Hypoelliptic heat kernel inequalities on Lie groups

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature” takes on the value −∞$-\infty$ at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting.