On gradient bounds for the heat kernel on the Heisenberg group

Details On gradient bounds for the heat kernel on the Heisenberg group On gradient bounds for the heat kernel on the Heisenberg group

2007-10-16

arxiv.org/abs/0710.3139v5

Journal of Functional Analysis 255, 8 (2008) 1905-1938

Mathematics

Probability

Minor corrections

Scientific paper

10.1016/j.jfa.2008.09.002

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.

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