Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
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平方函数估计、BMO 狄利克雷问题以及低维集上调和测度的绝对连续性
DOI：
10.2140/apde.2019.12.1597
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发表时间：
2018
期刊：
影响因子：
2.2
通讯作者：
Zihui Zhao
中科院分区：
文献类型：
--
作者：
S. Mayboroda;Zihui Zhao
在最近的工作中[DFM1，DFM2]G.David，J.Feneuil和第一作者推出了一个程序，致力于低维集合的调和测度的模拟。与经典情形下的椭圆型偏微分方程组类似，一类相关的偏微分方程组是由退化的线性方程给出的，退化程度适当地依赖于到边界的距离。
本文继续这一研究方向，从边值问题的可解性出发，重点研究了新定义的调和测度相对于Hausdorff测度的定量绝对连续性的判据。特别地，在A_Infty(Sigma)中的基本假设下，建立了具有低维边界的区域BMO中Dirichlet问题的平方函数估计和可解性。更一般地，证明了在所有具有Ahlfors正则边界的区域中，Dirichlet问题的BMO可解性是调和测度绝对连续的充要条件。
In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary.
The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, $\omega\in A_\infty(\sigma)$, in terms of solvability of boundary value problems. The authors establish, in particular, square function estimates and solvability of the Dirichlet problem in BMO for domains with lower-dimensional boundaries under the underlying assumption $\omega\in A_\infty(\sigma)$. More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.
影响因子：
1.7
作者：
David, Guy;Feneuil, Joseph;Mayboroda, Svitlana
通讯作者：
Mayboroda, Svitlana
具有 Ahlfors-David 正则边界的单边 NTA 域上的可修正性和椭圆测度
影响因子：
1.3
作者：
Akman, Murat;Badger, Matthew;Hofmann, Steve;Martell, Jose Maria
通讯作者：
Martell, Jose Maria