]. S. Aid98 and . Aida, Uniform positivity improving property, Sobolev inequalities, and spectral gaps, J. Funct

[. Aida and I. Shigekawa, Logarithmic Sobolev Inequalities and Spectral Gaps: Perturbation Theory, Journal of Functional Analysis, vol.126, issue.2, pp.448-475, 1994.
DOI : 10.1006/jfan.1994.1154
URL : http://doi.org/10.1006/jfan.1994.1154

C. Ané, Sur les inégalités de Sobolev logarithmiques Panoramas et Synthèses [Panoramas and Syntheses, pp.217-219, 2000.

D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, pp.552978-552981, 2014.

I. [. Bobkov, M. Gentil, and . Ledoux, Hypercontractivity of Hamilton???Jacobi equations, Journal de Math??matiques Pures et Appliqu??es, vol.80, issue.7, pp.669-696, 2001.
DOI : 10.1016/S0021-7824(01)01208-9

F. [. Bobkov and . Götze, Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities, Journal of Functional Analysis, vol.163, issue.1, pp.1-28, 1999.
DOI : 10.1006/jfan.1998.3326
URL : http://doi.org/10.1006/jfan.1998.3326

A. [. Cattiaux, L. M. Guillin, and . Wu, A note on Talagrand's transportation inequality and logarithmic Sobolev inequality. Probab. Theory Related Fields 148, pp.285-304, 2010.

F. [. Chafaï and . Malrieu, On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.46, issue.1, pp.72-96, 2010.
DOI : 10.1214/08-AIHP309

P. Diaconis and L. Saloff-coste, Logarithmic Sobolev inequalities for finite Markov chains, The Annals of Applied Probability, vol.6, issue.3
DOI : 10.1214/aoap/1034968224

]. N. Goz10 and . Gozlan, Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Probab, Stat, vol.463, pp.708-739, 2010.

]. N. Goz+14, C. Gozlan, P. Roberto, P. Samson, and . Tetali, Kantorovich duality for general transport costs and applications

]. M. Led01a and . Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, Séminaire de Probabilités, pp.167-194, 2001.

M. Ledoux, The concentration of measure phenomenon Mathematical Surveys and Monographs, pp.181-181, 2001.

]. B. Mau91 and . Maurey, Some deviation inequalities, Geom. Funct. Anal, vol.1, issue.2, pp.188-197, 1991.

A. [. Menz and . Schlichting, Poincar?? and logarithmic Sobolev inequalities by decomposition of the energy landscape, The Annals of Probability, vol.42, issue.5, pp.1809-1884, 2014.
DOI : 10.1214/14-AOP908

E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound, Duke Mathematical Journal, vol.154, issue.2, pp.207-239, 2010.
DOI : 10.1215/00127094-2010-038
URL : http://arxiv.org/abs/0902.1560

F. Otto and C. Villani, Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality, Journal of Functional Analysis, vol.173, issue.2, pp.361-400, 2000.
DOI : 10.1006/jfan.1999.3557

]. G. Roy07 and . Royer, An initiation to logarithmic Sobolev inequalities SMF/AMS Texts and Monographs . Translated from the 1999 French original by Donald Babbitt, p.119, 2007.

]. A. Sch12 and . Schlichting, The Eyring-Kramers formula for Poincaré and logarithmic Sobolev inequalities, 2012.

[. Wang and J. Wang, Functional inequalities for convolution probability measures, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.52, issue.2
DOI : 10.1214/14-AIHP659
URL : http://arxiv.org/abs/1308.1713

D. Zimmermann, Bounds for logarithmic Sobolev constants for Gaussian convolutions of compactly supported measures

]. D. Zim13 and . Zimmermann, Logarithmic Sobolev inequalities for mollified compactly supported measures, J. Funct. Anal, vol.2656, pp.1064-1083, 2013.

]. D. Zim14b and . Zimmermann, Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on R

J. Bardetfr, L. , U. De-rouen, A. De-l-'université, and . Bp, e-mail: jean-baptiste.bardet(AT)univ-rouen, TECHNOPÔLE DU MADRILLET, vol.12

M. Florent, L. De, . Et, and . Théorique, mail: florent.malrieu(AT)univ-tours, FÉDÉRATION DENIS POISSON (FR CNRS 2964)

Z. Pierre-andré, e-mail: pierre-andre.zitt(AT)u-pem