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arXiv · 1607.02741

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

Michel Bonnefont·Djalil Chafaï·Ronan Herry
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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.

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Michel Bonnefont, Djalil Chafaï, Ronan Herry. 2020-09-09. On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group. https://doi.org/10.5802/afst.1633

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