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Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures. (English. French summary) Zbl 1270.26016
Summary: 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. G. Menz and F. Otto [“Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential”, Preprint (2011)] 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.

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
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[1] S. G. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999) 1903-1921. · Zbl 0964.60013 · doi:10.1214/aop/1022677553
[2] S. Bobkov and M. Ledoux. From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000) 1028-1052. · Zbl 0969.26019 · doi:10.1007/PL00001645
[3] T. Bodineau and B. Helffer. The log-Sobolev inequalities for unbounded spin systems. J. Funct. Anal. 166 (1999) 168-178. · Zbl 0972.82035 · doi:10.1006/jfan.1999.3419
[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. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[5] D. Cordero-Erausquin. On Berndtsson’s generalization of Prékopa’s theorem. Math. Z. 249 (2005) 401-410. · Zbl 1079.32020 · doi:10.1007/s00209-004-0704-6
[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. · Zbl 1179.60068 · doi:10.1214/07-AIHP200 · eudml:78025
[7] O. Guédon. Kahane-Khinchine type inequalities for negative exponent. Mathematika 46 (1999) 165-173. · Zbl 0965.26011 · doi:10.1112/S002557930000766X
[8] L. Hörmander. \(L^{2}\) estimates and existence theorems for the \(\bar{\partial}\) operator. Acta Math. 113 (1965) 89-152. · Zbl 0158.11002 · doi:10.1007/BF02391775
[9] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995) 541-559. · Zbl 0824.52012 · doi:10.1007/BF02574061 · eudml:131379
[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. · Zbl 1022.60087 · doi:10.1016/S0246-0203(02)01108-1 · numdam:AIHPB_2002__38_5_739_0 · eudml:77731
[11] E. H. Lieb and M. Loss. Analysis , 2nd edition. Amer. Math. Soc., Providence, RI, 2001. · Zbl 0966.26002
[12] G. Menz and F. Otto. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Preprint, 2011. · Zbl 1282.60096
[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. · Zbl 1109.60013 · doi:10.1016/j.jfa.2006.10.002
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