Skip to main content
Log in

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

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Google Scholar 

  2. Birnbaum, Z., Orlicz, W.: Über die verallgemeinerung des begriffes der zueinan-der konjugierten potenzen. Studia Math. 3, 1–67 (1931)

    Google Scholar 

  3. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)

    Google Scholar 

  4. Calderón, A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099 (1965)

    Google Scholar 

  5. Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31(1), 7–16 (1982)

    Google Scholar 

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

    Google Scholar 

  7. Cianchi, A.: Strong and weak type inequalities for some classical operators in Orlicz spaces. J. Lond. Math. Soc. 60(2), 187–202 (1999)

    Google Scholar 

  8. Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certain Espaces Homogenes. Lecture Notes in Mathematics, No. 242. Springer, Berlin (1971)

    Google Scholar 

  9. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 831, 569–645 (1977)

    Google Scholar 

  10. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)

    Google Scholar 

  11. Coifman, R., Lions, P., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)

    Google Scholar 

  12. Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Springer, Berlin (2009)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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))

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  20. Genebashvili, I., Gogatishvili, A., Kokilashvili, V., Krbec, M.: Weight Theory for Integral Transforms on Spaces of Homogeneous Type. Longman, Harlow (1998)

    Google Scholar 

  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)

    Google Scholar 

  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)

  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)

    Google Scholar 

  24. Guliyev, V.S., Deringoz, F., Hasanov, S.G.: Riesz potential and its commutators on Orlicz spaces. J. Inequal. Appl. 2017, 75 (2017)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  27. Krasnoselskii, M.A., Rutickii, YaB: Convex Functions and Orlicz Spaces (English translation). P. Noordhoff Ltd., Groningen (1961)

    Google Scholar 

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

    Google Scholar 

  29. Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5(3), 587–602 (2001)

    Google Scholar 

  30. Nakai, E.: On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type. Sci. Math. Jpn. 54, 473–487 (2001)

    Google Scholar 

  31. Nakai, E.: The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Studia Math. 176(1), 1–19 (2006)

    Google Scholar 

  32. Nakai, E.: Recent topics of fractional integrals. Sugaku Expo. 20(2), 215–235 (2007)

    Google Scholar 

  33. O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)

    Google Scholar 

  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))

  35. Peetre, J.: On the theory of \({\cal{L}}_{p,\lambda }\). J. Funct. Anal. 4, 71–87 (1969)

    Google Scholar 

  36. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. M. Dekker Inc., New York (1991)

    Google Scholar 

  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)

    Google Scholar 

  38. Sugano, S., Tanaka, H.: Boundedness of fractional integral operators on generalized Morrey spaces. Sci. Math. Jpn. 58(3), 531–540 (2003)

    Google Scholar 

  39. Torchinsky, A.: Interpolation of operators and Orlicz classes. Studia Math. 59, 177–207 (1976)

    Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fatih Deringoz.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guliyev, V.S., Deringoz, F. A characterization for fractional integral and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type. Anal.Math.Phys. 9, 1991–2019 (2019). https://doi.org/10.1007/s13324-019-00295-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-019-00295-w

Keywords

Mathematics Subject Classification

Navigation