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Extrapolation results of Lions-Peetre type

  • Fernando CobosEmail author
  • Thomas Kühn
Article

Abstract

We establish compactness results for extrapolation constructions which correspond to the well-known Lions-Peetre compactness theorems of interpolation theory. Applications are given to compactness of certain limiting Sobolev embeddings.

Mathematics Subject Classification (2000)

47B06 46B70 46E35 46E30 

Notes

Acknowledgments

The authors have been supported in part by the Spanish Ministerio de Economía y Competitividad (MTM2010-15814).

References

  1. 1.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    Butzer, P.L., Berens, H.: Semi-Groups of Operators and Approximation. Springer, New York (1967)CrossRefzbMATHGoogle Scholar
  4. 4.
    Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Cobos, F., Cwikel, M., Matos, P.: Best possible compactness results of Lions-Peetre type. Proc. Edinb. Math. Soc. 44, 153–172 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cobos, F., Fernández-Cabrera, L.M., Triebel, H.: Abstract and concrete logarithmic interpolation spaces. J. Lond. Math. Soc. 70, 231–243 (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cobos, F., Kühn, T.: Extrapolation estimates for entropy numbers. J. Funct. Anal. 263, 4009–4033 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cobos, F., Kühn, T., Schonbek, T.: One-sided compactness results for Aronszajn-Gagliardo functors. J. Funct. Anal. 106, 274–313 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cobos, F., Peetre, J.: Interpolation of compactness using Aronszajn-Gagliardo functors. Isr. J. Math. 68, 220–240 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cwikel, M.: Complex interpolation spaces, a discrete definition and reiteration. Indiana Univ. Math. J. 27, 1005–1009 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Edmunds, D.E., Triebel, H.: Logarithmic spaces and related trace problems. Functiones et Approximatio 26, 189–204 (1998)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51, 131–148 (2000)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Fiorenza, A., Karadzhov, G.E.: Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwendungen 23, 657–681 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fiorenza, A., Rakotoson, J.M.: Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications. Calc. Var. 25, 187–203 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119, 129–143 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Jawerth, B., Milman, M.: Extrapolation theory with applications. Mem. Am. Math. Soc. 89(440), (1991)Google Scholar
  18. 18.
    Karadzhov, G.E., Milman, M.: Extrapolation theory: new results and applications. J. Approx. Theory 133, 38–99 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kühn, T., Schonbek, T.: Extrapolation of entropy numbers. Contemp. Math. 445, 195–206 (2007)CrossRefGoogle Scholar
  20. 20.
    Lions, J.-L., Peetre, J.: Sur une classe d’espaces d’interpolation. Inst. Hautes Études Sci. Publ. Math. 19, 5–68 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Milman, M.: Extrapolation and Optimal Decompositions, Lecture Notes in Math. vol. 1580. Springer, Berlin (1994)Google Scholar
  22. 22.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)Google Scholar
  23. 23.
    Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)CrossRefzbMATHGoogle Scholar
  24. 24.
    Triebel, H.: Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces. Proc. Lond. Math. Soc. 66, 589–618 (1993)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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