A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Let X = {X-1, X-2,..., X-m} be a system of smooth vector fields in R-n satisfying the Hormander's finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Caratheodory space G associated to system X(1/integral(BR) K(x) dx integral(BR) vertical bar u vertical bar(t) K (x) dx)(1/t) <= C R (1/integral(BR) 1/K(x) dx integral(BR) vertical bar Xu vertical bar(2)/K(x) dx)(1/2),where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt's class A(2) and Gehring's class G tau, where tau is a suitable exponent related to the homogeneous dimension.