Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

TL;DR

This paper revisits Lipschitz approximation techniques in NTA domains, removing the need for surface measure regularity, and explores harmonic measure's absolute continuity, extending classical planar results to higher dimensions.

Contribution

It shows Lipschitz graph constructions in NTA domains do not require surface measure regularity, enabling new analysis of harmonic measure and surface measure relationships.

Findings

Lipschitz approximation methods are valid without upper Ahlfors regularity.

A partial F. and M. Riesz theorem analogue is established for NTA domains in space.

Wolff snowflakes have infinite surface measure.

Abstract

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence every Wolff snowflake has infinite surface measure.