Mass Transport and Variants of the Logarithmic Sobolev Inequality
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Adams, R.A.: General logarithmic Sobolev inequalities and Orlicz embeddings. J. Func. Anal. 34, 292–303 (1979)
Agueh, M., Ghoussoub, N., Kang, X.: Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14, 215–244 (2004)
Aida, S.: Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158, 152–185 (1998)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2005)
Ané, C., Blachère, S., Chafai, D., Fougère, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Panorama et Synthèses, vol. 10, S.M.F., Paris (2002)
Artstein, S., Klartag, B., Milman, V.: The Santalo point of a function, and a functional form of Santalo inequality. Mathematika 51, 33–48 (2004)
Bakry, D.: L’hypercontractivité et son utilisation en théorie des semi groupes. In: Ecole d’eté de Probabilités de Saint-Flour. Lect. Notes Math., vol. 1581, pp. 1–114. Springer, Berlin (1994)
Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. (Available at http://arxiv.org/abs/math/0703355 )
Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités XIX. Lect. Notes in Math., vol. 1123, pp. 179–206. Springer, Berlin (1985)
Bakry, D., Qian, Z.: Volume comparison theorems without Jacobi fields. In: Current Trends in Potential Theory. Theta Ser. Adv. Math., vol. 4, pp. 115–122. Theta, Bucharest (2005)
Bakry, D., Ledoux, M.: Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123, 259–281 (1996)
Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115(3), 463–482 (1994)
Barthe, F.: Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. (6) 10(3), 393–404 (2001)
Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22(3), 993–1067 (2006)
Barthe, F., Roberto, C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159(3), 481–497 (2003)
Barthe, F., Roberto, C.: Modified logarithmic Sobolev inequalities on ℝ. Preprint
Beckner, W.: A generalized Poincaré inequalities for Gaussian measures. Proc. Am. Math. Soc. 105, 397–400 (1989)
Beckner, W.: Geometric asymptotics and the logarithmic Sobolev inequality. Forum Math. 11(1), 105–137 (1999)
Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)
Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequality. J. Funct. Anal. 163, 1–28 (1999)
Bobkov, S.G., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001)
Bobkov, S.G., Houdré, C.: Isoperimetric constants for product probability measures. Ann. Probab. 25(1), 184–205 (1997)
Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Am. Math. Soc. 129(616) (1997)
Bobkov, S.G., Ledoux, M.: Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Relat. Fields 107(3), 383–400 (1997)
Bobkov, S.G., Ledoux, M.: From Brunn-Minkovsky to Brascamp-Lieb and to logarithmic Sobolev inequality. Geom. Funct. Anal. 10, 1028–1052 (2000)
Bobkov, S.G., Zegarlinski, B.: Entropy bounds and isoperimetry. Mem. Am. Math. Soc. 176(829), 1–69 (2005)
Bogachev, V.I., Kolesnikov, A.V.: On the Monge–Ampère equation in infinite dimensions. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(4), 547–572 (2005)
Caffarelli, L.A.: Monotonicity properties of optimal transportation and the FKG and related inequalities. Commun. Math. Phys. 214(3), 547–563 (2000)
Carlen, E., Loss, M.: Logarithmic Sobolev inequalities and spectral gaps. In: Recent Advances in the Theory and Applications of Mass Transport. Contemp. Math., vol. 353, pp. 53–60. Am. Math. Soc., Providence (2004)
Cattiaux, P.: Hypercontractivity for perturbed diffusion semigroups. Ann. Fac. Sci. Toulouse Math. 14(4), 609–628 (2005)
Chafaï, D.: Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities. J. Math. Kyoto Univ. 44(2), 325–363 (2004)
Cordero-Erausquin, D.: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257–269 (2002)
Cordero-Erausquin, D., Gangbo, W., Houdré, C.: Inequalities for generalized entropies and optimal transportation. In: Recent Advances in the Theory and Applications of Mass Transport. Contemp. Math., vol. 353, pp. 73–94. Am. Math. Soc., Providence (2004)
Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001)
Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Ann. Fac. Sci. Toulouse Math. (6) 15(4), 613–635 (2006)
Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004)
Gentil, I.: From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality. Preprint (2006)
Gentil, I., Guillin, A., Miclo, L.: Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Relat. Fields 133(3), 409–436 (2005)
Gentil, I., Guillin, A., Miclo, L.: Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23(1), 237–260 (2007)
Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXVI. Lecture Notes in Math., vol. 1801, pp. 1–134. Springer, Berlin (2003)
Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. In: Dell’Antonio, G., Mosco, U. (eds.) Dirichlet Forms. Lect. Notes Math., vol. 1563, pp. 54–88. Springer, Berlin (1993)
Kolesnikov, A.V.: Convexity inequalities and optimal transport of infinite-dimensional measures. J. Math. Pures Appl. 83(11), 1373–1404 (2004)
Kolesnikov, A.V.: Modified log-Sobolev inequalities and isoperimetry. Rend. Lincei Mat. Appl. 18(2), 179–208 (2007)
Kusuoka, S., Stroock, D.: Some boundedness properties of certain stationary diffusion semigroups. J. Func. Anal. 60, 243–264 (1985)
Latała, R., Oleskiewicz, K.: Between Sobolev and Poincaré. In: Geometric aspects of functional analysis. Lecture Notes in Math., vol. 1745, pp. 147–168. Springer, Berlin (2000)
Ledoux, M.: A simple analytic proof of an inequality by P. Buser. Proc. Am. Math Soc. 121, 951–959 (1994)
Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. Am. Math. Soc., Providence (2001)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979). Function spaces
Naor, A., Peres, Y., Schramm, O., Sheffield, S.: Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J. 134(1), 165–197 (2006)
Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems. V. I, II. Springer, New York (1998)
Roberto, C., Zegarliński, B.: Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243(1), 28–66 (2007)
Röckner, M., Wang, F.Y.: Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185, 564–603 (2001)
Ros, A.: The isoperimetric problem. In: Global Theory of Minimal Surfaces. Clay Math. Proc., vol. 2, pp. 175–209. Am. Math. Soc., Providence (2005)
Rothaus, O.S.: Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64, 296–313 (1985)
Rosen, J.: Sobolev inequalities for weight spaces and supercontractivity. Trans. Am. Math. Soc. 222, 367–376 (1976)
Schmuckenscläger, M.: Abstract Prékopa-Leindler type inequalities. Unpublished manuscript
Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)
Wang, F.-Y.: Logarithmic Sobolev inequalities on non-compact Riemannian manifolds. Probab. Theory Relat. Fields. 109, 417–424 (1997)
Wang, F.-Y.: Functional inequalities for empty essential spectrum. J. Func. Anal. 170, 219–245 (2000)
Wang, F.-Y.: Logarithmic Sobolev inequalities: conditions and counterexamples. J. Oper. Theory 46, 183–197 (2001)
Wang, F.-Y.: A generalization of Poincaré and log-Sobolev inequalities. Potential Anal. 22(1), 1–15 (2005)
Wang, F.-Y.: Orlicz-Poincaré inequalities. Preprint (2006) (Available at http://math.bnu.edu.cn/~wangfy/ )