The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition in uniform domains with Ahlfors regular boundaries. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular \(L^p\) data case. The first step is taken in Part I, where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant, using a novel application of techniques developed in geometric measure theory. In Part II we establish the final result, that is, the “large constant case”. The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

本文建立了一类PDE的解的性质与欧几里得空间中集合的几何之间的对应关系。我们解决了椭圆测度相对于表面测度的（定量）绝对连续性与边界的均匀可校正性是否相等的问题，在散度形式的最优类中，椭圆算子在Ahlfors的均匀域中满足合适的Carleson测度条件规则边界。可以将结果视为适合于奇数\（L ^ p \）数据情况的维纳准则的定量模拟。第一步是在第一部分中进行的，其中我们考虑了这样一种情况：系数上的所需Carleson测度条件保持足够小的常数，使用在几何测量理论中开发的技术的新颖应用。在第二部分中，我们确定了最终结果，即“大常数情况”。关键要素是强大的外推论证，它提供了一种自我完善的尺度不变小常数估计的一般途径，以及一种在域及其子域之间转移椭圆度量的定量绝对连续性的新机制。