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

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 $r(B_1)\le {\delta }_{1}r(B_0)$ such that, for some constant $\alpha >0$, $$\frac{\mu (B_1)}{r(B_1{)}^n}\ge \alpha \frac{\mu (B_0)}{r(B_0{)}^n}\mathrm{.}$$ (ii) For some $\epsilon >0$, $${\int }_{2B_0}\mathrm{\mid }R\mu -m_{\mu ,2B_0}(R\mu ){\mathrm{\mid }}^{2}d\mu \le \epsilon (\frac{\mu (B_0)}{r(B_0{)}^n}{)}^2\mu (B_0)\mathrm{.}$$ If ${\delta }_1$ is small enough, depending on $n$ and $\alpha$, and $\epsilon$ is small enough, then there exists a uniformly $n$-rectifiable set $\mathrm{\Gamma }$ and some $\tau >0$ such that $\mu (\mathrm{\Gamma }\cap B_0)\ge \tau \mu (B_0)\mathrm{.}$