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Limiting case of the boundedness of fractional integral operators on nonhomogeneous space

Journal of Inequalities and Applications volume 2006, Article number: 092470 (2006) Cite this article

Abstract

We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp.

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References

  1. Edmunds DE, Kokilashvili V, Meskhi A: Bounded and Compact Integral Operators, Mathematics and Its Applications. Volume 543. Kluwer Academic, Dordrecht; 2002:xvi+643. chapter 6 chapter 6

    Book  MATH  Google Scholar 

  2. García-Cuerva J, Gatto AE: Boundedness properties of fractional integral operators associated to non-doubling measures. Studia Mathematica 2004,162(3):245–261. 10.4064/sm162-3-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Han Y, Yang D: Triebel-Lizorkin spaces with non-doubling measures. Studia Mathematica 2004,162(2):105–140. 10.4064/sm162-2-2

    Article  MathSciNet  MATH  Google Scholar 

  4. Kokilashvili V: Weighted estimates for classical integral operators. In Nonlinear Analysis, Function Spaces and Applications, Vol. 4 (Roudnice nad Labem, 1990), Teubner-Texte Math.. Volume 119. Teubner, Leipzig; 1990:86–103.

    Chapter  Google Scholar 

  5. Kokilashvili V, Meskhi A: Fractional integrals on measure spaces. Fractional Calculus & Applied Analysis 2001,4(1):1–24.

    MathSciNet  MATH  Google Scholar 

  6. Nakai E: Generalized fractional integrals on Orlicz-Morrey spaces. In Banach and Function Spaces. Yokohama Publishers, Yokohama; 2004:323–333.

    Google Scholar 

  7. Nazarov F, Treil S, Volberg A: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. International Mathematics Research Notices 1998,1998(9):463–487. 10.1155/S1073792898000312

    Article  MathSciNet  MATH  Google Scholar 

  8. Nazarov F, Treil S, Volberg A: The-theorem on non-homogeneous spaces. Acta Mathematica 2003,190(2):151–239. 10.1007/BF02392690

    Article  MathSciNet  MATH  Google Scholar 

  9. Sawano Y: -valued extension of the fractional maximal operators for non-doubling measures via potential operators. International Journal of Pure and Applied Mathematics 2006,26(4):505–523.

    MathSciNet  MATH  Google Scholar 

  10. Sawano Y, Tanaka H: Morrey spaces for non-doubling measures. Acta Mathematica Sinica 2005,21(6):1535–1544. 10.1007/s10114-005-0660-z

    Article  MathSciNet  MATH  Google Scholar 

  11. Tolsa X: BMO,, and Calderón-Zygmund operators for non doubling measures. Mathematische Annalen 2001,319(1):89–149. 10.1007/PL00004432

    Article  MathSciNet  MATH  Google Scholar 

  12. Tolsa X: Painlevé's problem and the semiadditivity of analytic capacity. Acta Mathematica 2003,190(1):105–149. 10.1007/BF02393237

    Article  MathSciNet  MATH  Google Scholar 

  13. Tolsa X: Bilipschitz maps, analytic capacity, and the Cauchy integral. Annals of Mathematics 2005,162(3):1243–1304. 10.4007/annals.2005.162.1243

    Article  MathSciNet  MATH  Google Scholar 

  14. Trudinger NS: On imbeddings into Orlicz spaces and some applications. Journal of Mathematics and Mechanics 1967, 17: 473–483.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Yoshihiro Sawano & Hitoshi Tanaka

  2. Department of Mathematics Education, Faculty of Education, Okayama University, 3-1-1 Tsushima-naka, Okayama, 700-8530, Japan

    Takuya Sobukawa

Authors
  1. Yoshihiro Sawano
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  2. Takuya Sobukawa
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  3. Hitoshi Tanaka
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Correspondence to Yoshihiro Sawano.

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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.

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Sawano, Y., Sobukawa, T. & Tanaka, H. Limiting case of the boundedness of fractional integral operators on nonhomogeneous space. J Inequal Appl 2006, 092470 (2006). https://doi.org/10.1155/JIA/2006/92470

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  • DOI: https://doi.org/10.1155/JIA/2006/92470

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