• Bernoulli
  • Volume 25, Number 4A (2019), 3090-3108.

$\Phi$-entropy inequalities and asymmetric covariance estimates for convex measures

Van Hoang Nguyen

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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.

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Bernoulli, Volume 25, Number 4A (2019), 3090-3108.

Received: April 2018
Revised: September 2018
First available in Project Euclid: 13 September 2019

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$\Phi$-entropy inequalities asymmetric covariance estimates Beckner type inequalities Brascamp–Lieb type inequalities convex measures $L^{2}$-method of Hörmander Poincaré type inequalities semi-group


Nguyen, Van Hoang. $\Phi$-entropy inequalities and asymmetric covariance estimates for convex measures. Bernoulli 25 (2019), no. 4A, 3090--3108. doi:10.3150/18-BEJ1082.

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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.
  • [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.
  • [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.
  • [4] Bakry, D., Gentil, I. and Scheffer, G. (2018). Sharp Beckner-type inequalities for Cauchy and spherical distributions. Preprint. Available atarXiv:1804.03374.
  • [5] Beckner, W. (1989). A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 397–400.
  • [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. Paris 344 431–436.
  • [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.
  • [8] Bobkov, S.G. and Ledoux, M. (2009). Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 403–427.
  • [9] Bolley, F. and Gentil, I. (2010). Phi-entropy inequalities for diffusion semigroups. J. Math. Pures Appl. (9) 93 449–473.
  • [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. USA 107 16459–16464.
  • [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.
  • [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.
  • [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.
  • [14] Cattiaux, P., Guillin, A. and Wu, L.-M. (2011). Some remarks on weighted logarithmic Sobolev inequality. Indiana Univ. Math. J. 60 1885–1904.
  • [15] Chafaï, D. (2004). Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities. J. Math. Kyoto Univ. 44 325–363.
  • [16] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061–1083.
  • [17] Hörmander, L. (1965). $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator. Acta Math. 113 89–152.
  • [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.
  • [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.
  • [20] Scheffer, G. (2003). Local Poincaré inequalities in non-negative curvature and finite dimension. J. Funct. Anal. 198 197–228.