Fractional Poincaré inequalities for general measures

Clément Mouhot (1, 2) , Emmanuel Russ (3) , Yannick Sire (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 DAMTP - Department of Applied Mathematics and Theoretical Physics
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We prove a fractional version of Poincaré inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.

Mots clés: Gaffney estimate; Poincaré inequality; fractional derivative; Lévy operator; Ornstein-Uhlenbeck; fractional Sobolev space; off-diagonal estimate; Gaffney estimate.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Clément Mouhot, Emmanuel Russ, Yannick Sire. Fractional Poincaré inequalities for general measures. Journal de Mathématiques Pures et Appliquées, 2011, 95 (1), pp.72-84. ⟨10.1016/j.matpur.2010.10.003⟩. ⟨hal-00435240v2⟩