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 ∂tu = m i,j=1 Xi(aijXj u) + X0u, where A = (aij ) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, under the assumption that the operator m i=1 X2 i + X0 − ∂t is hypoelliptic and the vector fields X1, . . . , Xm and X0 − ∂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 bound of the solution u in terms of its L p loc norm. We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator m i,j=1 Xi(aijXj )+X0 −∂t . The bound is given in terms of the value function of an optimal control problem related to the vector fields X1, . . . , Xm and X0 − ∂t . Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients aij . © 2007 Elsevier Inc. All rights reserved