Skip to main content

Applications of interpolation with a function parameter to Lorentz, Sobolev and besov spaces

Contributed Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 1070)

Abstract

In this paper we show that interpolation with a function parameter is perfectly suited to identify interpolation spaces between two quasi-normed Lorentz spaces ∧p(φ) and, in the case of Banach spaces, between two Sobolev spaces \(W_{\Lambda ^p (\phi )}^m\). We deduce the A.P. Calderón theorem for these spaces. We prove the identity \((H_p^{\phi _0 } ,H_p^{\phi _1 } )_{f,q;K} = B_{p,q}^\psi\) where ψ is classically connected with φ0, φ1 and f, and in which the Sobolev space H φp and the Besov space B φp,q are constructed in the same way as in the classical case. Imbedding and trace theorems are given for these spaces, as well as equivalent norms on space H φp,q in connection with semi-groups and approximation theory.

Keywords

  • Banach Space
  • Sobolev Space
  • Function Parameter
  • Besov Space
  • Lorentz Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Diss. Mat. (Roszp. Mat.) 175, 5–67 (1980).

    MathSciNet  MATH  Google Scholar 

  2. Bergh, J., Löfström, J: Interpolation spaces. An introduction. Springer Verlag, Berlin, Heidelberg, New-York (1976).

    Google Scholar 

  3. Boyd, D.W.: The Hilbert transform on rearrangement-invariant spaces. Can. J. Math. 19, 599–616 (1967).

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Butzer, P.L., Berens, H.: Semi-groups of operators and approximation. Springer Verlag, Berlin, Heidelberg, New-York (1967).

    CrossRef  MATH  Google Scholar 

  5. Calderón, C.P., Milman, M.: Interpolation of Sobolev spaces. The real method. Ind. Math. J. (to appear).

    Google Scholar 

  6. Cwikel, M.: Monotonicity properties of interpolation spaces II. Ark. Mat. 19, 123–136 (1981).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. De Vore, R., Scherer, K.: Interpolation of linear operators on Sobolev spaces. Ann. Math. 109, 583–599 (1979).

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Gustavsson, J.: A function parameter in connection with interpolation of Banach spaces. Math. Scand. 42, 289–305 (1978).

    MathSciNet  MATH  Google Scholar 

  9. Heinig, P.H.: Interpolation of quasi-normed spaces involving weights. Can. Math. Soc., Conf. Proc. 1, 245–267 (1981).

    MathSciNet  MATH  Google Scholar 

  10. Holmstedt, T.: Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970).

    MathSciNet  MATH  Google Scholar 

  11. Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J. 47, 959–982 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Kalugina, T.F.: Interpolation of Banach spaces with a functional parameter. The reiteration theorem. Vestnik Moskov. Univ. Ser. I, Mat. Meh. 30, 6, 68–77 (1975).

    MathSciNet  MATH  Google Scholar 

  13. Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications I. Dunod (1968).

    Google Scholar 

  14. Maligranda, L.: Indices and interpolation. Inst. Math., Pol. Acad. Sci., 274, 1–70 (1983).

    MATH  Google Scholar 

  15. Merucci, C.: Interpolation réelle avec fonction paramètre: réitération et applications aux espaces ∧p(φ) (o < p ⩽ + ∞). C.R. Acad. Sci. Paris, I, 295, 427–430 (1982).

    MathSciNet  MATH  Google Scholar 

  16. Merucci, C.: Interpolation réelle avec fonction paramètre: dualité, réitération et applications. Thèse d'Etat, Nantes (1983).

    Google Scholar 

  17. Milman, M.: Interpolation of operators of mixed weak-strong type between rearrangement invariant spaces. Ind. Univ. Math. J. 28, 6, 985–992 (1979).

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Milman, M.: Interpolation of some concrete scales of spaces. Techn. Rep., Lund, August (1982).

    Google Scholar 

  19. Nilsson, P.: Reiteration theorems for real interpolation and approximation spaces. Ann. Mat. Pura Appl. 32, 291–330 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Peetre, J.: Zur Interpolation Von Operatorenräumen. Arch. Math. 21, 601–608 (1970).

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Peetre, J.: New thoughts on Besov spaces. Duke Univ. Math. Ser. I, Durham (1976).

    Google Scholar 

  22. Torchinsky, A.: The K-functional for rearrangement invariant spaces. Stud. Math. 64, 175–190 (1979).

    MathSciNet  MATH  Google Scholar 

  23. Triebel, H.: Interpolation theory, function spaces, differential operators. North. Holl. Math. Libr. 18 (1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Merucci, C. (1984). Applications of interpolation with a function parameter to Lorentz, Sobolev and besov spaces. In: Cwikel, M., Peetre, J. (eds) Interpolation Spaces and Allied Topics in Analysis. Lecture Notes in Mathematics, vol 1070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099101

Download citation

  • DOI: https://doi.org/10.1007/BFb0099101

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13363-6

  • Online ISBN: 978-3-540-38913-2

  • eBook Packages: Springer Book Archive