Russian Mathematics

, Volume 63, Issue 9, pp 12–21 | Cite as

Modified Fractional Hardy and Hardy-Littlewood Operators and Their Commutators

  • S. S. VolosivetsEmail author
  • B. I. GolubovEmail author


For modified fractional Hardy and Hardy-Littlewood operators and their commutators with symbol from a central mean oscillation space, considered as acting from a modified Herz space into another one, we find conditions of their boundedness. The sharpness of the result concerning commutators of the fractional Hardy-Littlewood operator is established.

Key words

modified Hardy and Hardy-Littlewood operators commutator modified Herz space CMOq space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors express their gratitude to the anonymous referee for helpful remarks which allow to improve the text of the paper.


  1. 1.
    Golubov, B., Efimov, A., Skvortsov, V. Walsh series and transforms. Theory and applications (Kluver Academic Publishers, Dordrecht, Boston, London, 1991).CrossRefGoogle Scholar
  2. 2.
    Hardy, G.H., Littlewood, J.E., Pólya, G. Inequalities (Cambridge University Press, Cambridge, 1952).zbMATHGoogle Scholar
  3. 3.
    Andersen, K.F., Muckenhoupt, B. “Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions”, Studia Math. 72 (1), 9–26 (1982).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Volosivets, S.S. “Identities of Titchmarsh Type for Generalized Hardy and Hardy-Littlewood Operators”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 13 (1(2)) (2013), 28–33 [in Russian].CrossRefGoogle Scholar
  5. 5.
    Golubov, B.I. “On dyadic analogues of Hardy and Hardy-Littlewood operators”, Siberian Math. J. 40 (6), 1051–1058 (1999).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Golubov, B.I. “On the Boundedness of Dyadic Hardy and Hardy-Littlewood Operators on the Dyadic Spaces H and BMO”, Analysis Mathematica 26 (4), 287–298 (2000) [in Russian].MathSciNetCrossRefGoogle Scholar
  7. 7.
    Volosivets, S.S. “On P-adic analogs of Hardy and Hardy-Littlewood operators”, East J. Approx. 11 (1), 57–72 (2005).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Volosivets, S.S. “Modified Hardy and Hardy-Littlewood operators and their behaviour in various spaces”, Izv. Math. 75 (1), 29–51 (2011).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, Y.Z., Lau, K.S. “Some new classes of Hardy spaces”, J. Funct. Anal. 84 (2), 255–278 (1989).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garcia-Cuerva, J. “Hardy spaces and Beurling algebras”, J. London Math. Soc. 39 (3), 499–513 (1989).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Long, S., Wang, J. “Commutators of Hardy operators”, J. Math. Anal. Appl. 274 (2), 626–644 (2002).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Herz, C. “Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms”, J. Math. Mech. 18 (4), 283–324 (1968).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Onneweer, C.W. “Generalized Lipschitz spaces and Herz spaces on certain totally disconnected groups”(in: Martingale theory in harmonic analysis and Banach spaces, Lect. Notes in Math. 939, 106–121, Springer, Berlin, (1981)).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gao, G., Zhong, Y. “Some estimates of Hardy operators and their commutators on Morrey-Herz spaces”, J. Math. Ineq. 11 (1), 49–58 (2017).MathSciNetCrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Saratov State UniversitySaratovRussia
  2. 2.Moscow Institute of Physics and Technology (State University)Moscow region, DolgoprudnyRussia

Personalised recommendations