Harmonic Measure Is Absolutely Continuous with Respect to the Hausdorff Measure on All Low-Dimensional Uniformly Rectifiable Sets
Harmonic Measure Is Absolutely Continuous with Respect to the Hausdorff Measure on All Low-Dimensional Uniformly Rectifiable Sets
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调和测度相对于所有低维均匀可整流集上的 Hausdorff 测度是绝对连续的
DOI：
10.1093/imrn/rnac109
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发表时间：
2022
影响因子：
1
通讯作者：
Mayboroda, S
中科院分区：
文献类型：
--
作者：
David, G;Mayboroda, S
在过去的20年里，调和分析、几何测度理论和偏微分方程（PDE）的界面取得了惊人的成就，最终确定了调和测度相对于（n− 1）维集合的Hausdorff测度绝对连续的充分必要条件。在一些非常非正式的术语中，问题如下。定义域边界的子集E的调和测度ωX（E）是布朗旅行者从X∈开始穿过集合E而不是其补集的概率。著名的1924年维纳准则确定了所有边界点的调和函数是连续的，因此，调和措施是经典定义良好。然而，定量信息，也就是说，所得到的概率是否与集合E的豪斯多夫测度合理相关的问题，换句话说，布朗旅行者是否根据它们的大小看到边界的部分，结果要微妙得多。在偏微分方程中，它等价于这样一个问题：对于某个p，Dirichlet边值问题是否适定，（而不是连续的）数据，与适当的依赖解的Lp大小的边界上的数据[27]。这是相当值得注意的是，关键的几何概念在这方面已经确定在1916年，Riesz [29]证明了在以可求长曲线为界的单连通平面区域中调和测度关于Lebesgue测度是绝对连续的。可求正性是一个集合可以被一个可数的Lipschitz图集合覆盖的性质，模是测度为零的子集。将这一结果扩展到更高的维度花了世纪的时间，并发展了调和分析、奇异积分和一致可求长集上的电晕分解技术。我们不打算在本介绍中提供详细的概述，但让我们提到，关键的里程碑可能是Dahlberg在[8]中对Lipschitz域的处理，然后在[23]和[20]中对具有一致可求长边界的2边和1边NTA域的结果，然后，最后，发现了最近在[1]中确定的必要和充分的几何条件。其中一个主要问题是，一致可求长性是不够的[2]，因为，此外，域必须表现出某种定量连通性，并且
Spectacular achievements of the past 20 years at the interface of harmonic analysis, geometric measure theory, and partial diffferential equations (PDEs) have finally identified the necessary and sufficient conditions for the absolute continuity of harmonic measure with respect to the Hausdorff measure of an (n− 1)-dimensional set. In some very informal terms, the problem is as follows. The harmonic measure of a subset E of the boundary of a domain, ωX (E), is the probability that a Brownian traveler, starting at X∈, would exit through the set E rather than its complement. The celebrated 1924 Wiener criterion has identified all boundary points where the harmonic functions are continuous and, hence, the harmonic measure is classically well defined. However, the quantitative information, that is, the question whether the resulting probability is reasonably related to the Hausdorff measure of the set E, in other words, whether the Brownian travelers see the portions of the boundary in accordance with their size, turned out to be much more delicate. In PDE terms, it is equivalent to the question whether for some p, the Dirichlet boundary value problem is well-posed with the Lp (rather than continuous) data, with the appropriate dependence of solutions on the Lp size of the data on the boundary [27].It is quite remarkable that the key geometric notion in this context was identified already in 1916, when Riesz [29] proved that the harmonic measure is absolutely continuous with respect to the Lebesgue measure in a simply connected planar domain bounded by a rectifiable curve. Rectifiability is the property that the set can be covered by a countable collection of Lipschitz graphs, modulo a subset of measure zero. Extending this result to higher dimensions took more than a century and a development of harmonic analysis, singular integrals, and corona decomposition techniques on uniformly rectifiable sets. We do not aim to provide a detailed overview in this introduction, but let us mention that the key milestones were perhaps Dahlberg’s treatment of Lipschitz domains in [8], then results on 2-sided and 1-sided NTA domains with uniformly rectifiable boundaries in [23] and [20], and then, finally, the discovery of necessary and sufficient geometric conditions that were recently identified in [1]. One of the main problems was that the uniform rectifiability is not sufficient [2], for, in addition, the domain has to exhibit some quantitative connectedness, and the
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