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

Carmelo Puliatti

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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 Tμf(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.

Original languageEnglish
Article number109376
Number of pages77
JournalJournal of Functional Analysis
Volume282
Issue number6
DOIs
Publication statusPublished - 15 Mar 2022

Keywords

  • Elliptic measure
  • Rectifiability
  • Singular integrals
  • Two-phase problems

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