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

Abstract

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 $B\subset \mathbb{R}^{n+1}$ such that $\mu(B)\approx r(B)^n$. This result involves a flatness condition, formulated in terms of the so-called $\beta_1$-number of $B$, and the $L^2(\mu|_B)$-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.