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

By Fedor Nazarov, Xavier Tolsa, Alexander Volberg. 2012

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We show that, given a set E Rn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform is bounded in L2(HnbE), then E is n-rectifiable. From this result we deduce that a compact set E Rn+1 with Hn(E) < 1 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/d606a20e62597cef0edf562a692927f3cdb32184