Gradient of the single layer potential and quantitative rectifiability for general Radon measures

We identify a set of sufficient local conditions under which a significant portion of a Radon measure $\mu$ on ${\mathbb{R}}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset $\mathbb{R}$^{n+1}$suchthat$\mu(B)\approx r(B)^n$. This result involves a flatness condition, formulated in terms of the so-called ${\beta }_1$-number of $B$, and the $L^2(\mu {\mathrm{\mid }}_B)$-boundedness, as well as a control on the mean oscillation on the ball, of the operator \begin{equation} T_\mu f(x)=\int \nabla_x$\mathcal{E}$(x,y)f(y),d\mu(y). \end{equation} Here $\mathcal{E}(\cdot ,\cdot )$ is the fundamental solution for a uniformly elliptic operator in divergence form associated with an $(n+1)\times (n+1)$ matrix with H"older continuous coefficients. This generalizes a work by Girela-Sarri'on and Tolsa for the $n$-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.