消失的余维一密度点处调和测度密度比的几乎单调性失效
Failure of almost monotonicity of harmonic measure density ratios at points of vanishing codimension-one density
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中文总结 AI 辅助
本文研究调和测度在密度消失点的密度比行为，证明了在满足liminf_{r→0} ω(B(x,r))/rⁿ=0的边界点处，该密度比不具几乎单调性，小尺度下振荡幅度任意大。
中文摘要 AI 辅助
本文研究调和测度在密度消失点处的密度比行为。给定任意开集Ω⊂ℝ^{n+1}及其调和测度ω，我们证明：在∂Ω上ω-几乎处处的点x处，当liminf_{r→0} ω(B(x,r))/rⁿ=0时，密度比ω(B(x,r))/rⁿ关于半径r不具有几乎单调性，因此在小尺度下呈现任意大的振荡。
英文摘要
In this paper we study the behavior of the density ratios of harmonic measure at points with vanishing density. Given an arbitrary open set $Ω\subset\mathbb R^{n+1}$ with harmonic measure $ω$, we show that at $ω$-almost every point $x\in\partialΩ$ where $\liminf_{r\to0}\frac{ω(B(x,r))}{r^n}=0$, the density ratio $\frac{ω(B(x,r))}{r^n}$ is not almost monotone with respect to the radius $r$, and therefore exhibits arbitrarily large oscillations at small scales.
