Failure of almost monotonicity of harmonic measure density ratios at points of vanishing codimension-one density
Luis Lloret, Xavier Tolsa
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 Ω⊂ Rn+1 with harmonic measure ω, we show that at ω-almost every point x∈∂Ω where r0ω(B(x,r))rn=0, the density ratio ω(B(x,r))rn is not almost monotone with respect to the radius r, and therefore exhibits arbitrarily large oscillations at small scales.
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