Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures

Carlen, Eric A.; Cordero-Erausquin, Dario; Lieb, Elliott H.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 1-12.

Une inégalité de Brascamp et Lieb donne une estimation sur la covariance entre deux fonctions par rapport à une mesure log-concave, qui est bornée par le produit des normes $L^2$ des gradients des fonctions, où l’amplitude du gradient est calculée en utilisant un produit scalaire égal à l’inverse de la matrice Hessienne du potentiel de la mesure log-concave. Menz et Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (2011) Preprint] ont prouvé une variante de ce résultat où les normes $L^2$ sont remplacées par des normes $L^1$ et $L^{\infty }$, mais seulement dans ${ℝ}^1$. Nous prouvons une généralisation de ces deux résultats, avec une extension de ces inégalités à des normes $L^p$ et $L^q$ dans ${ℝ}^n$, pour tout $n\ge 1$. Nous prouvons aussi une inégalité pour des intégrales de différences divisées de fonctions à l’aide des intégrales de leurs gradients.

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (2011) Preprint] proved a variant of this with the two $L^2$ norms replaced by $L^1$ and $L^{\infty }$ norms, but only for ${ℝ}^1$. We prove a generalization of both by extending these inequalities to $L^p$ and $L^q$ norms and on ${ℝ}^n$, for any $n\ge 1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.

Classification : 26D10
Mots clés : convexity, log-concavity, poincaré inequality

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