The Riesz transform and quantitative rectifiability for general Radon measures

Abstract

In this paper we show that if $\mu$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_\mu$ is bounded in $L^2(\mu)$, and $B\subset\mathbb R^{n+1}$ is a ball with $\mu(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $\delta>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,d\mu(x)\leq \delta\,\mu(B),$ (b) for some constant $\epsilon>0$ small enough, $\int_B |R_\mu1(x) - m_{\mu,B}(R_\mu1)|^2\,d\mu(x) \leq \epsilon \,\mu(B)$, where $m_{\mu,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to $\mu$; then there exists a uniformly $n$-rectifiable subset $\Gamma$, with $\mu(\Gamma\cap B)\gtrsim \mu(B)$, and so that $\mu|_\Gamma$ is absolutely continuous with respect to $H^n|_\Gamma$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.