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On extrapolation blowups in the scale


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.



  1. 1.

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

    MathSciNet  Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 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. 4.

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

    MathSciNet  MATH  Google Scholar 

  5. 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. 6.

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

    MathSciNet  MATH  Google Scholar 

  7. 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. 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. 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. 10.

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

    Google Scholar 

  11. 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. 12.

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

    Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 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. 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. 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. 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 

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Correspondence to Claudia Capone.

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Capone, C., Fiorenza, A. & Krbec, M. On extrapolation blowups in the scale. J Inequal Appl 2006, 74960 (2006).

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