Square functions, nontangential limits, and harmonic measure in codimension larger than 1

Abstract

We characterize the rectifiability (both uniform and not) of an Ahlfors regular set E of arbitrary codimension 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 sharp contrast with the usual setting of codimension 1, a converse to Dahlberg’s theorem must be false on lower-dimensional boundaries without additional assumptions.