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

Altres ajuts: Acord transformatiu CRUE-CSICAltres ajuts: Gobierno Vasco IT-1247-19We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball such that . This result involves a flatness condition, formulated in terms of the so-called -number of B, and the -boundedness, as well as a control on the mean oscillation on the ball, of the operator. Here is the fundamental solution for a uniformly elliptic operator in divergence form associated with an 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