Fractional Poincaré inequalities for general measures

Clément Mouhot, Emmanuel Russ, Yannick Sire

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19 Scopus citations

Abstract

We prove a fractional version of Poincaré inequalities in the context of Rn endowed with a fairly general measure. Namely we prove a control of an L2 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. © 2010 Elsevier Masson SAS.
Original languageEnglish (US)
Pages (from-to)72-84
Number of pages13
JournalJournal de Mathématiques Pures et Appliquées
Volume95
Issue number1
DOIs
StatePublished - Jan 2011
Externally publishedYes

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