Schauder estimates for parabolic nondivergence operators of Hörmander type

Let X1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmanderʹs rank condition in a bounded domain Ω of . Let be a symmetric, uniformly positive definite matrix of real functions defined in a domain . For operators of kind we prove local a-priori estimates of Schauder-type, in the natural (parabolic) Ck,α(U) spaces defined by the vector fields Xi and the distance induced by them. Namely, for aij, bi, c Ck,α(U) and U′ U, we prove u Ck+2,α(U′) c{ Hu Ck,α(U)+ u L∞(U)}.