if the ρ-variation for the Riesz transform with respect to μ is a bounded operator in L2(μ). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L2(μ) boundedness of the Riesz transform to the uniform rectifiability of μ.
Original languageEnglish
Pages (from-to)2267-2321
JournalJournal of the European Mathematical Society
Volume16
Issue number11
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • Calderón-Zygmund singular integrals
  • Riesz transform
  • Uniform rectifiability
  • ρ-variation and oscillation

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