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

TL;DR

This paper establishes a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group, inspired by classical methods and extending to Carnot groups, with comparisons to existing inequalities.

Contribution

It introduces a novel logarithmic Sobolev inequality involving a new gradient based on Brownian bridges, extending results to Carnot groups of rank two.

Findings

The inequality contains the optimal Gaussian case in two dimensions.

Comparison with existing inequalities shows improvements or generalizations.

Extension to homogeneous Carnot groups of rank two is achieved.

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.