Skip to main content

On extrapolation blowups in the scale

Abstract

Yano's extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.

[1234567891011121314151617]

References

  1. Bennett C, Rudnick K: On Lorentz-Zygmund spaces. Dissertationes Mathematicae (Rozprawy Matematyczne) 1980, 175: 1–67.

    MathSciNet  Google Scholar 

  2. Bennett C, Sharpley R: Interpolation of Operators, Pure and Applied Mathematics. Volume 129. Academic Press, Boston; 1988:xiv+469.

    Google Scholar 

  3. Capone C, Fiorenza A: On small Lebesgue spaces. Journal of Function Spaces and Applications 2005,3(1):73–89.

    MathSciNet  Article  MATH  Google Scholar 

  4. Carro MJ, Martín M: Extrapolation theory for the real interpolation method. Collectanea Mathematica 2002,53(2):165–186.

    MathSciNet  MATH  Google Scholar 

  5. Da Prato G, Zabczyk J: A note on stochastic convolution. Stochastic Analysis and Applications 1992,10(2):143–153. 10.1080/07362999208809260

    MathSciNet  Article  MATH  Google Scholar 

  6. Fiorenza A: Duality and reflexivity in grand Lebesgue spaces. Collectanea Mathematica 2000,51(2):131–148.

    MathSciNet  MATH  Google Scholar 

  7. Fiorenza A, Karadzhov GE: Grand and small Lebesgue spaces and their analogs. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and its Applications 2004,23(4):657–681.

    MathSciNet  Article  MATH  Google Scholar 

  8. Fiorenza A, Krbec M: On an optimal decomposition in Zygmund spaces. Georgian Mathematical Journal 2002,9(2):271–286.

    MathSciNet  MATH  Google Scholar 

  9. Fiorenza A, Rakotoson JM: New properties of small Lebesgue spaces and their applications. Mathematische Annalen 2003,326(3):543–561.

    MathSciNet  MATH  Google Scholar 

  10. Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge; 1952:xii+324.

    MATH  Google Scholar 

  11. Iwaniec T, Sbordone C: On the integrability of the Jacobian under minimal hypotheses. Archive for Rational Mechanics and Analysis 1992,119(2):129–143. 10.1007/BF00375119

    MathSciNet  Article  MATH  Google Scholar 

  12. Krasnosel'skiĭ MA, Rutickiĭ JaB: Convex Functions and Orlicz Spaces. P. Noordhoff, Groningen; 1961:xi+249.

    Google Scholar 

  13. Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin; 1983:iii+222.

    Google Scholar 

  14. Rao MM, Ren ZD: Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 146. Marcel Dekker, New York; 1991:xii+449.

    Google Scholar 

  15. Seidler J, Sobukawa T: Exponential integrability of stochastic convolutions. Journal of the London Mathematical Society. Second Series 2003,67(1):245–258. 10.1112/S0024610702003745

    MathSciNet  Article  MATH  Google Scholar 

  16. Yano S: Notes on Fourier analysis. XXIX. An extrapolation theorem. Journal of the Mathematical Society of Japan 1951, 3: 296–305. 10.2969/jmsj/00320296

    MathSciNet  Article  MATH  Google Scholar 

  17. Zygmund A: Trigonometric Series. Vols. I, II. 2nd edition. Cambridge University Press, New York; 1959:Vol. I. xii+383 pp.; Vol. II. vii+354.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Claudia Capone.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Capone, C., Fiorenza, A. & Krbec, M. On extrapolation blowups in the scale. J Inequal Appl 2006, 74960 (2006). https://doi.org/10.1155/JIA/2006/74960

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/JIA/2006/74960

Keywords

  • Current Investigation
  • Maximal Function
  • Lebesgue Space
  • Stochastic Analysis
  • Boundedness Property