Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 1–10 | Cite as

Weak Type Estimates of the Fractional Integral Operators on Morrey Spaces with Variable Exponents

  • Kwok-Pun HoEmail author


We show that when the infimum of the exponent function equals to 1, the fractional integral operator is a bounded operator from the Morrey space with variable exponent to the weak Morrey space with variable exponent.


Fractional integral operators Morrey spaces Weak Morrey spaces Variable exponent 

Mathematics Subject Classification

42B20 46E30 47B38 



The author would like to thank the reviewers for careful reading of the paper and valuable suggestions.


  1. 1.
    Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15, 195–208 (2008) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable \(L^{p}\) spaces. Rev. Mat. Iberoam. 23, 743–770 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Birkhäuser, Basel (2013) CrossRefzbMATHGoogle Scholar
  5. 5.
    de Almeida, M.F., Ferreira, L.C.F.: On the Navier-Stokes equations in the half-space with initial and boundary rough data in Morrey spaces. J. Differ. Equ. 254, 1548–1570 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\cdot)}\) and \(W^{k,p(\cdot)}\). Math. Nachr. 268, 31–43 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponent. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011) CrossRefzbMATHGoogle Scholar
  8. 8.
    Edmunds, D., Rákosník, J.: Sobolev embeddings with variable exponent. Stud. Math. 143, 267–293 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Edmunds, D., Rákosník, J.: Sobolev embeddings with variable exponent II. Math. Nachr. 246–247, 53–67 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ferreira, L.: On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier-Stokes equations. J. Math. Pures Appl. (9) 105, 228–247 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guliyev, V.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009, 503948 (2009). 20 pp. MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guliyev, V., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107, 285–304 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guliyev, V., Hasanov, J., Badalov, X.: Maximal and singular integral operators and their commutators on generalized weighted Morrey spaces with variable exponent. Math. Inequal. Appl. 21, 41–61 (2018) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ho, K.-P.: Atomic decomposition of Hardy spaces and characterization of \(\mathit{BMO}\) via Banach function spaces. Anal. Math. 38, 173–185 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ho, K.-P.: The fractional integral operators on Morrey spaces with variable exponent on unbounded domains. Math. Inequal. Appl. 16, 363–373 (2013) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ho, K.-P.: Vector-valued operators with singular kernel and Triebel-Lizorkin-block spaces with variable exponents. Kyoto J. Math. 56, 97–124 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ho, K.-P.: Fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents. J. Math. Soc. Jpn. 69, 1059–1077 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ho, K.-P.: Two-weight norm, Poincaré, Sobolev and Stein-Weiss inequalities on Morrey spaces. Publ. Res. Inst. Math. Sci. 53, 119–139 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ho, K.-P.: Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces. Sci. China Math. 60, 449–468 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ho, K.-P.: The Ahlfors-Beurling transform on Morrey spaces with variable exponents. Integral Transforms Spec. Funct. 29, 207–220 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kokilashvili, V., Meskhi, A.: Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armen. J. Math. 1, 18–28 (2008) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kokilashvili, V., Meskhi, A.: Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure. Complex Var. Elliptic Equ. 55, 923–936 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kokilashvili, V., Samko, S.: On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwend. 22, 899–910 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lin, H., Yang, D.: Endpoint estimates for multilinear fractional integrals with non-doubling measures. Acta Math. Appl. Sin. Engl. Ser. 30, 755–764 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Maio, C-X., Weak, Y.B-Q.: Morrey spaces and strong solutions to the Navier-Stokes equations. Sci. China Ser. A 50, 1401–1417 (2007) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Jpn. 60, 583–602 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mizuta, Y., Shimomura, T.: Continuity properties for Riesz potentials of functions in Morrey spaces of variable exponent. Math. Inequal. Appl. 13, 99–122 (2010) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials. J. Math. Soc. Jpn. 62, 707–744 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Morrey, C.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–104 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nakai, E.: Recent topics of fractional integrals. Sūgaku Expo. 20, 215–235 (2007) zbMATHGoogle Scholar
  32. 32.
    Nakai, E.: Orlicz-Morrey spaces and the Hardy-Littlewood maximal function. Stud. Math. 188, 193–221 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Olsen, P.: Fractional integration, Morrey spaces and Schrödinger equation. Commun. Partial Differ. Equ. 20, 2005–2055 (1995) CrossRefzbMATHGoogle Scholar
  34. 34.
    Peetre, J.: On the theory of \(\mathcal{L}_{p,\lambda}\) spaces. J. Funct. Anal. 4, 71–87 (1969) CrossRefGoogle Scholar
  35. 35.
    Samko, S.: Convolution and potential type operators in \(L^{p(x)}({\mathbb {R}} ^{n})\). Integral Transforms Spec. Funct. 7, 261–284 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sawano, Y., El-Shabrawy, S.R.: Weak Morrey spaces with applications. Math. Nachr. 291, 178–186 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sawano, Y., Sugano, S., Tanaka, H.: Orlicz-Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tanaka, H.: Morrey spaces and fractional operators. J. Aust. Math. Soc. 88, 247–259 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tang, L.: Endpoint estimates for multilinear fractional integrals. J. Aust. Math. Soc. 84, 419–429 (2008) MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Information TechnologyThe Education University of Hong KongHong KongChina

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