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

• Published in: Journal of functional analysis Vol. 282; no. 6; p. 109376
• Main Author: Puliatti, Carmelo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2022
• Subjects: Elliptic measure; Rectifiability; Singular integrals; Two-phase problems; 28A75; Rectifiability; Elliptic measure; 42B20; Singular integrals; 35J15; 42B37; Two-phase problems
• ISSN: 0022-1236; 1096-0783; 1096-0783
• DOI: 10.1016/j.jfa.2021.109376

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 operatorTμf(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.