Mathematische Zeitschrift

, 264:279 | Cite as

Distortion of mappings and L q, p -cohomology

  • Vladimir Gol’dshtein
  • Marc Troyanov


We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the L q,p -cohomology of these manifolds. We apply these results to the L q,p -cohomology of a manifold with a cusp.


Lq,p-cohomology Differential forms Distortion of mappings 

Mathematics Subject Classification (2000)

58A12 30C66 


  1. 1.
    Gol’dshtein, V.M., Kuz’minov, V.I., Shvedov, I.A.: Differential forms on Lipschitz Manifolds. Siberian Math. J. 23(2), 16–30 (1982). English translation in Siberian Math. J. 23(2), 151–161 (1982)Google Scholar
  2. 2.
    Gol’dshtein, V.M., Kuz’minov, V.I., Shvedov, I.A.: L p-cohomology of warped cylinder. Siberian Math. J. 31(6), 55–63 (1990). English translation in Siberian Math. J. 31(6), 716–727 (1990)Google Scholar
  3. 3.
    Gol’dshtein V., Gurov L., Romanov A.: Homeomorphisms that induce monomorphisms of Sobolev spaces. Israel J. Math. 91(1), 31–60 (1995)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gol’dshtein, V.M., Romanov, A.S.: Transformations that preserve Sobolev spaces 25(3), 382–388 (1984)Google Scholar
  5. 5.
    Gol’dshtein, V., Troyanov, M.: The L pq-cohomology of SOL. Ann. Fac. Sci. Toulouse Vii(4) (1998)Google Scholar
  6. 6.
    Gol’dshtein V., Troyanov M.: Sobolev inequality for differential forms and L q,p-cohomology. J. Geom. Anal. 16(4), 597–631 (2006)MATHMathSciNetGoogle Scholar
  7. 7.
    Gol’dshtein, V., Troyanov, M.: A conformal de Rham complex. arXiv:0711.1286Google Scholar
  8. 8.
    Gol’dshtein, V., Troyanov, M.: On the naturality of exterior differential arXiv:0801.4295. Math. Rep. Can. Acad. Sci. (to appear)Google Scholar
  9. 9.
    Gromov, M.: Asymptotic Invariants of Infinite Groups in Geometric Group Theory, vol. 2. London Math. Soc. Lecture Notes 182, Cambridge University Press (1992)Google Scholar
  10. 10.
    Heinonen J., Koskela P.: Sobolev mappings with integrable distortion. Arch. Rat. Mech. Anal. 125, 81–97 (1993)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hencl S., Koskela P.: Mapping of finite distortion: openess and discretness for quasilight mappings. Ann. Inst. H. Poincaré 22, 331–342 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kauhanen J., Koskela P., Maly J.: Mappings of finite distortion: discretness and openness. Arch. Rat. Mech. Anal. 160(2), 135–151 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kopylov Y.A.: L q,p-Cohomology and normal solvability. Arch. Math. 89(1), 87–96 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kopylov, Y.A.: L p,q-Cohomology of warped cylinders. arXiv:0803.3298v1Google Scholar
  15. 15.
    Lelong-Ferrand L.: Etude d’une classe d’applications liées à des homomorphismes d’algèbres de fonctions et généralisant les quasi-conformes. Duke Math. J. 40, 163–186 (1973)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lück W.: L 2-Invariants: Theory and Applications to Geometry and K-Theory. Springer, Berlin (2002)Google Scholar
  17. 17.
    Maz’ya V.G.: Spaces. Springer, Heidelberg (1985)Google Scholar
  18. 18.
    Maz’ya V.G., Shaposhnikova T.: Theory of Multipliers in Spaces of Differentiable Functions. Pitman, London (1985)MATHGoogle Scholar
  19. 19.
    Pansu P.: Difféomorphismes de p-dilatation bornées. Ann. Acad. Sc. Fenn. 223, 475–506 (1997)MathSciNetGoogle Scholar
  20. 20.
    Pansu, P.: Cohomologie L p, espaces homogènes et pincement. Preprint, Orsay (1999)Google Scholar
  21. 21.
    Pansu, P.: L p-cohomology and pinching. In: Rigidity in Dynamics and Geometry (Cambridge, 2000), pp. 379–389, Springer, Berlin (2002)Google Scholar
  22. 22.
    Pansu P.: Cohomologie L p en degré 1 des espaces homogé nes. J. Potential Anal. 27, 151–165 (2007)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pansu, P.: Cohomologie L p et pincement. Comment. Math. Helvetici. (to appear)Google Scholar
  24. 24.
    Reiman M.: Uber harmonishe Kapazität und quasikonforme Abbildungen in Raum. Comm. Math. Helv. 44, 284–307 (1969)Google Scholar
  25. 25.
    Reshetnyak, Yu.G.: Space Mappings with Bounded distortion. Translations of Mathematical Monographs, vol. 73, American Mathematical Society (1985)Google Scholar
  26. 26.
    Troyanov M., Vodop’yanov S.K.: Liouville type theorem for mappings with bounded co-distortion. Ann. Inst. Fourier 52(6), 1754–1783 (2002)MathSciNetGoogle Scholar
  27. 27.
    Troyanov, M.: On the Hodge decomposition in \({\mathbb{R}^n}\) . arXiv:0710.5414.Google Scholar
  28. 28.
    Vodop’yanov S.K.: Topological and geometrical properties of mappings with an integrable Jacobian in Sobolev classes. Siberian Math. J. 41(4), 19–39 (2000)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Vodop’yanov S.K., Gol’dshtein V.M.: Quasiconformal mappings and spaces of mfubctions with generalized first derivatives. Siberian Math. J. 17(3), 515–531 (1976)MATHGoogle Scholar
  30. 30.
    Vodop’yanov S.K., Ukhlov A.D.: Sobolev spaces and (p, q)-quasiconformal mappings of Carnot groups. Siberian Math. J. 39(4), 776–795 (1998)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion UniversityBeer ShevaIsrael
  2. 2.Section de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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