On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Abstract

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

References in corpus (1)

• On gradient bounds for the heat kernel on the Heisenberg group

Cited by in corpus (5)

• A note on gradient estimates for the heat semigroup on nonisotropic Heisenberg groups
• Sub-Riemannian geometry on some step-two Carnot groups
• A nonlocal elliptic problem on a Heisenberg group
• The log-Sobolev inequality for spin systems of higher order interactions
• Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups