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\(\Phi\)-entropy inequalities and asymmetric covariance estimates for convex measures. (English) Zbl 1428.62189

Summary: In this paper, we use the semi-group method and an adaptation of the \(L^2\)-method of Hörmander to establish some \(\Phi\)-entropy inequalities and asymmetric covariance estimates for the strictly convex measures in \(\mathbb{R}^n\). These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The \(\Phi\)-entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.

MSC:

60E15 Inequalities; stochastic orderings
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[1] Arnaudon, M., Bonnefont, M. and Joulin, A. (2018). Intertwinings and generalized Brascamp-Lieb inequalities. Rev. Mat. Iberoam.34 1021-1054. · Zbl 1428.60112 · doi:10.4171/RMI/1014
[2] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math.1123 177-206. Berlin: Springer. · Zbl 0561.60080
[3] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Cham: Springer. · Zbl 1376.60002
[4] Bakry, D., Gentil, I. and Scheffer, G. (2018). Sharp Beckner-type inequalities for Cauchy and spherical distributions. Preprint. Available atarXiv:1804.03374. · Zbl 1471.60045
[5] Beckner, W. (1989). A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc.105 397-400. · Zbl 0677.42020
[6] Blanchet, A., Bonforte, M., Dolbeault, J., Grillo, G. and Vázquez, J.-L. (2007). Hardy-Poincaré inequalities and applications to nonlinear diffusions. C. R. Math. Acad. Sci. Paris344 431-436. · Zbl 1190.35119
[7] Blanchet, A., Bonforte, M., Dolbeault, J., Grillo, G. and Vázquez, J.L. (2009). Asymptotics of the fast diffusion equation via entropy estimates. Arch. Ration. Mech. Anal.191 347-385. · Zbl 1178.35214 · doi:10.1007/s00205-008-0155-z
[8] Bobkov, S.G. and Ledoux, M. (2009). Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab.37 403-427. · Zbl 1178.46041 · doi:10.1214/08-AOP407
[9] Bolley, F. and Gentil, I. (2010). Phi-entropy inequalities for diffusion semigroups. J. Math. Pures Appl. (9) 93 449-473. · Zbl 1193.47046 · doi:10.1016/j.matpur.2010.02.004
[10] Bonforte, M., Dolbeault, J., Grillo, G. and Vázquez, J.L. (2010). Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proc. Natl. Acad. Sci. USA107 16459-16464. · Zbl 1256.35026 · doi:10.1073/pnas.1003972107
[11] Bonnefont, M., Joulin, A. and Ma, Y. (2016). A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions. ESAIM Probab. Stat.20 18-29. · Zbl 1355.60103 · doi:10.1051/ps/2015019
[12] Brascamp, H.J. and Lieb, E.H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal.22 366-389. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[13] Carlen, E.A., Cordero-Erausquin, D. and Lieb, E.H. (2013). Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures. Ann. Inst. Henri Poincaré Probab. Stat.49 1-12. · Zbl 1270.26016 · doi:10.1214/11-AIHP462
[14] Cattiaux, P., Guillin, A. and Wu, L.-M. (2011). Some remarks on weighted logarithmic Sobolev inequality. Indiana Univ. Math. J.60 1885-1904. · Zbl 1270.26017 · doi:10.1512/iumj.2011.60.4405
[15] Chafaï, D. (2004). Entropies, convexity, and functional inequalities: On \(\Phi \) -entropies and \(\Phi \) -Sobolev inequalities. J. Math. Kyoto Univ.44 325-363. · Zbl 1079.26009
[16] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math.97 1061-1083. · Zbl 0318.46049 · doi:10.2307/2373688
[17] Hörmander, L. (1965). \(L^2\) estimates and existence theorems for the \(\bar{\partial}\) operator. Acta Math.113 89-152. · Zbl 0158.11002
[18] Menz, G. and Otto, F. (2013). Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab.41 2182-2224. · Zbl 1282.60096 · doi:10.1214/11-AOP715
[19] Nguyen, V.H. (2014). Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa’s theorem. J. Funct. Anal.266 931-955. · Zbl 1292.26055 · doi:10.1016/j.jfa.2013.11.003
[20] Scheffer, G. (2003). Local Poincaré inequalities in non-negative curvature and finite dimension. J. Funct. Anal.198 197-228. · Zbl 1019.58011
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