Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

Abstract

() defined in some bounded domain and assume that the Xi satisfy Hörmander's rank condition of some step r in , and . We extend to this nonsmooth context some results which are well known for smooth Hörmander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption , Poincaré's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow us to draw some consequences about second order differential operators modeled on these nonsmooth Hörmander's vector fields: