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On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 2, pp. 335-355.

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.

Dans cette note, nous obtenons une inégalité de Sobolev logarithmique nouvelle pour le noyau de la chaleur sur le groupe de Heisenberg. La preuve est inspirée de la méthode historique de Leonard Gross à base de théorème limite central pour une marche aléatoire. Ici la nature non commutative des incréments produit un nouveau gradient qui fait intervenir naturellement un pont brownien sur le groupe de Heisenberg. Cette nouvelle inégalité contient l’inégalité de Sobolev logarithmique optimale pour la mesure gaussienne en deux dimensions. Nous comparons cette nouvelle inégalité avec l’inégalité sous-elliptique de Hong-Quan Li et avec les inégalités plus récentes de Fabrice Baudoin et Nicola Garofalo obtenues avec un critère de courbure généralisé. Enfin nous étendons notre inégalités au cas des groupes de Carnot homogène de rang deux.

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DOI: 10.5802/afst.1633
Classification: 22E30,  35R03,  35A23,  60J65
Keywords: Heisenberg group, Heat kernel, Brownian Motion, Poincaré inequality, Logarithmic Sobolev inequality, Random Walk, Central Limit Theorem
Michel Bonnefont 1; Djalil Chafaï 2; Ronan Herry 3

1 Institut de Mathématiques de Bordeaux, Université de Bordeaux, France
2 CEREMADE, Université Paris-Dauphine, PSL, IUF, France
3 Université du Luxembourg et Université Paris-Est Marne-la-Vallée, France
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Michel Bonnefont; Djalil Chafaï; Ronan Herry. On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 2, pp. 335-355. doi : 10.5802/afst.1633. https://afst.centre-mersenne.org/articles/10.5802/afst.1633/

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