Skip to main content
SpringerLink
Log in

Singular and fractional integral operators on Campanato spaces with variable growth conditions

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

Let X be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we prove boundedness of singular and fractional integral operators on Campanato spaces over X with variable growth conditions. The function spaces contain generalized Lipschitz spaces with variable exponent as special cases. Moreover, by using the function spaces, we can deal with functions which are L p-functions locally on one subset in X, BMO-functions locally on one another subset and Lip α -functions locally on the other one. Our results are new even for ℝn case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative Sur Certains Espaces Homogenes. Lecture Notes in Math., vol. 242. Springer, Berlin (1971)

    MATH  Google Scholar 

  2. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Diening, L.: Maximal function on generalized Lebesgue spaces L p(⋅). Math. Inequal. Appl. 7(2), 245–253 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(⋅) and W k,p(⋅). Math. Nachr. 268, 31–43 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Diening, L., Ružička, M.: Calderón-Zygmund operators on generalized Lebesgue spaces L p(⋅) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)

    MATH  MathSciNet  Google Scholar 

  6. Eridani, Gunawan, H., Nakai, E.: On generalized fractional integral operators. Sci. Math. Jpn. 60, 539–550 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potential space of variable exponent. Math. Nachr. 279, 1463–1473 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embeddings for variable exponent Riesz potentials on metric spaces. Ann. Acad. Sci. Fenn. Math. 31(2), 495–522 (2006)

    MATH  MathSciNet  Google Scholar 

  9. Gatto, A.E., Vági, S.: Fractional integrals on spaces of homogeneous type. In: Analysis and Partial Differential Equations, pp. 171–216. Dekker, New York (1990)

    Google Scholar 

  10. Gatto, A.E., Segovia, C., Vági, S.: On fractional differentiation and integration on spaces of homogeneous type. Rev. Mat. Iberoam. 12(1), 111–145 (1996)

    MATH  Google Scholar 

  11. Lerner, A.K.: Some remarks on the Hardy-Littlewood maximal function on variable L p spaces. Math. Z. 251, 509–521 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Macías, R.A.: C. Segovia Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    Article  MATH  Google Scholar 

  13. Macías, R.A., Segovia, C.: Singular integrals on generalized Lipschitz and Hardy spaces. Stud. Math. 65(1), 55–75 (1979)

    MATH  Google Scholar 

  14. Mizuta, Y., Shimomura, T.: Sobolev’s inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity. J. Math. Anal. Appl. 311, 268–288 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Jpn. 60(2), 583–602 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Nakai, E.: Pointwise multipliers for functions of weighted bounded mean oscillation. Stud. Math. 105, 105–119 (1993)

    MATH  MathSciNet  Google Scholar 

  17. Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nakai, E.: Pointwise multipliers on weighted BMO spaces. Stud. Math. 125, 35–56 (1997)

    MATH  MathSciNet  Google Scholar 

  19. Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5(3), 587–602 (2001)

    MATH  MathSciNet  Google Scholar 

  20. Nakai, E.: On generalized fractional integrals on the weak Orlicz spaces, BMO φ , the Morrey spaces and the Campanato spaces. In: Function Spaces, Interpolation Theory and Related Topics, pp. 389–401. de Gruyter, Berlin (2002)

    Google Scholar 

  21. Nakai, E.: The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Stud. Math. 176, 1–19 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nakai, E.: A generalization of Hardy spaces H p by using atoms. Acta Math. Sinica 24, 1243–1268 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Jpn. 37(2), 207–218 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nakai, E., Yabuta, K.: Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math. Jpn. 46, 15–28 (1997)

    MATH  MathSciNet  Google Scholar 

  25. Peetre, J.: On the theory of ℒp,λ spaces. J. Funct. Anal. 4, 71–87 (1969)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka, 582-8582, Japan

    Eiichi Nakai

Authors
  1. Eiichi Nakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eiichi Nakai.

Additional information

Dedicated to Professor Yoshihiro Mizuta on his sixtieth birthday.

The author was partly supported by Grant-in-Aid for Scientific Research (C), No. 20540167, Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakai, E. Singular and fractional integral operators on Campanato spaces with variable growth conditions. Rev Mat Complut 23, 355–381 (2010). https://doi.org/10.1007/s13163-009-0022-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-009-0022-y

Mathematics Subject Classification (2000)

Search

Navigation