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

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 $\mu(\Gamma\cap B_0) \geq\tau\,\mu(B_0).$