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In this note, we derive a new logarithmic Sobolev inequality for the heatkernel on the Heisenberg group. The proof is inspired from the historicalmethod of Leonard Gross with the Central Limit Theorem for a random walk. Herethe non commutative nature of the increments produces a new gradient whichnaturally involves a Brownian bridge on the Heisenberg group. This newinequality contains the optimal logarithmic Sobolev inequality for the Gaussiandistribution in two dimensions. We compare this new inequality with thesub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the morerecent inequality of Fabrice Baudoin and Nicola Garofalo obtained using ageneralized curvature criterion. Finally, we extend this inequality to the caseof homogeneous Carnot groups of rank two.

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