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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 thestandard Poincar\'e inequality. The assumption on the measure is the fact thatit satisfies the classical Poincar\'e inequality, so that our result is animprovement of the latter inequality. Moreover we also quantify the tightnessat infinity provided by the control on the fractional derivative in terms of aweight growing at infinity. The proof goes through the introduction of thegenerator of the Ornstein-Uhlenbeck semigroup and some careful estimates of itspowers. To our knowledge this is the first proof of fractional Poincar\'einequality for measures more general than L\'evy measures.

Fractional Poincar\'e inequalities for general measures