Hypoelliptic heat kernel inequalities on Lie groups

Abstract

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) 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 −∞ at points of degeneracy of the semi-Riemannian metric. 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 given by the sum of squares of left invariant vector fields. In particular, "L p -type" gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate implies a Poincaré estimate in this context.

Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg

“…2. Dans [21,23], les estimations (1.4) avec 1 < w < +∞ ont été obtenues dans le cadre des groupes nilpotents par une méthode probabiliste et par une méthode analytique respectivement. Cependant, dans leurs preuves (indépendantes), ils ont tous utilisé des propriétés spéciales du système de Hörmander des champs de vecteurs invariants à gauche sur les groupes de Lie.…”

“…De plus, Melcher a donné dans [23] quelques familles des fonctions sur lesquelles l'estimation précédente avec w = 1 est satisfaite.…”

Duality on gradient estimates and Wasserstein controls

“…Our work is strongly motivated by recent development on gradient estimates on a Lie group endowed with a sub-Riemannian structure [5,8,12,13,22,27]. To explain a consequence of our duality, we deal with the 3-dimensional Heisenberg group here.…”

Geometry of subelliptic diffusions

“…Connections between the probabilistic behaviour of subelliptic diffusions and analytic properties of the corresponding heat semigroups, most directly expressed in functional inequalities, have attracted a lot of attention [17,48,43,3]. For instance, denoting by P t f the (minimal) heat semigroup generated by…”

“…holds pointwise for any t > 0, see [17,48]. Note that the squared norm of the horizontal gradient ∇ hor f is given by…”