Good elliptic operators on Cantor sets

TL;DR

This paper demonstrates that certain elliptic operators on purely unrectifiable Cantor sets in the plane can have harmonic measures that are absolutely continuous and proportional to the Hausdorff measure, contrary to classical expectations.

Contribution

It introduces elliptic operators on unrectifiable Cantor sets with harmonic measures absolutely continuous to Hausdorff measure, expanding understanding of measure behavior on fractal sets.

Findings

Existence of elliptic operators with absolutely continuous harmonic measure on Cantor sets.

Harmonic measure is essentially proportional to the Hausdorff measure on these sets.

Contradicts the classical notion that unrectifiable sets cannot support such measures.

Abstract

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in ${\mathbb{R}}^2$ whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.