The Hardy–Littlewood Maximal Operator on Discrete Morrey Spaces

  • Hendra Gunawan
  • Christopher SchwankeEmail author


We discuss the Hardy–Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman–Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.


Discrete Morrey spaces Hardy–Littlewood maximal operator Riesz potential 

Mathematics Subject Classification

42B35 46B45 46A45 



  1. 1.
    Berezhnoi, E.I.: A discrete version of local Morrey spaces. Izv. Math. 81, 1 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. 7(3–4), 273–279 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gunawan, H., Eridani, Nakai, E.: On generalized fractional integral operators. Scientiae Math. Japon. Online 10, 307–318 (2014)Google Scholar
  4. 4.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93(1), 107–115 (1971)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gunawan, H., Kikianty, E., Schwanke, C.: Discrete Morrey spaces and their inclusion properties. Math. Nachr. 291(8–9), 1283–1296 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Magyar, A., Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis: spherical averages. Ann. Math. 155, 189–208 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis (Sendai, 1990), pp. 183–189, ICM-90 Satell. Conf. Proc., Springer, Tokyo (1991)Google Scholar
  8. 8.
    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pierce, L.B.: Discrete analogues in harmonic analysis. Ph.D. Dissertation, Princeton University (2009)Google Scholar
  10. 10.
    Stein, E.M., Wainger, S.: Discrete analogues of singular Radon transform. Bull. Am. Math. Soc. 23, 537–544 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis I: \(\ell ^2\) estimates for singular Radon transforms. Am. J. Math. 21, 1291–1336 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis II: fractional integration. J. d’Analyse Math. 80, 335–355 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Stein, E.M., Wainger, S.: Two discrete fractional integral operators revisited. J. d’Analyse Math. 87, 451–479 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBandung Institute of TechnologyBandungIndonesia
  2. 2.Department of MathematicsLyon CollegeBatesvilleUSA

Personalised recommendations