Fundamental solutions and local solvability for nonsmooth Hörmander’s operators

The authors consider operators of the form L=\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \mathbb{R}^{p} where X_{0},X_{1},\ldots,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 the authors 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 the authors prove that \gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of \gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,\alpha} estimates on this solution.