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

We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain $Ω\subset \R^{n}$ with $n\geq 3$, with dimensional Ahlfors regular boundary $\partialΩ$ of dimension $s$ with $n-1-δ_0 \leq s\leq n-1$, that is uniformly non flat. Here $δ_0$ 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 $δ_0$ parameter.

💡 Research Summary

The paper investigates the relationship between harmonic measure and the geometry of the boundary of a domain Ω ⊂ ℝⁿ (n ≥ 3). The authors focus on domains whose boundary ∂Ω is Ahlfors–David s‑regular with dimension s satisfying n − 1 − δ₀ ≤ s ≤ n − 1, where δ₀ is a small positive constant depending on the quantitative non‑flatness parameter β and the regularity constant C₁. The main result is that under a uniform non‑flatness condition—expressed by the inequality

b_β^{ω_Ω}(x,r) := inf_V