Schauder estimates for parabolic nondivergence operators of Hörmander type

Let X1,X2,…,XqX1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω   of RnRn. Let A={aij(t,x)}i,j=1q be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂R×ΩU⊂R×Ω. For operators of kindH=∂t−∑i,j=1qaij(t,x)XiXj−∑i=1qbi(t,x)Xi−c(t,x) we prove local a-priori estimates of Schauder-type, in the natural (parabolic) Ck,α(U)Ck,α(U) spaces defined by the vector fields XiXi and the distance induced by them. Namely, for aijaij, bibi, c∈Ck,α(U)c∈Ck,α(U) and U′⋐UU′⋐U, we prove‖u‖Ck+2,α(U′)⩽c{‖Hu‖Ck,α(U)+‖u‖L∞(U)}.‖u‖Ck+2,α(U′)⩽c{‖Hu‖Ck,α(U)+‖u‖L∞(U)}.