论文标题
低维整体集合谐波度量的绝对连续性
Absolute continuity of the harmonic measure on low dimensional rectifiable sets
论文作者
论文摘要
我们考虑一个均匀的校正设置$γ\ subset \ mathbb r^n $ dimension $ d <n-1 $。通过在补充$ω= \ mathbb r^n \setMinusγ$，Guy David，Svitlana Moyboroda上使用脱名式椭圆运算符，作者对$γ$介绍了谐波测度的概念。 我们在本文中证明，对$γ$的这种谐波度量满足了$ a^\ infty $ property，这是谐波措施，$ d $ - d $ dimension hausdorff y Menate $γ$的措施以一种定量和规模不变的方式相互持续。因此，我们给出了戴维（David）和梅博罗达（Mayboroda）的最新定理的另一种证据，这本身将霍夫曼（Hofmann）和马特尔（Martell）的结果扩展到了均匀校正设置$γ$不是编成编辑1的情况下。 证明非常简单 - 特别是不遵循David和Mayboroda或Hofmann和Martell所使用的路线 - 但对于$ d <n -1 $的情况而言是特定的。
We consider a uniformly rectifiable set $Γ\subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $Ω= \mathbb R^n \setminus Γ$, Guy David, Svitlana Mayboroda, and the author introduced a notion of harmonic measure on $Γ$. We prove in the present article that this harmonic measure on $Γ$ satisfies the $A^\infty$-property, that is the harmonic measure and the $d$-dimension Hausdorff measure on $Γ$ are mutually absolutely continuous in a quantitative and scale invariant way. Thus, we give an alternate proof of a recent theorem of David and Mayboroda, which itself extends a result of Hofmann and Martell to the case where the uniformly rectifiable set $Γ$ is not of codimension 1. The proof is surprisingly simple - in particular does not follow the route used by David and Mayboroda, or by Hofmann and Martell - but is specific to the case when $d<n-1$.
