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

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

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 1, page 1-12
  • ISSN: 0246-0203

Abstract

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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 ∞ 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 ≥ 1 . We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.

How to cite

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Carlen, Eric A., Cordero-Erausquin, Dario, and Lieb, Elliott H.. "Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 1-12. <http://eudml.org/doc/271994>.

@article{Carlen2013,
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 $n\ge 1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.},
author = {Carlen, Eric A., Cordero-Erausquin, Dario, Lieb, Elliott H.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {convexity; log-concavity; poincaré inequality; Poincaré inequality},
language = {eng},
number = {1},
pages = {1-12},
publisher = {Gauthier-Villars},
title = {Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures},
url = {http://eudml.org/doc/271994},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Carlen, Eric A.
AU - Cordero-Erausquin, Dario
AU - Lieb, Elliott H.
TI - Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 1
EP - 12
AB - 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 $n\ge 1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.
LA - eng
KW - convexity; log-concavity; poincaré inequality; Poincaré inequality
UR - http://eudml.org/doc/271994
ER -

References

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  1. [1] S. G. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab.27 (1999) 1903–1921. Zbl0964.60013MR1742893
  2. [2] S. Bobkov and M. Ledoux. From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal.10 (2000) 1028–1052. Zbl0969.26019MR1800062
  3. [3] T. Bodineau and B. Helffer. The log-Sobolev inequalities for unbounded spin systems. J. Funct. Anal.166 (1999) 168–178. Zbl0972.82035MR1704666
  4. [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. Zbl0334.26009MR450480
  5. [5] D. Cordero-Erausquin. On Berndtsson’s generalization of Prékopa’s theorem. Math. Z.249 (2005) 401–410. Zbl1079.32020MR2115450
  6. [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. Zbl1179.60068MR2521405
  7. [7] O. Guédon. Kahane–Khinchine type inequalities for negative exponent. Mathematika46 (1999) 165–173. Zbl0965.26011MR1750653
  8. [8] L. Hörmander. L 2 estimates and existence theorems for the ∂ ¯ operator. Acta Math.113 (1965) 89–152. Zbl0158.11002MR179443
  9. [9] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom.13 (1995) 541–559. Zbl0824.52012MR1318794
  10. [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. Zbl1022.60087MR1931585
  11. [11] E. H. Lieb and M. Loss. Analysis, 2nd edition. Amer. Math. Soc., Providence, RI, 2001. Zbl0873.26002MR1817225
  12. [12] G. Menz and F. Otto. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Preprint, 2011. Zbl1282.60096MR3098070
  13. [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. Zbl1109.60013MR2291434

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