New criteria for the rectifiability of Radon measures in terms of Riesz transforms

TL;DR

This paper establishes new criteria linking the boundedness of Riesz transforms to the rectifiability of Radon measures, providing conditions under which measures are supported on rectifiable sets.

Contribution

It introduces novel conditions involving measure density and oscillation of Riesz transforms that guarantee measure rectifiability.

Findings

Bounded Riesz transform implies measure is supported on rectifiable sets.

Small oscillation of Riesz transforms within a ball indicates rectifiability.

New quantitative criteria for rectifiability based on Riesz transform behavior.

Abstract

In this paper we explore the connection between quantitative rectifiability of measures and the $L^2$ boundedness of the codimension one Riesz transform. Among other things, we prove the following. Let $\mu$ be a Radon measure in $\mathbb R^{n+1}$ with growth of degree $n$ such that the $n$-dimensional Riesz transform $R_\mu$ is bounded in $L^2(\mu)$, and let $B_0\subset\mathbb R^{n+1}$ be a suitably doubling ball such that: (i) There exists some (small) ball $B_1$ centered in $B_0$ with $r(B_1)\leq \delta_1 r(B_0)$ such that, for some constant $\alpha>0$, $$\frac{\mu(B_1)}{r(B_1)^n}\geq \alpha\,\frac{\mu(B_0)}{r(B_0)^n}.$$ (ii) For some $\epsilon>0$, $$\int_{2B_0} |R\mu - m_{\mu,2B_0}(R\mu)|^2\,d\mu\leq \epsilon\,\bigg(\frac{\mu(B_0)}{r(B_0)^n}\bigg)^2\,\mu(B_0).$$ If $\delta_1$ is small enough, depending on $n$ and $\alpha$, and $\epsilon$ is small enough, then there exists a uniformly $n$-rectifiable set $\Gamma$ and some $\tau>0$ such that $\mu(\Gamma\cap B_0) \geq\tau\,\mu(B_0).$