Fractional Poincaré inequalities for general measures

Details Fractional Poincaré inequalities for general measures Fractional Poincaré inequalities for general measures

2009-11-24

arxiv.org/abs/0911.4563v2

Journal de Math\'ematiques Pures et Appliqu\'es (2010) To appear

Mathematics

Analysis of PDEs

18 pages. Version 2 includes corrections of 2 misprints and an additionnal reference [BBCG08] providing a weaker condition (1.

Scientific paper

We prove a fractional version of Poincar\'e 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 Poincar\'e inequality. The assumption on the measure is the fact that it satisfies the classical Poincar\'e 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 Poincar\'e inequality for measures more general than L\'evy measures.

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