[Paper Review] A sufficient geometric criterion for quantitative absolute continuity of harmonic measure

Steve Hofmann, José María Martell|arXiv (Cornell University)|Dec 11, 2017

Advanced Harmonic Analysis Research23 references7 citations

TL;DR

This paper establishes a sufficient geometric condition for the quantitative absolute continuity of harmonic measure with respect to surface measure on the boundary of an open set $Ω \subset \mathbb{R}^{n+1}$, $n \geq 2$, with uniformly rectifiable boundary. It shows that if $\Omega$ satisfies a weak local John condition, then harmonic measure is weak-$A_\infty$ with respect to surface measure, providing a geometric criterion for the regularity of harmonic measure in non-connected domains.

ABSTRACT

Let $\Omega\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that harmonic measure for $\Omega$ is weak-$A_\infty$ with respect to surface measure on $\partial\Omega$, provided that $\Omega$ satisfies a certain weak version of a local John condition.

Motivation & Objective

• To determine geometric conditions under which harmonic measure is absolutely continuous with respect to surface measure on the boundary of a domain.
• To extend the theory of harmonic measure to domains that are not necessarily connected, focusing on the role of boundary geometry.
• To identify a weak local John condition as sufficient for the weak-$A_\infty$ property of harmonic measure.
• To provide a quantitative geometric criterion for the absolute continuity of harmonic measure in the context of uniformly rectifiable sets.

Proposed method

• The analysis relies on the geometric structure of the boundary, specifically its uniform rectifiability, which ensures the existence of Ahlfors regular measures and bilipschitz charts.
• A weak local John condition is introduced as a geometric control on the connectivity of the domain, ensuring that points in $\Omega$ can be connected to a boundary point via a curve with controlled decay of distance to the boundary.
• The proof uses comparison principles for harmonic functions and the John condition to control the oscillation of harmonic functions near the boundary.
• The weak-$A_\infty$ property is established by showing that the harmonic measure satisfies a weak-type testing condition relative to surface measure.
• Key estimates are derived using the Whitney decomposition of the domain and testing with Whitney cubes to control the density of harmonic measure.
• The argument leverages the fact that uniformly rectifiable sets support a doubling measure and admit a weak geometric Poincaré inequality, enabling the use of $A_\infty$-type testing.

Experimental results

Research questions

• RQ1 Under what geometric conditions on the domain $\Omega$ is harmonic measure weak-$A_\infty$ with respect to surface measure on $\partial\Omega$?
• RQ2 Can the weak-$A_\infty$ property of harmonic measure be established for non-connected domains with uniformly rectifiable boundaries?
• RQ3 How does a weak local John condition relate to the absolute continuity of harmonic measure in the presence of uniform rectifiability?
• RQ4 What is the role of the boundary's geometric structure in ensuring quantitative control over the density of harmonic measure?
• RQ5 Can the weak-$A_\infty$ property be characterized purely by a geometric condition on the domain's connectivity and boundary regularity?

Key findings

• Harmonic measure is weak-$A_\infty$ with respect to surface measure on $\partial\Omega$ whenever $\Omega$ satisfies a weak local John condition and $\partial\Omega$ is uniformly rectifiable.
• The result holds even when $\Omega$ is not connected, extending previous results that required connectivity.
• The weak local John condition provides a geometric substitute for stronger connectivity assumptions, allowing the analysis of more general domains.
• The proof establishes quantitative control on the density of harmonic measure relative to surface measure via testing conditions.
• The result confirms that uniform rectifiability of the boundary, combined with a weak geometric control on the domain, is sufficient for the weak-$A_\infty$ property.
• The paper provides a sufficient geometric criterion that is both intrinsic and verifiable on the boundary and domain structure, without requiring global regularity.