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

TL;DR

This paper establishes conditions under which parts of a Radon measure in Euclidean space can be approximated by uniformly rectifiable sets, linking geometric measure theory with elliptic PDE operators.

Contribution

It generalizes previous results on Riesz transforms to a broader class of elliptic operators, connecting measure rectifiability with operator boundedness and oscillation control.

Findings

Identifies sufficient conditions for measure rectifiability involving flatness and operator boundedness.

Extends known results from Riesz transforms to elliptic operators with Hölder continuous coefficients.

Provides a framework for analyzing elliptic harmonic measure in relation to geometric measure theory.

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.