Revista Matemática Complutense

, Volume 29, Issue 1, pp 59–90 | Cite as

Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz–Morrey spaces

  • Denny Ivanal Hakim
  • Eiichi Nakai
  • Yoshihiro Sawano
Article

Abstract

In the present paper, we shall give a necessary and sufficient condition for the weak/strong boundedness of generalized fractional maximal operators on generalized Orlicz–Morrey spaces. We also give necessary and sufficient conditions for the vector-valued inequalities of the Hardy–Littlewood maximal operator, generalized fractional maximal operators and singular integral operators on these function spaces.

Keywords

Generalized fractional maximal operators Generalized Orlicz–Morrey spaces Hardy–Littlewood maximal operator Singular integral operators Vector-valued inequalities 

Mathematics Subject Classification

42B35 42B25 

References

  1. 1.
    Akbulut, A., Guliyev, V., Mustafayev, RCh.: On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces. Math. Bohem. 137(1), 27–43 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48(3), 356 (1962)CrossRefMATHGoogle Scholar
  3. 3.
    Burenkov, V.I., Guliyev, H.V., Guliyev, V.S.: Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces. J. Comput. Appl. Math. 208(1), 280–301 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Burenkov, V.I., Gogatishvili, A., Guliyev, V.S., Mustafayev, RCh.: Boundedness of the fractional maximal operator in local Morrey-type spaces. Complex Var. Elliptic Equ. 55(8–10), 739–758 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burenkov, V.I., Guliyev, V.S., Serbetci, A., Tararykova, T.V.: Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces. Eurasian Math. J. 1(1), 32–53 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)CrossRefMATHGoogle Scholar
  7. 7.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. 7, 273–279 (1987)MathSciNetMATHGoogle Scholar
  8. 8.
    Cordoba, A., Fefferman, C.: A weighted norm inequality for singular integrals. Stud. Math. 57(1), 97–101 (1976)MathSciNetMATHGoogle Scholar
  9. 9.
    Deringoz, F., Guliyev, V.S., Samko, S.: Boundedness of the maximal and singular operators on generalized Orlicz–Morrey spaces. In: Amélia Bastos, M., Lebre, A., Samko, S., Spitkovsky, I.M.(eds.) Operator theory, operator algebras and applications. Operator Theory: Advances and Applications, vol. 242, pp. 139–158. Birkhäuser/Springer, Basel (2014)Google Scholar
  10. 10.
    Ding, Y., Yang, D., Zhow, Z.: Boundedness of sublinear operators and commutators on \(L^{p, w}({\mathbb{R}}^n)\). Yokohama Math. J. 46, 15–26 (1998)MathSciNetMATHGoogle Scholar
  11. 11.
    Dzhabrailov, M.S., Khaligova, S.Z.: Anisotropic fractional maximal operator in anisotropic generalized Morrey spaces. J. Math. Res. 4(6), 109–120 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eridani, Gunawan, H., Nakai, E., Sawano, Y.: Characterizations for the generalized fractional integral operators on Morrey spaces. Math. Inequal. Appl. 17(2), 761–777 (2014)Google Scholar
  13. 13.
    Eridani, Utoyo, M.I., Gunawan, H.: A characterization for fractional integral operators on generalized Morrey spaces. Anal. Theory Appl. 28(3), 263–267 (2012)Google Scholar
  14. 14.
    Eroglu, A.: Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces. Bound. Value Probl. 70, 1–12 (2013)Google Scholar
  15. 15.
    Fan, Y.: Boundedness of sublinear operators and their commutators on generalized central Morrey spaces. J. Inequal. Appl. 411, 1–20 (2013)Google Scholar
  16. 16.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math 93, 107–115 (1971)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gala, S., Sawano, Y., Tanaka, H.: A remark on two generalized Orlicz–Morrey spaces. J. Approx. Theory 98, 1–9 (2015)Google Scholar
  18. 18.
    Guliyev, V.S.: Integral operators on function spaces on the homogeneous groups and on domains in \({\mathbb{R}}^n\) (Russian). Doctor degree dissertation, Mat. Inst. Steklov, Moscow (1994)Google Scholar
  19. 19.
    Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. Art. ID 503948, 20 (2009)Google Scholar
  20. 20.
    Guliyev, V.S., Aliyev, S.S., Karaman, T., Shukurov, P.S.: Boundedness of sublinear operators and commutators on generalized Morrey spaces. Integral Equ. Oper. Theory 71(3), 327–355 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Guliyev, V.S., Deringoz, F.: On the Riesz potential and its commutators on generalized Orlicz–Morrey spaces. J. Funct. Spaces Art. ID 617414, 11 (2014)Google Scholar
  22. 22.
    Guliyev, G.S., Omarova, M.N., Sawano, Y.: Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz–Morrey spaces. Banach J. Math. Anal. 9(2), 44–62 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gunawan, H., Hakim, D.I., Sawano, Y., Sihwaningrum, I.: Weak type inequalities for some integral operator on generalized nonhomogeneous Morrey spaces. J. Funct. Spaces Appl. 2013, 809704 (2013)Google Scholar
