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From log Sobolev to Talagrand: A quick proof
1. | Laboratoire J. A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice, France |
2. | Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Heat Flow and calculus over spaces with Ricci curvature bounded from below - the compact case, To appear in Rend. Acc. Naz. Lnice, in memory of E. Magenes. |
[2] |
_______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, Preprint (2011), arXiv:1106.2090. |
[3] |
_______, Density of lipschitz functions and equivalence of weak gradients in metric measure spaces, Preprint (2011), arXiv:1111.3730 . |
[4] |
S. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80 (2001), 669-696.
doi: 10.1016/S0021-7824(01)01208-9. |
[5] |
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517.
doi: 10.1007/s000390050094. |
[6] |
N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Accepted paper at CPAM (2011), arXiv:1008.1319. |
[7] |
N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab., 37 (2009), 2480-2498.
doi: 10.1214/09-AOP470. |
[8] |
N. Gozlan and C. Léonard, Transport inequalities. A survey, Markov Process. Rel. Fields, 16 (2010), 635-736. |
[9] |
N. Gozlan, C. Roberto and P.-M. Samson, Characterization of Talagrand's transport-entropy inequalities in metric spaces, Preprint 2011. |
[10] |
J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232.
doi: 10.1090/S0273-0979-07-01140-8. |
[11] |
J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl. (9), 88 (2007), 219-229.
doi: 10.1016/j.matpur.2007.06.003. |
[12] |
F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[13] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.
doi: 10.1006/jfan.1999.3557. |
[14] |
M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.
doi: 10.1007/BF02249265. |
[15] |
C. Villani, "Optimal Transport," Grundlehren der Mathematischen Wissenschaften. [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009, Old and new.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Heat Flow and calculus over spaces with Ricci curvature bounded from below - the compact case, To appear in Rend. Acc. Naz. Lnice, in memory of E. Magenes. |
[2] |
_______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, Preprint (2011), arXiv:1106.2090. |
[3] |
_______, Density of lipschitz functions and equivalence of weak gradients in metric measure spaces, Preprint (2011), arXiv:1111.3730 . |
[4] |
S. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80 (2001), 669-696.
doi: 10.1016/S0021-7824(01)01208-9. |
[5] |
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517.
doi: 10.1007/s000390050094. |
[6] |
N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Accepted paper at CPAM (2011), arXiv:1008.1319. |
[7] |
N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab., 37 (2009), 2480-2498.
doi: 10.1214/09-AOP470. |
[8] |
N. Gozlan and C. Léonard, Transport inequalities. A survey, Markov Process. Rel. Fields, 16 (2010), 635-736. |
[9] |
N. Gozlan, C. Roberto and P.-M. Samson, Characterization of Talagrand's transport-entropy inequalities in metric spaces, Preprint 2011. |
[10] |
J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232.
doi: 10.1090/S0273-0979-07-01140-8. |
[11] |
J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl. (9), 88 (2007), 219-229.
doi: 10.1016/j.matpur.2007.06.003. |
[12] |
F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[13] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.
doi: 10.1006/jfan.1999.3557. |
[14] |
M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.
doi: 10.1007/BF02249265. |
[15] |
C. Villani, "Optimal Transport," Grundlehren der Mathematischen Wissenschaften. [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009, Old and new.
doi: 10.1007/978-3-540-71050-9. |
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