Failure of almost monotonicity of harmonic measure density ratios at points of vanishing codimension-one density

Abstract

In this paper we study the behavior of the density ratios of harmonic measure at points with vanishing density. Given an arbitrary open set $\Omega \subset {\mathbb{R}}^{n+1}$ with harmonic measure $\omega$, we show that at $\omega$-almost every point $x\in \partial \Omega$ where ${\mathrm{liminf}}_{r\to 0}\frac{\omega (B(x,r))}{r^n}=0$, the density ratio $\frac{\omega (B(x,r))}{r^n}$ is not almost monotone with respect to the radius $r$, and therefore exhibits arbitrarily large oscillations at small scales.