Extension of Hardy–Littlewood–Sobolev Inequalities for Riesz Potentials on Hypergroups


We establish in this paper the Hardy–Littlewood–Sobolev inequalities for the Riesz potentials on Morrey spaces over commutative hypergroups. As a consequence, we are also able to get Olsen-type inequality on the same spaces. Here, the condition of upper Ahlfors n-regular by identity is assumed to obtain the inequalities.

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


  1. 1.

    Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bloom, W.R., Heyer, H.: Harmonic analysis of probability measures on hypergroups. de Gruyter, Berlin (1995)

    Google Scholar 

  3. 3.

    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. 7, 273–279 (1987)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    García-Cuerva, J., Gatto, A.E.: Boundedness properties of fractional integral operators associated to non-doubling measures. Stud. Math. 162, 245–261 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    García-Cuerva, J., Martell, J.M.: Two weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana Univ. Math. J. 50, 1241–1280 (2001)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Gunawan, H., Eridani: Fractional integrals and generalized Olsen inequalities. Kyungpook Math. J. 49(1), 31–39 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gunawan, H.: A note on the generalized fractional integral operators. J. Indones. Math. Soc. (MIHMI) 9(1), 39–43 (2003)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Guliyev, V., Sawano, Y.: Linear and sublinear operators on generalized Morrey spaces with non-doubling measures. Publ. Math. Debr. 83(3), 303–327 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guo, Y.: Generalized Olsen inequality and Schrödinger type elliptic equations. Georgian Math. J. (2019). https://doi.org/10.1515/gmj-2019-2059

    Article  Google Scholar 

  10. 10.

    Hajibayov, M.G.: Boundedness in Lebesgue spaces of Riesz potentials on commutative hypergroups. Glob. J. Math. Anal. 3(1), 18–25 (2015)

    Article  Google Scholar 

  11. 11.

    Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. Math. Z. 27, 565–606 (1927)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jewett, R.I.: Spaces with an abstract convolution of measure. Adv. Math. 18, 1–101 (1975)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kurata, K., Nishigaki, S., Sugano, S.: Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators. Proc. Am. Math. Soc. 128, 1125–1134 (2002)

    Article  Google Scholar 

  14. 14.

    Morrey, C.B.: Functions of several variables and absolute continuity. Duke Math. J. 6, 187–215 (1940)

    MathSciNet  Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 16.

    Nakai, E.: Generalized fractional integrals on generalized Morrey spaces. Math. Nachr. 287(2–3), 339–351 (2014)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Nakai, E., Sadasue, G.: Martingale Morrey–Campanato spaces and fractional integrals. J. Funct. Spaces Appl. 2012, 29 (2012). (Article ID 673929)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Olsen, P.A.: Fractional integration, Morrey spaces and a Schrödinger equation. Commun. Partial Differ. Equ. 20, 2005–2055 (1995)

    Article  Google Scholar 

  19. 19.

    Ruzhansky, M., Suragan, D., Yessirkegenov, N.: Hardy–Littlewood, Bessel–Riesz, and fractional integral operators in anisotropic Morrey and Companato spaces. Fract. Calc. Appl. Anal. 21(3), 577–612 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(6), 1535–1544 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Sawano, Y., Shimomura, T.: Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value \(1\) over non-doubling measure spaces. J. Inequal. Appl. 2013, 1–19 (2013)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 263(12), 6481–6503 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Sihwaningrum, I., Gunawan, H., Nakai, E.: Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces. Math. Nachr. 291, 1400–1417 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sihwaningrum, I., Suryawan, H.P., Gunawan, H.: Fractional integral operators and Olsen inequalities on non-homogeneous spaces. Aust. J. Math. Anal. Appl. 7(1), 14 (2010)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Sihwaningrum, I., Sawano, Y.: Weak and strong type estimates for fractional integral operator on Morrey spaces over metric measure spaces. Eurasian Math. J. 4, 76–81 (2013)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Sobolev, S.L.: On a theorem in functional analysis (Russian). Mat. Sob. 46, 471–497 (1938). (English translation in Amer. Math. Soc. Transl. ser. 2, 34, 39–68 (1963))

    Google Scholar 

Download references


We are indebted to the referees for their useful comments on the earlier version of this paper. The first and second authors are supported by Directorate General of Higher Education, Grant/Award Number: P/1721/IJN23T4PN/2019. The third author is supported by P3MI-ITB Program 2020.

Author information



Corresponding author

Correspondence to Idha Sihwaningrum.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sihwaningrum, I., Maryani, S. & Gunawan, H. Extension of Hardy–Littlewood–Sobolev Inequalities for Riesz Potentials on Hypergroups. Mediterr. J. Math. 17, 203 (2020). https://doi.org/10.1007/s00009-020-01645-w

Download citation


  • Ahlfors condition
  • Riesz potential
  • Morrey spaces
  • commutative hypergroup
  • Olsen inequality

Mathematics Subject Classification

  • 42B20
  • 26A33
  • 47B38
  • 47G10