A Criterion for the Absolute Continuity of the Harmonic Measure Associated with an Elliptic Operator

for some l > 0 and for all x E 0 and C; E R" . We also assume a;j = aj;. For such operators, the Dirichlet problem is solvable in n if and only if it is solvable for the Laplace operator, according to a theorem of Littman, Stampacchia and Weinberger [I]. This means that if n ~ R" is a sufficiently nice bounded region (the unit ball, B, is an example) and J is a given continuous function on the boundary of 0, then there exists a unique function u, continuous on n, so that L(u) = 0 in nand u = J on on. Let us assume, for convenience, that the origin belongs to O. Then the mapping J E ~(on) -> u(O) is a positive linear functional so there exists a unique nonnegative measure w on 00 such that for every J E ~(on), r J dw = u(O). Jun This measure w is called the harmonic measure associated to L. It is often important for applications to know whether or not w is absolutely continuous with respect to the surface measure da on the boundary of n. If this is the case, it is also of interest to know how nice the Radon-Nikodym derivative dw/da (the Poisson kernel) is. In recent years, several results have been found to answer these questions. First, according to a result of Caffarelli, Fabes, and Kenig [2] there exist elliptic