论文标题
Carleson测量估计值，电晕分解和椭圆操作员的扰动
Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity
论文作者
论文摘要
令$ω$为一个开放式套装，具有满足开瓶器条件的AHLFORS-DAVID常规边界。当$ω$以某种定量形式连接时，人们可以确定，对于任何具有有界系数的实际椭圆运算符，椭圆度度量的定量绝对连续性等同于所有有界的null解决方案都满足Carleson测量估计值的事实。反过来，在相同的情况下，这些等效属性在Fefferman-Kenig-pipher扰动下是稳定的。但是，没有连接性，就没有可用的Fefferman-Kenig-pipher扰动结果。在本文中，我们与与椭圆度度量相关的电晕分解工作，并表明它等同于有限的空溶液满足部分/弱的Carleson测量估计值，或者绿色函数与Corona Wentes中与边界的距离相当的事实。这种特征具有深远的后果。我们将Fefferman-Kenig-Pipher的扰动扩展到非连接的设置。对于Laplacian，这些电晕分解或等效地，部分/弱的Carleson测量估计值足够有意义，可以表征边界的均匀重构性。结果，我们获得了集合的边界，如果有界的空溶液对于任何Fefferman-kenig-pipher扰动，则可以统一地整理。对于Kenig-Pipher算子，在系数矩阵的换位或对称性下，表征的任何特性都是稳定的。结果，我们获得了Carleson测量的估计值，即在且仅当开放式设置的边界均匀整合时，就会发生满足$ l^1 $ kenig-pipher条件的非对称变量运算符的有界空溶液。我们的结果概括了先前的工作在定量连通性的设置中。
Let $Ω$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $Ω$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, we work with a corona decomposition associated with the elliptic measure and show that it is equivalent to the fact that bounded null solutions satisfy partial/weak Carleson measure estimates, or to the fact that the Green function is comparable to the distance to the boundary in the corona sense. This characterization has profound consequences. We extend Fefferman-Kenig-Pipher's perturbation to non-connected settings. For the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary. As a consequence, we obtain that the boundary of the set is uniformly rectifiable if bounded null solutions for any Fefferman-Kenig-Pipher perturbation of the Laplacian satisfy Carleson measure estimates. For Kenig-Pipher operators any of the properties of the characterization is stable under transposition or symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates for bounded null-solutions of non-symmetric variable operators satisfying an $L^1$-Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Our results generalize previous work in settings where quantitative connectivity.
