Principal values for Riesz transforms and rectifiability

TL;DR

This paper characterizes n-rectifiability of sets in Euclidean space via the existence of principal value limits of Riesz transforms, supported by sharp estimates on these transforms on Lipschitz graphs.

Contribution

It establishes a new criterion for rectifiability based on the principal value existence of Riesz transforms, with precise bounds on their L^2 norms on Lipschitz graphs.

Findings

E is n-rectifiable iff Riesz transform principal values exist almost everywhere.

Provides sharp L^2 estimates for Riesz transforms on Lipschitz graphs.

Connects geometric measure theory with harmonic analysis through Riesz transforms.

Abstract

Let $E\subset R^d$ with $H^n(E)<\infty$, where H^n stands for the $n$-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit $$\lim_{\ve\to0}\int_{y\in E:|x-y|>\ve} \frac{x-y}{|x-y|^{n+1}} dH^n(y)$$ exists H^n-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the $L^2$ norm of the n-dimensional Riesz transforms on Lipschitz graphs.