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 $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E}f(x)={\int }_E\frac{x-y}{|x-y{|}^{n+1}}f(y)d{H}^n(y)$$ 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.

Correction of some typos. To appear in Publicacions Matematiques