Skip to main content
SpringerLink
Log in

The Hölder-Poincaré duality for L q,p -cohomology

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We prove the following version of Poincaré duality for reduced L q,p -cohomology: For any 1 < q, p < ∞, the L q,p -cohomology of a Riemannian manifold is in duality with the interior L p',q'-cohomology for 1/p + 1/p′ = 1/q + 1/q′ = 1.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Brezis H.: Analyse fonctionnelle, Théorie et applications. Dunod, Paris (1999)

    Google Scholar 

  2. Carron G.: Cohomologie L 2 et parabolicité. J. Geom. Anal. vol. 15(3), 391–404 (2005)

    MathSciNet  Google Scholar 

  3. Eichhorn J.: Global analysis on open manifolds. Nova Science Publishers Inc., New York (2007)

    MATH  Google Scholar 

  4. Gol’dshtein V.M., Reshetnyak Yu.G.: Quasiconformal mappings and sobolev spaces. Kluwer Academic Publishers, Dordrecht (1990)

    MATH  Google Scholar 

  5. Gol’dshtein V., Ramm A.: Compactness of the embedding operators for rough domains. Math. Inequal. Appl. 4(1), 127–141 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Gol’dshtein V.M., Kuz’minov V.I., Shvedov I.A.: Differential forms on lipschitz manifolds. Siberian Math. J. 23(2), 16–30 (1984)

    MathSciNet  Google Scholar 

  7. Gol’dshtein V.M., Kuz’minov V.I., Shvedov I.A.: A Property of de Rham Regularization Operators. Siberian Math. J. 25(2), 104–111 (1984)

    MathSciNet  Google Scholar 

  8. Gol’dshtein V.M., Kuz’minov V.I., Shvedov I.A.: Dual spaces of spaces of differential forms. Siberian Math. J. 54(1), 35–43 (1986)

    Article  MathSciNet  Google Scholar 

  9. Gol’dshtein V., Troyanov M.: The Kelvin-Nevanlinna-Royden criterion for p−parabolicity. Math. Z. 232, 607–619 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gol’dshtein V., Troyanov M.: Sobolev inequality for differential forms and L q,p -cohomology. J. Geom. Anal. 16(4), 597–631 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Gol’dshtein V., Troyanov M.: A conformal de Rham complex. J. Geom. Anal. 20, 651–669 (2010). doi:10.1007/s12220-010-9119-2

    Article  MathSciNet  MATH  Google Scholar 

  12. Gol’dshtein V., Troyanov M.: Distortion of mappings and L q,p -cohomology. Math. Zeitschrift 264, 279–293 (2010). doi:10.1007/s00209-008-0463-x

    Article  MathSciNet  MATH  Google Scholar 

  13. Gol’dshtein V., Troyanov M.: A short proof of the Hölder-Poincaré duality for L p -cohomology. Rend. Sem. Mat. Univ. Padova 124, 179–184 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Gromov, M.: Asymptotic invariants of infinite groups. In: Niblo, G., Roller, M. (eds.) Geometric Group Theory. London Mathematical Society Lecture Note Series 182, University Press, Cambridge (1993)

  15. Holopainen I.: Volume growth, Green’s functions, and parabolicity of ends. Duke Math. J. 97(2), 319–346 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kopylov Y.: L p, q -cohomology and normal solvability. Arch. Math. 89(1), 87–96 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kopylov Y.: L p,q -cohomology of some warped cylinders. Ann. Math. Blaise Pascal 16(2), 321–338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuz’minov V.I., Shvedov I.A. Homological aspects of the theory of Banach complexes. Sib. Math. J. 40(4), 754–763 (1999) (translation from Siberian Mat. Zh. 40(4), 893–904 (1999))

  19. Li X.D.: On the strong L p-Hodge decomposition over complete Riemannian manifolds. J. Funct. Anal. 257(11), 36173646 (2009)

    Article  Google Scholar 

  20. Li X.D.: Sobolev inequalities on forms and L q,p -cohomology on complete Riemannian Manifolds. J. Geom. Anal. 20(2), 354–387 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maz’ya V.: Sobolev Spaces. Springer Verlag, Berlin (1985)

    Google Scholar 

  22. Pansu P.: Cohomologie L p et pincement. Comment. Math. Helv. 83(1), 327–357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pansu, P.: L p-cohomology of symmetric spaces. Geometry, analysis and topology of discrete groups, 305326, Advanced Mathematics Lecture Notes (ALM), vol. 6, International Press, Somerville (2008)

  24. Pigola S., Rigoli M., Setti A.: Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Progress in Mathematics, vol. 266. Birkhuser Verlag, Basel (2008)

    Google Scholar 

  25. Pigola, S., Setti, A., Troyanov, M.: The topology at infinity of a manifold supporting an L q,p-Sobolev inequality. arXiv:1007.1761

  26. Poborchii S., Maz’ya V.G.: Differentiable functions on bad domains. World Scient, Singapore (1997)

    Google Scholar 

  27. Troyanov M.: Parabolicity of manifolds. Siberian Adv. Math. 9, 125–150 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Troyanov M.: On the Hodge decomposition in \({{\mathbb R^n}}\) . Mosc. Math. J. 9(4), 899926 (2009)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer Sheva, Israel

    Vladimir Gol’dshtein

  2. Section de Mathamatiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Marc Troyanov

Authors
  1. Vladimir Gol’dshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Troyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Troyanov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gol’dshtein, V., Troyanov, M. The Hölder-Poincaré duality for L q,p -cohomology. Ann Glob Anal Geom 41, 25–45 (2012). https://doi.org/10.1007/s10455-011-9269-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-011-9269-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation