FROM LOG SOBOLEV TO TALAGRAND: A QUICK PROOF

Nicola Gigli (1) , Michel Ledoux (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincare' inequality.

Mots clés: Metric measure spaces; log-Sobolev inequality; Talagrand inequality

Domaines: Géométrie métrique [math.MG]; Analyse fonctionnelle [math.FA]

Nicola Gigli, Michel Ledoux. FROM LOG SOBOLEV TO TALAGRAND: A QUICK PROOF. Discrete and Continuous Dynamical Systems - Series A, 2013, pp.dcds.2013.33.xx. ⟨hal-00769384⟩