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Let X 1 , X 2 , . . . , X q be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω of R n . Let A = {a ij (t, x)} q i,j =1 be a symmetric, uniformly positive definite matrix of real functions defined in a domain U ⊂ R × Ω. For operators of kindwe prove local a-priori estimates of Schauder-type, in the natural (parabolic) C k,α (U ) spaces defined by the vector fields X i and the distance induced by them. Namely, for a ij , b i , c ∈ C k,α (U ) and U U , we prove u C k+2,α (U ) c H u C k,α (U ) + u L ∞ (U ) .

We consider a class of degenerate Ornstein-Uhlenbeck operators in R N , of the kindTo get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in R N +1 and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).

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