  This paper discusses the existence of gradient estimates for second order
hypoelliptic heat kernels on manifolds. It is now standard that such
inequalities, in the elliptic case, 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 associated to the operator. For this reason, the
standard proofs for the elliptic theory fail in the hypoelliptic setting.
  This paper presents recent results for hypoelliptic operators. Malliavin
calculus methods transfer the problem to one of determining certain infinite
dimensional estimates. Here, the underlying manifold is a Lie group, and the
hypoelliptic operators are invariant under left translation. In particular,
"L^p-type" gradient estimates hold for p\in(1,\infty), and the p=2 gradient
estimate implies a Poincar\'e estimate in this context.
