A Green function characterization of uniformly rectifiable sets of any codimension

Abstract

In this paper, we obtain a unified characterization of uniformly rectifiable sets of {\it any codimension} in terms of a Carleson estimate on the second derivatives of the Green function. When restricted to domains with boundaries of codimension 1, our result generalizes a previous result of Azzam for the Laplacian to more general elliptic operators. For domains with boundaries of codimension greater than 1, our result is completely new.

Made corrections to some typos and minor mistakes in the first version. 38 pages. To appear in Adv.Math

References in corpus (4)

• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators
• Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem
• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Carleson measure estimates for the Green function