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The present paper, along with its companion [Hofmann, Martell, Mayboroda,Toro, Zhao, arXiv:1710.06157], establishes the correspondence between theproperties of the solutions of a class of PDEs and the geometry of sets inEuclidean space. We settle the question of whether (quantitative) absolutecontinuity of the elliptic measure with respect to the surface measure anduniform rectifiability of the boundary are equivalent, in an optimal class ofdivergence form elliptic operators satisfying a suitable Carleson measurecondition. The result can be viewed as a quantitative analogue of the Wienercriterion 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 casein which the desired Carleson measure condition on the coefficients holds withsufficiently small constant. In this paper we establish the final, generalresult, that is, the "large constant case". The key elements of our approachare a powerful extrapolation argument, which provides a general pathway toself-improve scale-invariant small constant estimates, as well as a newmechanism to transfer quantitative absolute continuity of elliptic measurebetween a domain and its subdomains.

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case