Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range


We prove that for operators satistying weighted inequalities with Ap weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x|αw(x) for wAp. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the Hardy-Littlewood maximal operator, and by S. Nakamura and Y. Sawano for Calderón-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain Aq condition.

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


  1. 1.

    Adams, D.R.: Morrey spaces, Lecture notes in applied and numerical harmonic analysis. Birkhäuser/Springer, Cham (2015)

    Google Scholar 

  2. 2.

    Cruz-Uribe, D.V., Martell, J.M., Pérez, C.: Weights, extrapolation and the theory of Rubio de Francia operator theory: Advances and applications, vol. 215. Basel, Birkhäuser (2011)

    Google Scholar 

  3. 3.

    Duoandikoetxea, J., Rosenthal, M.: Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings. J. Geom. Anal. 28, 3081–3108 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Johnson, R., Neugebauer, C.J.: Change of variable results for Ap, and reverse Hölder RHr-classes. Trans. Amer. Math. Soc. 328, 639–666 (1991)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Knese, G., McCarthy, J.E., Moen, K.: Unions of Lebesgue spaces and A1 majorants. Pacific J. Math. 280, 411–432 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Nakamura, S., Sawano, Y.: The singular integral operator and its commutator on weighted Morrey spaces. Collect. Math. 68, 145–174 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350, 56–72 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Samko, N.: On a Muckenhoupt-type condition for Morrey spaces. Mediterr. J. Math. 10, 941–951 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

    Tanaka, H.: Two-weight norm inequalities on Morrey spaces. Ann. Acad. Sci. Fenn. Math. 40, 773–791 (2015)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Javier Duoandikoetxea.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author is supported by the grants MTM2014-53850-P of the Ministerio de Economía y Competitividad (Spain) and grant IT-641-13 of the Basque Gouvernment.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Duoandikoetxea, J., Rosenthal, M. Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range. Potential Anal 53, 1255–1268 (2020).

Download citation


  • Morrey spaces
  • Muckenhoupt weights
  • Hardy-Littlewood maximal operator
  • Calderón-Zygmund operators

Mathematics Subject Classification (2010)

  • 42B35
  • 42B25
  • 46E30
  • 42B20