Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability

TL;DR

This paper proves that for certain disjoint domains, mutual absolute continuity of harmonic measures ensures boundary rectifiability and surface measure absolute continuity, linking harmonic analysis and geometric measure theory.

Contribution

It establishes a new connection between harmonic measure mutual absolute continuity and boundary rectifiability under non-degeneracy conditions.

Findings

Mutual absolute continuity of harmonic measures implies boundary rectifiability.

Harmonic measures are absolutely continuous with respect to surface measure.

Results apply to disjoint domains with non-degenerate boundaries.

Abstract

We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.