The riesz transform, rectifiability, and removability for Lipschitz harmonic functions

Fedor Nazarov, Xavier Tolsa, Alexander Volberg

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)

Abstract

We show that, given a set E ⊂ ℝn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform (Equation Presented) is bounded in L2 (Hn90E), then E is n-rectifiable. From this result we deduce that a compact set E ⊂ ℝn+1 with Hn(E) < ∞ is removable for Lipschitz harmonic functions if and only if it is purely n-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.
Original languageEnglish
Pages (from-to)517-532
JournalPublicacions Matematiques
Volume58
Issue number2
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • Lipschitz harmonic functions
  • Rectifiability
  • Riesz transform

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