HAL: hal-00179865, version 5

arXiv: 0710.3139

DOI: 10.1016/j.jfa.2008.09.002

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

Available versions:

v1 (2007-10-16)

v2 (2007-11-21)

v3 (2008-03-26)

v4 (2008-05-07)

v5 (2008-09-05)

On gradient bounds for the heat kernel on the Heisenberg group

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

(2008-10-15)

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.

1:	Institut de Mathématiques de Toulouse (IMT)
	Université Paul Sabatier - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées de Toulouse – CNRS : UMR5219
2:	Physiopathologie et Toxicologie Expérimentales (UPTE)
	INRA : UR0181 – Ecole Nationale Vétérinaire de Toulouse

Subject

:

Mathematics/Probability

Keyword(s) : Heat kernel – Heisenberg group – functional inequalities – hypoelliptic diffusions

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From: Djalil Chafai <>

Submitted on: Friday, 5 September 2008 12:41:43

Updated on: Tuesday, 25 November 2008 18:26:15