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 μ on Rⁿ⁺¹ with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball B⊂Rⁿ⁺¹ such that μ(B)≈r(B)ⁿ. This result involves a flatness condition, formulated in terms of the so-called β₁-number of B, and the L²(μ|_B)-boundedness, as well as a control on the mean oscillation on the ball, of the operator T_μf(x)=∫∇_xE(x,y)f(y)dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.