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Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

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Abstract

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated.

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References

  1. Chen, M. F.:Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci. Chin. (A), 44(4), 409–418 (2001)

    MATH  Google Scholar 

  2. Chen, M. F.:Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. Chin. (A), 42(8), 805–815 (1999)

    MATH  Google Scholar 

  3. Chen, M. F., Wang, F. Y.:Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc., 349(3), 1239–1267( 1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kantorovich, L. V., Akilov, G. P.:Functional Analysis (2ed), Pergamon Press, Oxford, New York (1982)

  5. Bobkov, S. G., Götze, F.:Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal., 163, 1–28 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mao, Y. H.:The logarithmic Sobolev inequalities for birth-death process and diffusion process on the line. Chin. J. Appl. Prob. Statis., 18(1), 94–100 (2002)

    Google Scholar 

  7. Mao, Y. H.:Nash inequalities for Markov processes in dimension one. Acta Math. Sin. Eng. Ser., 18(1), 147–156 (2002)

    MATH  Google Scholar 

  8. Mao, Y. H.: On empty essential spectrum for Markov processes in dimension one, preprint (2000)

  9. Mao, Y. H.: On super-contractivity for Markov semigroups, preprint (2001)

  10. Gross, L.: Logarithmic Sobolev inequalities and contractivity of semigroups. LNM., 1563, (1993)

  11. Wang, F. Y.: Functional inequalities for empty essential spectrum. J. Funct. Anal., 170, 219–245 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gong, F. Z., Wang, F. Y.: Functional inequalities for uniformly integrable semigroups and application to essential spectrums. Forum. Math., 14, 293–313 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Muckenhoupt, B.: Hardy’s inequality with weights. Studia. Math., (XLIV), 31–38 (1972)

    MathSciNet  Google Scholar 

  14. Mazya, V. G.: Sobolev Spaces, Springer-Valerg (1985)

  15. Opic, B., Kufner, A.: Hardy-type Inequalities, Longman, New York (1990)

  16. Chen, M. F.:Explicit bounds of the first eigenvalue. Sci. in China, Ser. A, 43(10), 1051–1059 (2000)

    MATH  Google Scholar 

  17. Chen, M. F.: Logarithmic Sobolev inequality for symmetric forms. Sci. Chin, 43(6), 601–608 (2000)

    MATH  Google Scholar 

  18. Wang, F. Y., Sobolev type inequalities for general symmetric forms. Proc. Amer. Math. Soc., 128(12), 3675–3682 (2001)

    Article  Google Scholar 

  19. Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces, Marcel Dekker Inc., New York (1991)

  20. Rothaus, O. S.: Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal., 64, 296–313 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, Beijing, 100875, P. R. China

    Mu Fa Chen

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  1. Mu Fa Chen
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Correspondence to Mu Fa Chen.

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Research supported in part by NSFC (No. 10121101), 973 Project and RFDP

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Chen, M.F. Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line. Acta Math Sinica 18, 417–436 (2002). https://doi.org/10.1007/s10114-002-0176-8

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  • DOI: https://doi.org/10.1007/s10114-002-0176-8

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