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

We consider a class of hypoelliptic ultraparabolic operators in the form L = mΣj=1 Xj2 + X0 - &partt, under the assumption that the vector fields X1, Xm and X0 - &partt are invariant with respect to a suitable homogeneous Lie group. We show that if u,v are two solutions of Lu = 0 on N ×]0, T[ and u(&middot, 0) = ψ, then each of the following conditions: |u(x, t) - v(x, t)| can be bounded by Mexp (c|x| 2), or both u and v are non negative, implies u = v. We use a technique which relies on a pointwise estimate of the fundamental solution of L.