A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator
Robert A. Fefferman · Journal of the American Mathematical Society · 1989
for some A > O and for all x E Q and 4 E R n . We also assume a =a For such operators, the Dirichlet problem is solvable in Q if and only if it is solvable for the Laplace operator, according to a theorem of Littman, Stampacchia and Weinberger [1]. This means that if Q C R n is a sufficiently nice bounded region (the unit ball, B, is an example) and f is a given continuous function on the boundary of Q, then there exists a unique function u, continuous on Q, so that L(u) = 0 in Q and u = f on aQ. Let us assume, for convenience, that the origin belongs to Q. Then the mapping f C f(aQ) -* u(O) is a positive linear functional so there exists a unique nonnegative measure w on aQ such that for every f E (aQ),