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, such that the n-dimensional Riesz transform ${{\mathcal {R}}}_\mu$ 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

  $$\begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned}$$

• (b) for some constant ${\varepsilon }>0$ small enough,

  $$\begin{aligned}\int _{B} |{{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1)|^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned}$$

  where $m_{\mu ,B}({{\mathcal {R}}}_\mu 1)$ stands for the mean of ${{\mathcal {R}}}_\mu 1$ on B with respect to $\mu$,

then there exists a uniformly n-rectifiable set $\Gamma$, with $\mu (\Gamma \cap B)\gtrsim \mu (B)$, and such that $\mu |_\Gamma$ is absolutely continuous with respect to ${{\mathcal {H}}}^n|_\Gamma$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.