Skip to main content

Advertisement

SpringerLink
Log in

Capacitary Criteria for Poincaré-Type Inequalities

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

The Poincaré-type inequality is a unification of various inequalities including the F-Sobolev inequalities, Sobolev-type inequalities, logarithmic Sobolev inequalities, and so on. The aim of this paper is to deduce some unified upper and lower bounds of the optimal constants in Poincaré-type inequalities for a large class of normed linear (Banach, Orlicz) spaces in terms of capacity. The lower and upper bounds differ only by a multiplicative constant, and so the capacitary criteria for the inequalities are also established. Both the transient and the ergodic cases are treated. Besides, the explicit lower and upper estimates in dimension one are computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barthe, F., Cattiaux, P. and Roberto, C.: ‘Interpolated inequalities between exponential and Guassian, Orlicz hypercontractivity and isoperimetry’, arXiv:math.PR/0407219, 2004.

  2. Barthe, F. and Roberto, C.: ‘Sobolev inequalities for probability measures on the real line’, Studia Math. 159(3) (2003), 481–497.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  5. Chen, M.F.: ‘Variational formulas of Poincaré-type inequalities in Banach spaces of functions on the line’, Acta Math. Sin. Eng. Ser. 18(3) (2002), 417–436.

    Google Scholar 

  6. Chen, M.F.: ‘Variational formulas of Poincaré-type inequalities for birth–death processes’, Acta Math. Sin. Eng. Ser. 19(4) (2003), 625–644.

    Google Scholar 

  7. Chen, M.F. and Wang, F.Y.: ‘Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap’, Abstract. Chin. Sci. Bulletin 43(18) (1998), 1516–1519. Ann. Probab. 28(1) (2000), 235–257.

    Google Scholar 

  8. Deuschel, J.D. and Stroock, D.W.: Large Deviations, Academic Press, New York, 1989.

    Google Scholar 

  9. Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994.

  10. Fukushima, M. and Uemura, T.: ‘Capacitary bounds of measures and ultracontracitivity of time changed processes’, J. Math. Pure Appl. 82(5) (2003), 553–572.

    Google Scholar 

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

    Google Scholar 

  12. Gross, L.: Logarithmic Sobolev Inequalities and Contractivity of Semigroups, Lecture Notes in Math. 1563, 1993.

  13. Kaimanovich, V.A.: ‘Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators’, Potential Anal. 1 (1992), 61–82.

    Google Scholar 

  14. Ma, Z.M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer, 1992.

  15. Mao, Y.H.: ‘The logarithmic Sobolev inequalities for birth–death process and diffusion process on the line’, Chin. J. Appl. Probab. Statist. 18(1) (2002), 94–100.

    Google Scholar 

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

    Google Scholar 

  17. Maz’ya, V.G.: Sobolev Spaces, Springer, 1985.

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

    Google Scholar 

  19. Opic, B. and Kufner, A.: Hardy-type Inequalities, Longman, New York, 1990.

    Google Scholar 

  20. Rao, M.M. and Ren, Z.D.: Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.

    Google Scholar 

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

    Google Scholar 

  22. Vondraçek, Z.: ‘An estimate for the L2-norm of a quasi continuous function with respect to a smooth measure’, Arch. Math. 67 (1996), 408–414.

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Mu-Fa Chen

Authors
  1. Mu-Fa Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mu-Fa Chen.

Additional information

Mathematics Subject Classifications (2000)

60J55, 31C25, 60J35, 47D07.

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

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, MF. Capacitary Criteria for Poincaré-Type Inequalities. Potential Anal 23, 303–322 (2005). https://doi.org/10.1007/s11118-005-2609-3

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-005-2609-3

Keywords

Search

Navigation