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

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.