Hypoelliptic heat kernel inequalities on Lie groups

Details Hypoelliptic heat kernel inequalities on Lie groups Hypoelliptic heat kernel inequalities on Lie groups

2005-08-22

arxiv.org/abs/math/0508420v2

Stochastic Processes and their Applications, 118, no 3 p 368-388, 2008

Mathematics

Analysis of PDEs

22 pages, 0 figures; final journal version

Scientific paper

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.

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