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=∑i,j=1mXi(aijXju)+X0u, where A=(aij)A=(aij) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, under the assumption that the operator ∑i=1mXi2+X0−∂t is hypoelliptic and the vector fields X1,…,XmX1,…,Xm and X0−∂tX0−∂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 Lloc∞ bound of the solution u   in terms of its Llocp norm.We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator ∑i,j=1mXi(aijXj)+X0−∂t. The bound is given in terms of the value function of an optimal control problem related to the vector fields X1,…,XmX1,…,Xm and X0−∂tX0−∂t. Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients aijaij.