Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

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

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Abstract

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 $\mathbb{R}^{1}$. We prove a generalization of both by extending these inequalities to $L^{p}$ and $L^{q}$ norms and on $\mathbb{R}^{n}$, for any $n\geq1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.

Résumé

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 $\mathbb{R}^{1}$. Nous prouvons une généralisation de ces deux résultats, avec une extension de ces inégalités à des normes $L^{p}$ et $L^{q}$ dans $\mathbb{R}^{n}$, pour tout $n\geq1$. Nous prouvons aussi une inégalité pour des intégrales de différences divisées de fonctions à l’aide des intégrales de leurs gradients.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 1-12.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470123

Digital Object Identifier
doi:10.1214/11-AIHP462

Mathematical Reviews number (MathSciNet)
MR3060145

Zentralblatt MATH identifier
1270.26016

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Convexity Log-concavity Poincaré inequality

Citation

Carlen, Eric A.; Cordero-Erausquin, Dario; Lieb, Elliott H. Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 1--12. doi:10.1214/11-AIHP462. https://projecteuclid.org/euclid.aihp/1359470123


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References

  • [1] S. G. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999) 1903–1921.
  • [2] S. Bobkov and M. Ledoux. From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000) 1028–1052.
  • [3] T. Bodineau and B. Helffer. The log-Sobolev inequalities for unbounded spin systems. J. Funct. Anal. 166 (1999) 168–178.
  • [4] H. J. Brascamp and E. H. Lieb. On extensions of the Brunn–Minkovski and Prékopa–Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (1976) 366–389.
  • [5] D. Cordero-Erausquin. On Berndtsson’s generalization of Prékopa’s theorem. Math. Z. 249 (2005) 401–410.
  • [6] N. Grunewald, F. Otto, C. Villani and M. G. Westdickenberg. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302–351.
  • [7] O. Guédon. Kahane–Khinchine type inequalities for negative exponent. Mathematika 46 (1999) 165–173.
  • [8] L. Hörmander. $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator. Acta Math. 113 (1965) 89–152.
  • [9] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995) 541–559.
  • [10] C. Landim, G. Panizo and H. T. Yau. Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. Henri Poincaré Prob. Stat. 38 (2002) 739–777.
  • [11] E. H. Lieb and M. Loss. Analysis, 2nd edition. Amer. Math. Soc., Providence, RI, 2001.
  • [12] G. Menz and F. Otto. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Preprint, 2011.
  • [13] F. Otto and M. G. Reznikoff. A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243 (2007) 121–157.