Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one

Abstract

In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in a sharp contrast with the usual setting of co-dimension one, a converse to the Dahlberg's theorem (see [Da] and [DFM2]) must be false on lower dimensional boundaries without additional assumptions.

38 pages. Final version with revisions suggested by referees. To appear in Duke Math Journal

References in corpus (2)

• Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality
• Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

Cited by in corpus (3)

• Harmonic Measure and the Analyst's Traveling Salesman Theorem
• The Green function with pole at infinity applied to the study of the elliptic measure
• An $\alpha$-number characterization of $L^p$ spaces on uniformly rectifiable sets