Skip to content

Advertisement

  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20062006:092470

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

  • Received: 13 April 2006
  • Accepted: 12 June 2006
  • Published:

Abstract

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

Keywords

  • Integral Operator
  • Fractional Integral Operator
  • Nonhomogeneous Space

[123456789101112131415]

Authors’ Affiliations

(1)
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku Tokyo, 153-8914, Japan
(2)
Department of Mathematics Education, Faculty of Education, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

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 6View ArticleMATHGoogle 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/sm162-3-5MathSciNetView ArticleMATHGoogle Scholar
  3. Han Y, Yang D: Triebel-Lizorkin spaces with non-doubling measures. Studia Mathematica 2004,162(2):105–140. 10.4064/sm162-2-2MathSciNetView ArticleMATHGoogle 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.View ArticleGoogle Scholar
  5. Kokilashvili V, Meskhi A: Fractional integrals on measure spaces. Fractional Calculus & Applied Analysis 2001,4(1):1–24.MathSciNetMATHGoogle 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/S1073792898000312MathSciNetView ArticleMATHGoogle Scholar
  8. Nazarov F, Treil S, Volberg A: The -theorem on non-homogeneous spaces. Acta Mathematica 2003,190(2):151–239. 10.1007/BF02392690MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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-zMathSciNetView ArticleMATHGoogle Scholar
  11. Tolsa X: BMO, , and Calderón-Zygmund operators for non doubling measures. Mathematische Annalen 2001,319(1):89–149. 10.1007/PL00004432MathSciNetView ArticleMATHGoogle Scholar
  12. Tolsa X: Painlevé's problem and the semiadditivity of analytic capacity. Acta Mathematica 2003,190(1):105–149. 10.1007/BF02393237MathSciNetView ArticleMATHGoogle Scholar
  13. Tolsa X: Bilipschitz maps, analytic capacity, and the Cauchy integral. Annals of Mathematics 2005,162(3):1243–1304. 10.4007/annals.2005.162.1243MathSciNetView ArticleMATHGoogle Scholar
  14. Trudinger NS: On imbeddings into Orlicz spaces and some applications. Journal of Mathematics and Mechanics 1967, 17: 473–483.MathSciNetMATHGoogle 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

Copyright

Advertisement