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

TL;DR

This paper links the geometric property of rectifiability of lower-dimensional sets to the behavior of a regularized distance function and harmonic measure, revealing new insights into the analysis of sets with higher co-dimension.

Contribution

It provides a new characterization of rectifiability via regularized distance functions and explores the harmonic measure associated with a degenerate elliptic operator in higher co-dimension sets.

Findings

Characterization of rectifiability through regularized distance behavior

Identification of a degenerate elliptic operator solution in the complement of the set

Counterexample to the converse of Dahlberg's theorem in higher co-dimension

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.