Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators

We consider a class of hypoelliptic ultraparabolic operators in the form L = Sigma X-m(j=1)j(2) + X-0 - partial derivative(t), under the assumption that the vector fields X-1,...,X-m and X-0 - partial derivative(t) are invariant with respect to a suitable homogeneous Lie group G. We show that if u,v are two solutions of Lu = 0 on R-N x ]0,T[ and u(.,0) = phi, then each of the following conditions: vertical bar u(x,t) - v(x,t)vertical bar can be bounded by M exp (c vertical bar x vertical bar(2)(G)), or both u and v are non negative, implies u equivalent to v. We use a technique which relies on a pointwise estimate of the fundamental solution of L.