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

We consider a set of smooth vector fields X_1,…,X_m and X0−∂t satisfying the Hoermander's hyipoellipticity condition, under the assumption that X_1,…,X_m and X0−∂t are invariant with respect to a suitable homogeneous Lie group. We consider the second order partial differential equations in divergence form, X_i (aij X_j) + X0−∂t, where A=(aij) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, and we prove an L^infty source bound of the solution u in terms of its L^1 norm, by adaptingt the Moser's iterative methods to the non-Euclidean geometry of the Lie group.We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator X_i (aij X_j) + X0−∂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 X0−∂t. Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients aij.