Fundamental solutions and local solvability for nonsmooth Hörmander'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 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 $C_{X,loc}^{2,\alpha }$ estimates on this solution.

75 pages, LaTeX