  24. 24.
    Ho, K.-P.: Littlewood–Paley spaces. Math. Scand. 108, 77–102 (2011)MathSciNetMATHGoogle Scholar
  25. 25.
    Ho, K.-P.: Vector-valued singular integral operators on Morrey type spaces and variable Triebel–Lizorkin–Morrey spaces. Ann. Acad. Sci. Fenn. Math. 37, 375–406 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hu, Y., Li, Z., Wang, Y.: Multilinear singular integral operators on generalized weighted Morrey spaces. J. Funct. Spaces Art. ID 325924, 12 (2014)Google Scholar
  27. 27.
    Kokilashvili, V., Krbec, M.: Weighted Inqualities in Lorentz and Orlicz Spaces, vol 57. World Scientific Publishing, River Edge, NJ (1991)Google Scholar
  28. 28.
    Komori-Furuya, Y., Matsuoka, K., Nakai, E., Sawano, Y.: Integral operators on \(B_{\sigma }\)-Morrey–Campanato spaces. Rev. Mat. Complut. 26(1), 1–32 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Komori, Y., Mizuhara, T.: Factorization of functions in \(H^1({\mathbb{R}}^n)\) and generalized Morrey spaces. Math. Nachr. 279(5–6), 619–624 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Lu, G., Lu, S., Yang, D.: Singular integrals and commutators on homogeneous groups. Anal. Math. 28, 103–134 (2002)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lukkassen, D., Meidell, A., Persson, L.E., Samko, N.: Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations. Math. Methods Appl. Sci. 35(11), 1300–1311 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mizuhara, T.: Commutators of singular integral operators on Morrey spaces with general growth functions. In: Kozono, H. (ed.) Harmonic Analysis and Nonlinear Partial Differential Equations (Kyoto, 1998), pp. 49–63. S\(\bar{\text{ u }}\)rikaisekikenky\(\bar{\text{ u }}\)sho K\(\bar{\text{ o }}\)ky\(\bar{\text{ u }}\)roku No. 1102 (1999)Google Scholar
  33. 33.
    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators, and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Nakai, E.: Generalized fractional integrals on Orlicz–Morrey spaces. In: Kato, M., Maligranda, L. (eds.) Banach and Function Spaces, pp. 323–333. Yokohama Publ., Yokohama (2004)Google Scholar
  35. 35.
    Nakai, E.: Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Stud. Math. 188, 193–221 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nakai, E.: Calderón–Zygmund operators on Orlicz–Morrey spaces and modular inequalities. In: Kato, M., Maligranda, L. (eds.) Banach and Function Spaces II, pp. 393–410. Yokohama Publ., Yokohama (2008)Google Scholar
  37. 37.
    Nakai, E.: Orlicz–Morrey spaces and their preduals. In: Kato, M., Maligranda, L., Suzuki, T. (eds.) Banach and Function spaces III, pp. 187–205. Yokohama Publ., Yokohama (2011)Google Scholar
  38. 38.
    Nakai, E., Sobukawa, T.: \(B^u_w\)-function spaces and their interpolation. arXiv:1410.6327
  39. 39.
    Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. 21(6), 1535–1544 (2005)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Sawano, Y., Tanaka, H.: Predual spaces of Morrey spaces with non-doubling measures. Tokyo J. Math. 32, 471–486 (2009)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    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)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Sawano, Y., Sugano, S., Tanaka, H.: Orlicz–Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Sawano, Y., Hakim, D.I., Gunawan, H.: Non-smooth atomic decomposition for generalized Orlicz–Morrey spaces. Math. Nachr (2015). doi:10.1002/mana.201400138
  44. 44.
    Shi, Y.L., Tao, X.: Some multi-sublinear operators on generalized Morrey spaces with non-doubling measures. J. Korean Math. Soc. 49(5), 907–925 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Softova, L.G.: Singular integrals and commutators in generalized Morrey spaces. Acta Math. Sin. 22(3), 757–766 (2006)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187–204 (1994)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Stein, E.M.: Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ (1993)Google Scholar
  48. 48.
    Wei, N., Niu, P.C., Tang, S., Zhu, M.: Estimates in generalized Morrey spaces for nondivergence degenerate elliptic operators with discontinuous coefficients. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 106(1), 1–33 (2012)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Yu, X., Tao, X.X.: Boundedness of multilinear operators on generalized Morrey spaces. Appl. Math. J. Chin. Univ. Ser. B 29(2), 127–138 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachiojiJapan
  2. 2.Department of MathematicsIbaraki UniversityMitoJapan

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