On gradient bounds for the heat kernel on the Heisenberg group
By D. Bakry, Fabrice Baudoin, Michel Bonnefont, Djalil CHAFAÏ. 2007
116 citations recorded by the source corpus.
It is known that the couple formed by the two dimensional Brownian motion and its Levy 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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/4d4393bcd7c24b98da629ef3b1c9a7b37f600b63
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In Applied Mathematics
- Fabrice Baudoin
- Luigi Ambrosio
- Kazumasa Kuwada
- Michel Bonnefont
- Nicola Gigli
- Patricia Alonso-Ruiz
- Luke G. Rogers
- Li Chen
Citations in the corpus
- Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality (2011, background): “rom the diffusion operator and fails to hold even for simple subelliptic diffusion operators like the sub-Laplacian on the Heisenberg group (see [J]). However in the past few years, numerous works like [BBBC], [BBBQ], [BB], [Bo], [DM], [E], [IKZ], [Li] and [Q] have shown on some examples that the heat”
- Duality on gradient estimates and Wasserstein controls (2009, background): “1) is established in the following cases; • The case q = 1 with K1(t) ≡ K for some K > 0 on groups of type H [13] (including the Heisenberg group of arbitrary dimension, see [5, 22] also).”
- Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “icci curvature (in that case C = 1, see [95]), some metric graphs like the Walsh spider (see [18, Example 5.1] and also [18, Theorem 5.4])), the Heisenberg group and more generally H-type groups (see [12,30]). The quasi Bakry-Emery curvature condition, while stronger than the weak Bak´ ry-Emery cur-”
- The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds (2008, methodology): “[7], [ 5 ] and the references therein): H can be represented as R 3 endowed with the polynomial”