On gradient bounds for the heat kernel on the Heisenberg group

Abstract

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.

Minor corrections

References in corpus (4)

• Entropies, convexity, and functional inequalities
• Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue
• Hypoelliptic heat kernel inequalities on Lie groups
• Isoperimetry and Rough Path Regularity

Cited by in corpus (35)

• Precise estimates for the subelliptic heat kernel on H-type groups
• Gradient estimates for the subelliptic heat kernel on H-type groups
• Subelliptic Li-Yau estimates on three dimensional model spaces
• Generalized Curvature Condition for Subelliptic Diffusion Processes
• Logarithmic Sobolev inequalities for infinite dimensional Hörmander type generators on the Heisenberg group
• On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group
• Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups
• Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups
• Markov semigroups with hypocoercive-type generator in Infinite Dimensions: Ergodicity and Smoothing
• Coercive Inequalities and U-Bounds
• Dimension free Harnack inequalities on $\RCD(K, \infty)$ spaces
• A note on gradient estimates for the heat semigroup on nonisotropic Heisenberg groups
• Exponential ergodicity of infinite system of interating diffusions
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Diffusions under a local strong Hörmander condition. Part II: tube estimates
• Diffusions under a local strong Hörmander condition. Part I: density estimates
• Existence of equilibrium for Infinite System of Interacting Diffusions
• Hypoelliptic heat kernel inequalities on H-type groups
• Degenerate SDEs with Singular Drift and Applications to Heisenberg Groups
• Derivative Formula and Gradient Estimates for Gruschin Type Semigroups
• On gradient estimates of the heat semigroups on step-two Carnot groups
• Stochastic analysis on sub-Riemannian manifolds with transverse symmetries
• Ergodicity of Markov Semigroups with Hörmander type generators in Infinite Dimensions
• Positive curvature property for sub-Laplace on nilpotent Lie group of rank two
• Sub-Riemannian geometry on some step-two Carnot groups
• Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators
• Duality on gradient estimates and Wasserstein controls
• Log-Harnack Inequality for Gruschin Type Semigroups
• Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
• A Bakry-Émery criterion for weighted contractivity and $L^2$-Hardy inequalities
• Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups
• Hypoelliptic diffusions with singular drifts
• Derivative Formulae and Poincaré Inequality for Kohn-Laplacian Type Semigroups
• The subelliptic heat kernels on SL(2,R) and on its universal covering $\widetilde{SL(2,R)}$: integral representations and some functional inequalities
• From U-bounds to isoperimetry with applications to H-type groups