LP regularity of the Dirichlet problem for elliptic equations with singular drift

Let L-0 and L-1 be two elliptic operators in nondivergence form, with coefficients A(l) and drift terms b(l), l = 0, 1 satisfying sup vertical bar A(0) (Y) - A(1) (y)vertical bar(2) + delta (X)(2) vertical bar b(0) (Y) - b(1) (y)vertical bar(2)/delta(X) dX vertical bar Y-X vertical bar <= delta(X)/2 is a Carleson measure in a Lipschitz domain Omega subset of Rn+1, n >= 1, (here delta (X) = dist (X, partial derivative Omega)). If the harmonic measure d omega(L0) is an element of A infinity, then d omega(L1), is an element of A infinity. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases.