Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that Q(x)...... is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q.u) = b, we find sufficient conditions in order that Q....u has some first order smoothness. Specifically, if . is a bounded open set in Rn, we study when the components of Q....u belong to the first order Sobolev space W 1,2Q(.) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X i u has some first order smoothness if u is a weak solution in . of . ni=1 X .i X i u + b = 0. We do not assume that {X i } is a Hörmander collection of vector fields in .. The results signal ones for more general equations.