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arXiv · 1212.5431

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

Fedor Nazarov·Xavier Tolsa·Alexander Volberg
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Abstract

We show that, given a set $E\subset \mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset\mathbb R^{n+1}$ with $H^n(E)<\infty$ 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.

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Fedor Nazarov, Xavier Tolsa, Alexander Volberg. 2013-12-05. The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions. https://arxiv.org/abs/1212.5431

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