Skip to main content
SpringerLink
Log in

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We prove the boundedness of the maximal operator ℳΓ in the spaces L p(·)(Γ, ρ) with variable exponent p(t) and power weight ρ on an arbitrary Carleson curve under the assumption that p(t) satisfies the log-condition on Γ.

We prove also weighted Sobolev type L p(·)(Γ, ρ) → L q(·)(Γ, ρ)-theorem for potential operators on Carleson curves.

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. Maz’ya, V. G.: Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985

    Google Scholar 

  2. Genebashvili, I., Gogatishvili, A., Kokilashvili, V., Krbec, M.: Weight theory for integral transforms on spaces of homogeneous type, Pitman Monographs and Surveys, Pure and Applied mathematics: Longman Scientific and Technical, 1998, 422

  3. Kokilashvili, V., Samko, S.: Singular Integral Equations in the Lebesgue Spaces with Variable Exponent. Proc. A. Razmadze Math. Inst., 131, 61–78 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Fan, X.: Amemiya norm equals Orlicz norm in Musielak-Orlicz spaces. Acta Mathematica Sinica, English Series, 23(2), 281–288 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fan, X., Zhao, D: On the spaces L p(x)(Ω) and W m,p(x)(Ω). J. Math. Anal. Appl., 263(2), 424–446 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kováčik, O., Rákosník, J.: On spaces L p(x) and W k,p(x). Czechoslovak Math. J., 41(116), 592–618 (1991)

    MathSciNet  Google Scholar 

  7. Samko, S. G.: Differentiation and integration of variable order and the spaces L p(x), Proceed. of Intern. Conference „Operator Theory and Complex and Hypercomplex Analysis”, 12–17 December 1994, Mexico City, Mexico, Contemp. Math., Vol. 212, 203–219, 1998

    MathSciNet  Google Scholar 

  8. Fan, X.: The regularity of Lagrangians f(x, ξ) = |ξ|α(x) with Hölder exponents α(x). Acta Mathematica Sinica, New Series, 12(3), 254–261 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ružička, M.: Electroreological Fluids: Modeling and Mathematical Theory, Springer, Lecture Notes in Math., 2000, vol. 1748, 176

  10. Zhikov, V. V.: On some variational problems. Russian J. Math. Phys., 5(1), 105–116 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(·) and W k,p(·). Mathem. Nachrichten, 268, 31–43 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kokilashvili, V., Samko, S.: Maximal and fractional operators in weighted L p(x) spaces. Revista Matematica Iberoamericana, 20(2), 495–517 (2004)

    MathSciNet  Google Scholar 

  13. Khabazi, M.: Maximal operators in weighted L p(x) spaces. Proc. A. Razmadze Math. Inst., 135, 143–144, (2004)

    MATH  MathSciNet  Google Scholar 

  14. Diening, L.: Maximal functions on generalized Lebesgue spaces L p(x), Preprint Mathematische Fakultät, Albert-Ludwigs-Universität Freiburg, (02/2002, 16.01.2002), 1–6, 2002

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

    MATH  MathSciNet  Google Scholar 

  16. Böttcher, A., Karlovich, Yu.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Basel, Boston, Berlin: Birkhäuser Verlag, 1997, 397

    MATH  Google Scholar 

  17. Adams, R. A., Hedberg, L. I.: Function Spaces and Potential Theory, Springer, 1996

  18. Samko, S. G.: Convolution and potential type operators in L p(x). Integr. Transf. and Special Funct., 7(3–4), 261–284 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kokilashvili, V., Samko, S.: On Sobolev Theorem for the Riesz type Potentials in Lebesgue Spaces with Variable Exponent. Zeit. Anal. Anwend., 22(4), 899–910 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. Razmadze Mathematical Institute, International Black Sea University, Georgia, Germany

    V. Kokilashvili

  2. University of Algarve, Algarve, Portugal

    S. Samko

Authors
  1. V. Kokilashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Samko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Kokilashvili.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kokilashvili, V., Samko, S. Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta. Math. Sin.-English Ser. 24, 1775–1800 (2008). https://doi.org/10.1007/s10114-008-6464-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6464-1

Keywords

MR(2000) Subject Classification

Search

Navigation