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Potential Analysis

, 29:167 | Cite as

Modified Logarithmic Sobolev Inequalities on \(\mathbb{R}\)

  • F. Barthe
  • C. RobertoEmail author
Article

Abstract

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.

Keywords

Sobolev inequalities Concentration 

Mathematics Subject Classifications (2000)

26D10 60E15 

References

  1. 1.
    Ané, C., Blachère, S., Chafai, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques., vol. 10. Panoramas et Synthèses. S.M.F., Paris (2000)Google Scholar
  2. 2.
    Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and application to isoperimetry. Rev. Mat. Iberoamericana 22(3), 993–1067 (2006)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Barthe, F., Roberto, C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159(3), 481–497 (2003)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bobkov, S.G., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80(7), 669–696 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bobkov, S.G., Ledoux, M.: Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107, 383–400 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bobkov, S.G., Ledoux, M.: From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(5), 1028–1052 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bobkov, S.G., Zegarlinski, B.: Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176(829), x+69 (2005)MathSciNetGoogle Scholar
  9. 9.
    Cattiaux, P., Guillin, A.: On quadratic transportation cost inequalities. J. Math. Pures Appl. 86(4), 341–361 (2006)MathSciNetGoogle Scholar
  10. 10.
    Gentil, I., Guillin, A., Miclo, L.: Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133(3), 409–436 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gentil, I., Guillin, A., Miclo, L.: Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoamericana 23(1), 235–258 (2007)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Gozlan, N.: Characterizarion of Talagrand’s like transportation cost inequalities on the real line. ArXiv Preprint math.PR/0608241 (2006)Google Scholar
  13. 13.
    Gromov, M., Milman, V.: A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843–854 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kolesnikov, A.: Modified logarithmic Sobolev inequalities and isoperimetry. Rend. Lincei Mat. Appl. 18(2), 179–208 (2007)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXIII, number 1709 in Lecture Notes in Math., pp 120–216. Springer, Berlin (1999)Google Scholar
  16. 16.
    Miclo, L.: Quand est-ce que les bornes de Hardy permettent de calculer une constante de poincaré exacte sur la droite? Ann. Fac. Sci. Toulouse (2008, in press)Google Scholar
  17. 17.
    Muckenhoupt, B.: Hardy inequalities with weights. Studia Math. 44, 31–38 (1972)zbMATHMathSciNetGoogle Scholar
  18. 18.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Roberto, C.: Inégalités de hardy et de Sobolev logatihmiques. Thèse de doctorat de C. Roberto. PhD thesis, Université Paul Sabatier (2001)Google Scholar
  21. 21.
    Talagrand, M.: A new isoperimetric inequality and the concentration of measure phenomenon. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric Aspects of Functional Analysis, number 1469 in Lecture Notes in Math., pp. 94–124. Springer-Verlag, Berlin (1991)Google Scholar
  22. 22.
    Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81, 73–205 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Talenti, G.: Osservazioni sobre una classe di disuguaglianze. Rend. Sem. Mat. Fis. Milano 39, 171–185 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tomaselli, G.: A class of inequalities. Boll. Un. Mat. Ital. 21, 622–631 (1969)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Paul SabatierToulouse cedex 09France
  2. 2.Laboratoire d’Analyse et Mathématiques Appliquées- UMR 8050Universités de Marne la Vallée et de Paris 12-Val-de-MarneMarne la Vallée Cedex 2France

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