On Compactness of Operators in Variable Exponent Lebesgue Spaces

  • Stefan Samko
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)


We give a short discussion of known statements on compactness of operators in variable exponent Lebesgue spaces L p (·)(Ω, ϱ) and show that the existence of a radial integrable decreasing dominant of the kernel of a convolution operator guarantees its compactness in the space L p (·)(Ω, ϱ) whenever the maximal operator is bounded in this space, where |Ω| < ∞ and ϱ is an arbitrary weight such that L p (·)(Ω, ϱ) is a Banach function space. In the non-weighted case ϱ = 1 we also give a modification of this statement for Ω = ℝn.


Variable exponent spaces compact operator integral operators convolution operators radial decreasing dominants 

Mathematics Subject Classification (2000)

Primary 46E30 Secondary 47B38 47G10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Bennett and R. Sharpley. Interpolation of operators., volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988.Google Scholar
  2. [2]
    F. Cobos, T. Kühn and T. Schonbek. One-sided compactness results for Aronsjain-Gagliardo functors. J. Funct. Anal., 106:274–313, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D. Cruz-Uribe and A. Fiorenza. Approximate identities in variable L p spaces. Math. Nachr., 280(3):256–270, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    D. Cruz-Uribe, A. Fiorenza and C.J. Neugebauer. The maximal function on variable L p-spaces. Ann. Acad. Scient. Fennicae, Math., 28:223–238, 2003.zbMATHMathSciNetGoogle Scholar
  5. [5]
    M. Cwikel. Real and complex interpolation and extrapolation of compact operators. Duke Math. J., 65(2): 333–343, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    L. Diening. Maximal function on generalized Lebesgue spaces L (·). Math. Inequal. Appl., 7(2):245–253, 2004.zbMATHMathSciNetGoogle Scholar
  7. [7]
    L. Diening, P. Hästö and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands May 28–June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha, 38–58, 2005.Google Scholar
  8. [8]
    D.E. Edmunds, J. Lang and A. Nekvinda. On l p (x) norms. Proc. R. Soc. Lond. A 455: 219–225, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    K. Hayakawa. Interpolation by the real method preserves compactness of operators. J. Math. Soc. Japan, 21:189–199, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    N.K. Karapetiants and S.G. Samko. Equations with Involutive Operators. Birkhäuser, Boston, 2001.zbMATHGoogle Scholar
  11. [11]
    V. Kokilashvili. On a progress in the theory of integral operators in weighted Banach function spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held Milovy, Bohemian-Moravian Uplands May 28-June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha, 152–175, 2005.Google Scholar
  12. [12]
    V. Kokilashvili, N. Samko and S. Samko. The maximal operator in variable spaces L p (·)(Ω, ρ). Georgian Math. J., 13(1):109–125, 2006.zbMATHMathSciNetGoogle Scholar
  13. [13]
    V. Kokilashvili, N. Samko and S. Samko. The Maximal Operator in Weighted Variable Spaces L p (·). J. Function spaces and Appl., 5(3):299–317, 2007.zbMATHMathSciNetGoogle Scholar
  14. [14]
    V. Kokilashvili and S. Samko. The maximal operator in weighted variable Lebesgue spaces on metric spaces. Georgian Math. J., 15, no 4, 683–712, 2008.zbMATHMathSciNetGoogle Scholar
  15. [15]
    V. Kokilashvili and S. Samko. Singular Integral Equations in the Lebesgue Spaces with Variable Exponent. Proc. A. Razmadze Math. Inst., 131:61–78, 2003.zbMATHMathSciNetGoogle Scholar
  16. [16]
    V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted L p (x) spaces. Revista Matematica Iberoamericana, 20(2):493–515, 2004.zbMATHMathSciNetGoogle Scholar
  17. [17]
    V. Kokilashvili and S. Samko. Weighted boundedness of the maximal, singular and potential operators in variable exponent spaces. In A.A. Kilbas and S.V. Rogosin, editors, Analytic Methods of Analysis and Differential Equations, pages 139–164. Cambridge Scientific Publishers, 2008.Google Scholar
  18. [18]
    M.A. Krasnosel’skii. On a theorem of M. Riesz. Soviet Math. Dokl., 1:229–231, 1960.MathSciNetGoogle Scholar
  19. [19]
    M.A. Krasnosel’skii, P.P. Zabreiko, E.I. Pustyl’nik and P.E. Sobolevskii. Integral Operators in Spaces of Summable Functions. (Russian). Moscow: Nauka, 1968. 499 pages.Google Scholar
  20. [20]
    J.L. Lions and J. Peetre. Sur une classe d’espaces d’interpolation. Inst. Hautes Etudes Sci. Publ. Math., 19:5–66, 1964.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    A. Nekvinda. Hardy-Littlewood maximal operator on L p(x)(ℝn). Math. Inequal. and Appl., 7(2):255–265, 2004.zbMATHMathSciNetGoogle Scholar
  22. [22]
    A. Nekvinda. Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. and Appl., 337(2):1345–1365, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. Persson. Compact linear mappings between interpolation spaces. Ark. Mat., 5:215–219 (1964), 1964.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    V.S. Rabinovich and S.G Samko. Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integr. Eq. Oper. Theory, 60(4):507–537, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    H. Rafeiro and S. Samko. Dominated compactness theorem in Banach function spaces and its applications. Compl. Anal. Oper. Theory. 2(4): 669–681, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    H. Rafeiro and S. Samko. On a class of fractional type integral equations in variable exponent spaces. Fract. Calc. and Appl. Anal., (4):399–421, 2007.Google Scholar
  27. [27]
    M. Ružička. Electroreological Fluids: Modeling and Mathematical Theory. Springer, Lecture Notes in Math., 2000. vol. 1748, 176 pages.Google Scholar
  28. [28]
    S.G. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. and Spec. Funct, 16(5–6):461–482, 2005.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Stefan Samko
    • 1
  1. 1.Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal

Personalised recommendations