Square Functions and the $A_\infty$ Property of Elliptic Measures
  In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal Carleson measure estimate for bounded solutions of such operators implies a regularity result for the associated elliptic measure. 
C. Kenig, B. Kirchheim, J. Pipher and T.Toro
math.AP