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

TL;DR

This paper establishes a link between the boundedness of the Riesz transform on a set and its rectifiability, leading to a characterization of removable sets for Lipschitz harmonic functions in higher dimensions.

Contribution

It proves that bounded Riesz transform implies rectifiability and characterizes removable sets for Lipschitz harmonic functions in higher dimensions.

Findings

Bounded Riesz transform implies $n$-rectifiability.

Removability for Lipschitz harmonic functions characterized by pure unrectifiability.

Proves the higher-dimensional analog of Vitushkin's conjecture.

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.