Fundamental solutions and local solvability for nonsmooth H\"ormander's operators

Abstract

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 H\"ormander's vector fields of step r such that the highest order commutators are only H\"older 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 H\"older continuous f. We also prove $C_{X,loc}^{2,\alpha}$ estimates on this solution.