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.
Cite
@article{arxiv.1607.02741,
title = {On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group},
author = {Michel Bonnefont and Djalil Chafaï and Ronan Herry},
journal= {arXiv preprint arXiv:1607.02741},
year = {2020}
}