A characterization for fractional integral and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

  • Vagif S. Guliyev
  • Fatih DeringozEmail author


In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz–Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces.


Orlicz space Generalized Orlicz–Morrey space Maximal operator Fractional integral Commutator Spaces of homogeneous type 

Mathematics Subject Classification

42B20 42B25 42B35 



The research of V.S. Guliyev was partially supported by the grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement No. EIF-BGM-4-RFTF-1/2017-21/01/1) and by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008). The research of F. Deringoz was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.18.019).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Birnbaum, Z., Orlicz, W.: Über die verallgemeinerung des begriffes der zueinan-der konjugierten potenzen. Studia Math. 3, 1–67 (1931)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  4. 4.
    Calderón, A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31(1), 7–16 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Math. Appl. 7(7), 273–279 (1987)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cianchi, A.: Strong and weak type inequalities for some classical operators in Orlicz spaces. J. Lond. Math. Soc. 60(2), 187–202 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certain Espaces Homogenes. Lecture Notes in Mathematics, No. 242. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  9. 9.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 831, 569–645 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coifman, R., Lions, P., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Springer, Berlin (2009)zbMATHGoogle Scholar
  13. 13.
    Deringoz, F., Guliyev, V.S., Samko, S.G.: Boundedness of maximal and singular operators on generalized Orlicz–Morrey spaces. Oper Theory Oper Algebras Appl Ser Oper Theory Adv Appl 242, 1–24 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Characterizations for the Riesz potential and its commutators on generalized Orlicz–Morrey spaces. J. Inequal. Appl. 2016, 248 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Deringoz, F., Guliyev, V.S., Ragusa, M.A.: Intrinsic square functions on vanishing generalized Orlicz–Morrey spaces. Set Valued Var. Anal. 25(4), 807–828 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Eroglu, A., Guliev, V.S., Azizov, DzhV: Characterizations of fractional integral operators in generalized Morrey spaces on Carnot groups (Russian). Mat. Zametki 102(5), 789–804 (2017). (translation in Math. Notes 102 no. 5–6, 722–734 (2017)) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fu, X., Yang, D., Yuan, W.: Boundedness of multilinear commutators of Calderón–Zygmund operators on Orlicz spaces over non-homogeneous spaces. Taiwan. J. Math. 16, 2203–2238 (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gala, S., Ragusa, M.A., Sawano, Y., Tanaka, H.: Uniqueness criterion of weak solutions for the dissipative quasi-geostrophic equations in Orlicz–Morrey spaces. Appl. Anal. 93(2), 356–368 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gala, S., Guo, Z., Ragusa, M.A.: A remark on the regularity criterion of Boussinesq equations with zero heat conductivity. Appl. Math. Lett. 27, 70–73 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Genebashvili, I., Gogatishvili, A., Kokilashvili, V., Krbec, M.: Weight Theory for Integral Transforms on Spaces of Homogeneous Type. Longman, Harlow (1998)zbMATHGoogle Scholar
  21. 21.
    Guliyev, V.S., Mustafayev, R.C.: Fractional integrals in spaces of functions defined on spaces of homogeneous type (Russian). Anal. Math. 24(3), 181–200 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. Art. ID 503948, 20 pp (2009)Google Scholar
  23. 23.
    Guliyev, V.S., Aliyev, S.S., Karaman, T., Shukurov, P.S.: Boundedness of sublinear operators and commutators on generalized Morrey space. Integr. Equ. Oper. Theory 71(3), 327–355 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Riesz potential and its commutators on Orlicz spaces. J. Inequal. Appl. 2017, 75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guliyev, V.S., Omarova, M.N., Ragusa, M.A., Scapellato, A.: Commutators and generalized local Morrey spaces. J. Math. Anal. Appl. 457(2), 1388–1402 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Izuki, M., Sawano, Y.: Characterization of BMO via ball Banach function spaces. Vestn. St.-Peterbg. Univ. Mat. Mekh. Astron. 4(62), 78–86 (2017)MathSciNetGoogle Scholar
  27. 27.
    Krasnoselskii, M.A., Rutickii, YaB: Convex Functions and Orlicz Spaces (English translation). P. Noordhoff Ltd., Groningen (1961)Google Scholar
  28. 28.
    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5(3), 587–602 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nakai, E.: On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type. Sci. Math. Jpn. 54, 473–487 (2001)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nakai, E.: The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Studia Math. 176(1), 1–19 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nakai, E.: Recent topics of fractional integrals. Sugaku Expo. 20(2), 215–235 (2007)zbMATHGoogle Scholar
  33. 33.
    O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Orlicz, W.: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Acad. Polon. A, pp 207–220 (1932)(Reprinted in: Collected Papers, PWN, Warszawa, pp 217–230 (1988)) Google Scholar
  35. 35.
    Peetre, J.: On the theory of \({\cal{L}}_{p,\lambda }\). J. Funct. Anal. 4, 71–87 (1969)CrossRefGoogle Scholar
  36. 36.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. M. Dekker Inc., New York (1991)zbMATHGoogle Scholar
  37. 37.
    Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363(12), 6481–6503 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sugano, S., Tanaka, H.: Boundedness of fractional integral operators on generalized Morrey spaces. Sci. Math. Jpn. 58(3), 531–540 (2003)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Torchinsky, A.: Interpolation of operators and Orlicz classes. Studia Math. 59, 177–207 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.S.M. Nikolskii Institute of MathematicsRUDN UniversityMoscowRussia
  2. 2.Department of MathematicsDumlupinar UniversityKutahyaTurkey
  3. 3.Institute of Mathematics and MechanicsBakuAzerbaijan
  4. 4.Department of MathematicsAhi Evran UniversityKirsehirTurkey

Personalised recommendations