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 ) .

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We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

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We consider a family of vector fields