Interpolation between logarithmic Sobolev and Poincaré inequalities

Communications in Mathematical Sciences

Volumn 5, Issue 4, 2007, Pages 971-979

  Arnold, Anton ^a   Bartier, Jean Philippe ^b   Dolbeault, Jean ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

Functional inequalities; Hypercontractivity; Logarithmic Sobolev inequality; Poincar inequality; Spectral gap

Indexed keywords

EID: 39049090576     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2007.v5.n4.a12     Document Type: Article

References (16)

1. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Société Mathématique de France, 2000.
2. A. Arnold and J. Dolbeault, Refined convex Sobolev inequalities, J. Functional Analysis, 225(2), 337-351, 2005.
3. A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26(1-2), 43-100, 2001.
4. D. Bakry and M. Emery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sc. Paris Sér. I 299, 775-778, 1984.
5. F. Barthe, P. Cattiaux and C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana 22(3), 993-1067, 2006.
6. F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Studia Math., 159, 481-497, 2003.
7. W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105, 397-400, 1989.
8. S.G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Functional Analysis, 163, 1-28, 1999.
9. D. Chafaï, Entropies, convexity, and functional inequalities, J. Math. Kyoto Univ., 44, 325-363, 2004.
10. L. Gross, Logarithmic Sobolev inequalities, Amer. J. of Math., 97, 1061-1083, 1975.
11. R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric Aspects of Functional Analysis, Lecture Notes in Math., Springer, Berlin, 1745, 147-168, 2000.
12. M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math., 6, 305-366, 2000.
13. E. Nelson, The free Markoff field, J. Functional Analysis, 12, 211-227, 1973.
14. M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, Harcourt Brace Jovanovich Publishers, 1978.
15. F.Y. Wang, A generalization of Poincaré and log-Sobolev inequalities, Potential Analysis, 22, 1-15, 2005.
16. F.Y. Wang, Functional inequalities for empty essential spectrum, J. Functional Analysis, 170(1), 219-245, 2001.