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

We consider a family of vector fields Xi = Sigma(p)(j=1)b(ij) (x)partial derivative(xj) (i = 1, 2, ... , n; n < p) defined in some bounded domain Omega subset of R-p and assume that the X-i satisfy Hormander's rank condition of some step r in Omega, and b(ij) is an element of Cr-1(<(Omega)over bar>). We extend to this nonsmooth context some results which are well known for smooth Hormander'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 b(ij) is an element of Cr-1((Omega) over bar), Poincar'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 Hormander's vector fields: Sigma(n)(i,j=1) X-i(*)(a(ij)(x)X-j), where {a(ij)} is a uniformly elliptic matrix of L infinity(Omega) functions.