On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

2026/01/31 by Aritro Pathak
Mathematics · #math.AP #math.CA

paper · pdf

Corrected some minor issues; added a table in the notation section

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain Ω⊂ \Rⁿ with n≥ 3, with dimensional Ahlfors regular boundary ∂Ω of dimension s with n-1-δ₀ ≤ s≤ n-1, that is uniformly non flat. Here δ₀ is a small positive constant dependent on the parameters of the problem. Our novel construction relies on elementary geometric and potential theoretic considerations. We avoid the use of Riesz transforms and compactness arguments, and also give quantitative bounds on the δ₀ parameter.

Citations

• Cantor sets with absolutely continuous harmonic measure
• The dimension of harmonic measure on some AD-regular flat sets of fractional dimension
• One phase problem for two positive harmonic function: below the codimension 1 threshold
• Dimension drop for harmonic measure on Ahlfors regular boundaries
• A two-phase free boundary problem for harmonic measure and uniform rectifiability
• Reifenberg Flatness and Oscillation of the Unit Normal Vector
• Absolute continuity of harmonic measure for domains with lower regular boundaries
• Tangent measures and absolute continuity of harmonic measure
• Dimension of the harmonic measure of non-homogeneous Cantor sets

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• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in L^p implies uniform rectifiability