论文标题
相对于所有低维度均匀整流集的Hausdorff度量，谐波度量绝对连续
Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets
论文作者
论文摘要
最近表明，在现在有充分理解的其他拓扑约束的情况下，相对于具有$ n-1 $尺寸均匀整流边界的Hausdorff度量，谐波度量绝对连续。正如C. Bishop和P. Jones的反例所示，拓扑限制是必要的，虽然是温和的，并且这些结果尚无类似的类似物。 在本文中，我们表明，对于任何$ d <n-1 $，对于具有$ d $尺寸均匀校正边界的任何域而言，相对于边界的Hausdorff度量，适当的退化椭圆形操作员的椭圆度量绝对是连续的。没有与上述结果相反的拓扑或维度限制。
It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of now well understood additional topological constraints. The topological restrictions, while mild, are necessary, as the counterexamples of C. Bishop and P. Jones show, and no analogues of these results have been available for higher co-dimensional sets. In the present paper we show that for any $d<n-1$ and for any domain with a $d$-dimensional uniformly rectifiable boundary the elliptic measure of an appropriate degenerate elliptic operator is absolutely continuous with respect to the Hausdorff measure of the boundary. There are no topological or dimensional restrictions contrary to the aforementioned results.
