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

Fedor Nazarov, Xavier Tolsa, Alexander Volberg

Research output: Contribution to journalArticleResearchpeer-review

13 Citations (Scopus)

Abstract

We show that, given a set E ⊂ R n+1 with ffnite n-Hausdorff measure H n , if the n-dimensional Riesz transform RH n bEf(x) =/E/x-y/x-y/ n+1 f(y)H n (y)is bounded in L2(HnbE), then E is n-rectiáble. From this result we deduce that a compact set E ⊂ R n+1 with H n (E) < ∞ is removable for Lipschitz harmonic functions if and only if it is purely n-unrectiáble, 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
  • Rectiá bility
  • Riesz transform

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