  We consider divergence form elliptic operators L = - div A(x)\nabla, defined
in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is
bounded, measurable, uniformly elliptic, t-independent, and not necessarily
symmetric. We establish square function/non-tangential maximal function
estimates for solutions of the homogeneous equation Lu = 0, and we then combine
these estimates with the method of "\epsilon-approximability" to show that
L-harmonic measure is absolutely continuous with respect to surface measure
(i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant
sense: more precisely, that it belongs to the class A_\infty with respect to
surface measure (equivalently, that the Dirichlet problem is solvable with data
in L^p, for some p < \infty). Previously, these results had been known only in
the case n = 1.
