Fundamental solutions and local solvability for nonsmooth Hörmander's operators

We consider operators of the form $L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of R^p where X_0, X_1,...,X_n are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method we construct a local fundamental solution γ for L and provide growth estimates for γ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that γ also possesses second derivatives, and we deduce the local solvability of L, constructing, by means of γ, a solution to Lu=f with Hölder continuous f. We also prove $C_{X,loc}^{2,α}$ estimates on this solution.

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