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Positivity

pp 1–31 | Cite as

Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

  • Fatih Deringoz
  • Vagif S. Guliyev
  • Eiichi Nakai
  • Yoshihiro Sawano
  • Minglei Shi
Article
  • 22 Downloads

Abstract

In the present paper, we will characterize the boundedness of the generalized fractional integral operators \(I_{\rho }\) and the generalized fractional maximal operators \(M_{\rho }\) on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators \(M_{\rho }\) and \(I_{\rho }\) on generalized Orlicz–Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators \(M_{\rho }\) and \(I_{\rho }\) on generalized Orlicz–Morrey spaces.

Keywords

Generalized fractional maximal function Generalized fractional integral Orlicz spaces Generalized Orlicz-Morrey spaces 

Mathematical Subject Classification

42B20 42B25 42B35 46E30 

Notes

Acknowledgements

The research of F. Deringoz was partially supported by the Grant of Ahi Evran University Scientific Research Project (FEF.A4.18.019). The research of V. Guliyev was partially supported by the Grant of 1st Azerbaijan–Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1) and by the Ministry of Education and Science of the Russian Federation (the Agreement No. 02.a03.21.0008). Eiichi Nakai was supported by Grant-in-Aid for Scientific Research (B), No. 15H03621, Japan Society for the Promotion of Science. Yoshihiro Sawano was supported by Grant-in-Aid for Scientific Research (C) (16K05209), the Japan Society for the Promotion of Science and by Peoples Friendship University of Russia.

References

  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cianchi, A.: Strong and weak type inequalities for some classical operators in Orlicz spaces. J. Lond. Math. Soc. 60(1), 187–202 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Deringoz, F., Guliyev, V.S., Samko, S.: Boundedness of maximal and singular operators on generalized Orlicz–Morrey spaces. Oper. Theory Oper. Algebras Appl. Ser. Oper. Theory: Adv. Appl. 242, 139–158 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Deringoz, F., Guliyev, V.S., Hasanov, S.G.: A characterization for Adams type boundedness of the fractional maximal operator on generalized Orlicz–Morrey spaces. Integral Transform. Spec. Funct. 28(4), 284–299 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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).  https://doi.org/10.1186/s13660-016-1192-z
  6. 6.
    Eridani, A., Gunawan, H., Nakai, E., Sawano, Y.: Characterizations for the generalized fractional integral operators on Morrey spaces. Math. Inequal. Appl. 17(2), 761–777 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gala, S., Sawano, Y., Tanaka, H.: A remark on two generalized Orlicz–Morrey spaces. J. Approx. Theory 198, 1–9 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guliyev, V.S., Hasanov, S.G., Sawano, Y., Noi, T.: Non-smooth atomic decompositions for generalized Orlicz–Morrey spaces of the third kind. Acta Appl. Math. 145, 133–174 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A., Serbetci, A.: Generalized Fractional integral operators on generalized local Morrey spaces. J. Funct. Spaces (2015).  https://doi.org/10.1155/2015/594323 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hakim, D.I., Nakai, E., Sawano, Y.: Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz–Morrey spaces. Rev. Mat. Complut. 29(1), 59–90 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hästo, P.A.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269(12), 4038–4048 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kawasumi R., Nakai E.: Pointwise multipliers on weak Orlicz spaces. arXiv:1811.02858
  14. 14.
    Kokilashvili, V., Krbec, M.M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)CrossRefGoogle Scholar
  15. 15.
    Maligranda, L.: Orlicz spaces and interpolation, Seminários de Matemática, vol. 5 (1989)Google Scholar
  16. 16.
    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(3), 707–744 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nakai, E.: A characterization of pointwise multipliers on the Morrey spaces. Sci. Math. 3(3), 445–454 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nakai, E.: On generalized fractional integrals in the Orlicz spaces. In: Proceedings of the Second ISAAC Congress, Vol. 1, pp. 75–81. (Fukuoka, 1999), Int. Soc. Anal. Appl. Comput., 7, Kluwer Acad. Publ., Dordrecht (2000)Google Scholar
  19. 19.
    Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5, 587–602 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nakai, E.: On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type. Sci. Math. Jpn. 54, 473–487 (2001)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Nakai, E.: On generalized fractional integrals on the weak Orlicz spaces, \({\rm BMO}_{\phi }\), the Morrey spaces and the Campanato spaces. Function spaces, interpolation theory and related topics (Lund, 2000), de Gruyter. Berlin 2002, pp. 389–401 (2000)Google Scholar
  22. 22.
    Nakai, E.: Generalized fractional integrals on Orlicz–Morrey spaces. In: Banach and Function Spaces, pp. 323–333. (Kitakyushu, 2003), Yokohama Publishers, Yokohama (2004)Google Scholar
  23. 23.
    Nakai, E., Sumitomo, H.: On generalized Riesz potentials and spaces of some smooth functions. Sci. Math. Jpn. 54(3), 463–472 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    O’Neil, R.: Fractional integration in Orlicz spaces. I. Trans. Am. Math. Soc. 115, 300–328 (1965)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pustylnik, E.: Generalized potential type operators on rearrangement invariant spaces. Israel Math. Conf. Proc. 13(3), 161–171 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43, 663–683 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Dekker, New York (1991)zbMATHGoogle Scholar
  28. 28.
    Spanne, S.: Some function spaces defined using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa 19(3), 593–608 (1965)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sawano, Y.: Theory of Besov spaces, Development in Mathematics vol. 56Google Scholar
  30. 30.
    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)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sawano, Y., Sugano, S., Tanaka, H.: Orlicz–Morrey spaces and fractional operators. Potential Anal. 36(4), 517–556 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Weiss, G.: A note on Orlicz spaces. Portugal Math. 15, 35–47 (1950)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsAhi Evran UniversityKirsehirTurkey
  2. 2.Department of MathematicsDumlupinar UniversityKutahyaTurkey
  3. 3.Institute of Mathematics and MechanicsBakuAzerbaijan
  4. 4.S.M. Nikolskii Institute of Mathematics at RUDN UniversityMoscowRussia
  5. 5.Department of MathematicsIbaraki UniversityMitoJapan
  6. 6.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachiojiJapan

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