SpringerLink
Log in

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

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MathSciNet  MATH  Google Scholar 

  2. Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48(3), 356 (1962)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  6. Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Cordoba, A., Fefferman, C.: A weighted norm inequality for singular integrals. Stud. Math. 57(1), 97–101 (1976)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  11. Dzhabrailov, M.S., Khaligova, S.Z.: Anisotropic fractional maximal operator in anisotropic generalized Morrey spaces. J. Math. Res. 4(6), 109–120 (2012)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  14. Eroglu, A.: Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces. Bound. Value Probl. 70, 1–12 (2013)

  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. Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math 93, 107–115 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gala, S., Sawano, Y., Tanaka, H.: A remark on two generalized Orlicz–Morrey spaces. J. Approx. Theory 98, 1–9 (2015)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  24. Ho, K.-P.: Littlewood–Paley spaces. Math. Scand. 108, 77–102 (2011)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hu, Y., Li, Z., Wang, Y.: Multilinear singular integral operators on generalized weighted Morrey spaces. J. Funct. Spaces Art. ID 325924, 12 (2014)

  27. Kokilashvili, V., Krbec, M.: Weighted Inqualities in Lorentz and Orlicz Spaces, vol 57. World Scientific Publishing, River Edge, NJ (1991)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lu, G., Lu, S., Yang, D.: Singular integrals and commutators on homogeneous groups. Anal. Math. 28, 103–134 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

  35. Nakai, E.: Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Stud. Math. 188, 193–221 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  38. Nakai, E., Sobukawa, T.: \(B^u_w\)-function spaces and their interpolation. ar**v:1410.6327

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Sawano, Y., Tanaka, H.: Predual spaces of Morrey spaces with non-doubling measures. Tokyo J. Math. 32, 471–486 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sawano, Y., Sugano, S., Tanaka, H.: Orlicz–Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  45. Softova, L.G.: Singular integrals and commutators in generalized Morrey spaces. Acta Math. Sin. 22(3), 757–766 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187–204 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. Stein, E.M.: Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ (1993)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are thankful to anonymous referees for their careful reading of our paper and some fruitful comments. The second author was supported by Grant-in-Aid for Scientific Research (C), No. 24540159 and Grant-in-Aid for Scientific Research (B), No. 15H03621, Japan Society for the Promotion of Science. The third author was supported by Grant-in-Aid for Young Scientists (B), No. 24740085, Japan Society for the Promotion of Science.

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo, 192-0397, Japan

    Denny Ivanal Hakim & Yoshihiro Sawano

  2. Department of Mathematics, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

    Eiichi Nakai

Authors
  1. Denny Ivanal Hakim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eiichi Nakai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoshihiro Sawano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denny Ivanal Hakim.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hakim, D.I., Nakai, E. & Sawano, Y. Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz–Morrey spaces. Rev Mat Complut 29, 59–90 (2016). https://doi.org/10.1007/s13163-015-0178-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-015-0178-6

Keywords

Mathematics Subject Classification

Search

Navigation