Schauder estimates for parabolic nondivergence operators of Hormander type

Let X-1, X-2,..., X-q be a system of real smooth vector fields satisfying Hormander's rank condition in a bounded domain Omega of R-n. Let A = {a(ij)(t, x)}(i,j=1)(q) be a symmetric, uniformly positive definite matrix of real functions defined in a domain U subset of R x Omega. For operators of kind H = partial derivative t - Sigma(q)(i,j=1) (t, x) XiXj - Sigma(q)(i=1) 6i (t, x) X-i - c(t, x) we prove local a-priori estimates of Schauder-type, in the natural (parabolic) C-k,C-alpha(U) spaces defined by the vector fields X-i and the distance induced by them. Namely, for a(ij), b(i), c is an element of C-k,C-alpha (U) and U' subset of U, we prove parallel to u parallel to(Ck+2.alpha(U')) <= c{parallel to Hu parallel to(Ck.alpha(U)) + parallel to u parallel to(L infinity(U))}. (c) 2006 Elsevier Inc. All rights reserved.