Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators

We consider a class of second order ultraparabolic differential equations in the form partial derivative(t)u = Sigma X-m(i,j=1)i(a(ij)X(j)u) + X(0)u, where A = (a(ij)) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, under the assumption that the operator Sigma X-m(i=1)i(2) + X-0 - partial derivative(t) is hypoelliptic and the vector fields X-1,..., X-m and X-0 - partial derivative(t) are invariant with respect to a suitable homogeneous Lie group. We adapt the Moser's iterative methods to the non-Euclidean geometry of the Lie groups and we prove an L-loc(infinity) bound of the solution u in terms of its L-loc(p) norm. We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator Sigma X-m(i,j=1)i(a(ij)X(j)) + X-0 - partial derivative(t). The bound is given in terms of the value function of an optimal control problem related to the vector fields X-1,..., X-m and X-0 - partial derivative(t). Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients a(ij). (C) 2007 Elsevier Inc. All rights reserved.