Harmonic measure on sets of codimension larger than one

Abstract

We introduce a new notion of a harmonic measure for a d-dimensional set in Rn with d<n−1, that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.