Abstract
We show that, given a set $E\subset{\mathbb R}^{n+1}$ with finite $n$-Hausdorff measure${\mathcal H}^n$, if the $n$-dimensional Riesz transform
$$R_{{\mathcal H}^n{\lfloor} E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}}\,f(y)\,{\mathcal H}^n(y)$$
is bounded in $L^2({\mathcal H}^n{\lfloor} E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset{\mathbb R}^{n+1}$ with ${\mathcal H}^n(E)<\infty$ 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.
Citation
Fedor Nazarov. Xavier Tolsa. Alexander Volberg. "The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions." Publ. Mat. 58 (2) 517 - 532, 2014.
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