  The present paper, along with its companion [Hofmann, Martell, Mayboroda,
Toro, Zhao, arXiv:1710.06157], 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. The result can be viewed as a quantitative analogue of the Wiener
criterion adapted to the singular $L^p$ data case.
  The first step in this direction was taken in our previous paper [Hofmann,
Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], where we considered the case
in which the desired Carleson measure condition on the coefficients holds with
sufficiently small constant. In this paper we establish the final, general
result, that is, the "large constant case". The key elements of our approach
are a powerful extrapolation argument, which provides a general pathway to
self-improve scale-invariant small constant estimates, as well as a new
mechanism to transfer quantitative absolute continuity of elliptic measure
between a domain and its subdomains.
