Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

摘要

We consider a family of vector fields defined in some bounded domain of R^p,and we assume that they satisfy Hormander's rank condition of some step r, andthat their coefficients have r-1 continuous derivatives. We extend to thisnonsmooth context some results which are well-known for smooth Hormander'svector fields, namely: some basic properties of the distance induced by thevector fields, the doubling condition, Chow's connectivity theorem, and, underthe stronger assumption that the coefficients belong to C^{r-1,1}, Poincare'sinequality. By known results, these facts also imply a Sobolev embedding. Allthese tools allow to draw some consequences about second order differentialoperators modeled on these nonsmooth Hormander's vector fields.