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

Abstract: 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.