Schauder estimates for parabolic nondivergence operators of Hörmander type

Let $X_{1},X_{2},\ldots,X_{q}$ be a system of real smooth vector fields satisfying H\"{o}rmander's rank condition in a bounded domain $\Omega$ of $\mathbb{R}^{n}$. Let $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ be a symmetric, uniformly positive definite matrix of real functions defined in a domain $U\subset\mathbb{R}\times\Omega$. For operators of kind \[ H=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}\left( t,x\right) X_{i}X_{j}-\sum _{i=1}^{q}b_{i}\left( t,x\right) X_{i}-c\left( t,x\right) \] we prove local a-priori estimates of Schauder-type, in the natural (parabolic) $C^{k,\alpha}\left( U\right) $ spaces defined by the vector fields $X_{i}$ and the distance induced by them. Namely, for $a_{ij},b_{i},c\in$ $C^{k,\alpha}\left( U\right) $ and $U^{\prime}\Subset U,$ we prove% \[ \left\Vert u\right\Vert _{C^{k+2,\alpha}\left( U^{\prime}\right) }\leqslant c\left\{ \left\Vert Hu\right\Vert _{C^{k,\alpha}\left( U\right) }+\left\Vert u\right\Vert _{L^{\infty}\left( U\right) }\right\} . \]