Résumé en anglais

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 commutativity 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 show that this new inequality is close to the symmetrized version of the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li on the Heisenberg group, seen as a weighted inequality. We show furthermore that a semigroup approach can produce such weighted inequalities.