On gradient bounds for the heat kernel on the Heisenberg group

Dominique Bakry (1) , Fabrice Baudoin (1) , Michel Bonnefont (1) , Djalil Chafai (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UPTE - Physiopathologie et Toxicologie Expérimentales

It is known that the couple formed by the two dimensional Brownian motion and its Lévy 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.

Mots clés: Heat kernel; Heisenberg group; functional inequalities; hypoelliptic diffusions

Domaines: Probabilités [math.PR]

Dominique Bakry, Fabrice Baudoin, Michel Bonnefont, Djalil Chafai. On gradient bounds for the heat kernel on the Heisenberg group. Journal of Functional Analysis, 2008, 255 (8), pp.1905-1938. ⟨10.1016/j.jfa.2008.09.002⟩. ⟨hal-00179865v5⟩