Fundamental solutions and local solvability for nonsmooth Hormander's operators

We consider operators sum of squareo vector fields in a bounded domain where the vector fields are nonsmooth Hormander's vector fields of step r such that the highest order commutators are only holder continuous. Applying Levi's parametrix method we construct a local fundamental solution Gamma for L and provide growth estimates for Gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that Gamma also possesses second derivatives, and we deduce the local solvability of L, constructing, by means of Gamma, a solution to Lu = f with holder continuous f. We also prove C^2,alpha_loc estimates on this solution.