Generalized Sobolev-Morrey estimates for hypoelliptic operators on homogeneous groups

Let G = (R-N, o, delta(lambda)) be a homogeneous group, Q is the homogeneous dimension of G, X-0, X-1,..., X-m be left invariant real vector fields on G and satisfy Hormander's rank condition on R-N. Assume that X-1,..., X-m (m <= N - 1) are homogeneous of degree one and X-0 is homogeneous of degree two with respect to the family of dilations (delta(lambda))(lambda>0). Consider the following hypoelliptic operator with drift on G