Skip to main content
SpringerLink
Log in

On local Poincaré via transportation

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

It is shown that curvature-dimension bounds CD(N,K) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L 1-Poincaré-inequality provided (X,d) has m-almost surely no branching points.

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. Ambrosio L. and Tilli P. (2004). Topics on analysis in metric spaces. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Cheeger J. and Colding T.H. (1996). Lower bounds on Ricci curvature and the almost rigidity of warped. Ann. Math. (2) 144(1): 189–237

    Article  MATH  MathSciNet  Google Scholar 

  3. Cordero-Erausquin D., McCann R.J. and Schmuckenschläger M. (2001). A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2): 219–257

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheeger J. (1999). Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3): 428–517

    Article  MATH  MathSciNet  Google Scholar 

  5. Heinonen J. (2001). Lectures on analysis on metric spaces. Universitext. Springer, New York

    Google Scholar 

  6. Hajłasz P. and Koskela P. (2000). Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688): x+101

    Google Scholar 

  7. Lott, J., Villani, C.: Ricci curvature for metric measure spaces via optimal transportation. Ann. Math. (to appear)

  8. Ohta, S.-I.: On measure contraction property of metric measure spaces. Comment. Math. Helv. (to appear)

  9. Otsu Y. and Shioya T. (1994). The Riemannian structure of Alexandrov spaces. J. Differ. Geom. 39(3): 629–658

    MATH  MathSciNet  Google Scholar 

  10. Ranjbar-Motlagh, A.: On the Poincaré inequality for abstract spaces. Preprint, Sharif University of Technology, Teheran (2002)

  11. Sturm K.-T. and Renesse M.-K. (2004). Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math. 58(7): 923–940

    Article  Google Scholar 

  12. Saloff-Coste L. (2002). Aspects of Sobolev-type inequalities. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  13. Sturm K.-T. (1996). Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75(3): 273–297

    MATH  MathSciNet  Google Scholar 

  14. Sturm K.-T. (1998). Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26(1): 1–55

    Article  MATH  MathSciNet  Google Scholar 

  15. Sturm K.-T. (2006). On the geometry of metric measure spaces I. Acta Math. 196(1): 65–131

    Article  MATH  MathSciNet  Google Scholar 

  16. Sturm K.-T. (2006). On the geometry of metric measure spaces II. Acta Math. 196(1): 133–177

    Article  MATH  MathSciNet  Google Scholar 

  17. Villani C. (2003). Topics in optimal transportation. American Mathematical Society, Providence

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Max-K. von Renesse

Authors
  1. Max-K. von Renesse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Max-K. von Renesse.

Additional information

Work supported by the Alexander von Humboldt-Foundation (AvH).

Rights and permissions

Reprints and permissions

About this article

Cite this article

von Renesse, MK. On local Poincaré via transportation. Math. Z. 259, 21–31 (2008). https://doi.org/10.1007/s00209-007-0206-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-007-0206-4

Keywords

Search

Navigation