Skip to main content
SpringerLink
Log in

Mixed Morrey spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman–Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, K.F., John, R.T.: Weighted inequalities for vector-valued maximal functions and singular integrals. Stud. Math. 69, 19–31 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bagby, R.J.: An extended inequality for the maximal function. Proc. Am. Math. Soc. 48, 419–422 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benedek, A., Panzone, R.: The spaces \(L^{P}\), with mixed norm. Duke Math. J. 28, 301–324 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Curbera, G.P., García-Cuerva, J., Martell, J.M., Pérez, C.: Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 20 203(1), 256–318 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duoandikoetxea, J.: Fourier analysis. Translated and revised from the 1995 Spanish original by D. Cruz-Uribe. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathmatics, vol. 249. Springer, New York (2008)

    MATH  Google Scholar 

  10. Jessen, B., Marcinkiewicz, J., Zygmund, A.: Note on the differentiability of multiple integrals. Fund. Math. 25, 217–234 (1935)

    Article  MATH  Google Scholar 

  11. Liu, F., Torres, R., Xue, Q., Yabuta, K.: Multilinear strong maximal operators on mixed Lebesgue spaces (Preprint)

  12. Morrey Jr., C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  13. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Muckenhoupt, B., Hunt, R., Weeden, R.: Weighted norm inequalities for the conjugate function and the Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peetre, J.: On the theory of \({\cal{L}}_{p, \lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)

    Article  Google Scholar 

  16. Sawano, Y., Hakim, D.I., Gunawan, H.: Non-smooth atomic decomposition for generalized Orlicz–Morrey spaces. Math. Nachr. 288(14–15), 1741–1775 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  19. Stöckert, B.: Ungleichungen von Plancherel–Polya–Nikol’skij typ in gewichteten \(L_p^{\Omega }\)- Räumen mit gemischten Norm. Math. Nach. 86, 19–32 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tanaka, H.: Personal communication

  21. Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278, 904–917 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Professor Yoshihiro Sawano for enthusiastic guidance and be also grateful to Professor Hitoshi Tanaka for his kind suggestion on the fractional integral operators. Furthermore, the author thanks the anonymous referee for his/her comments on this paper, which improved readability.

Author information

Authors and Affiliations

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

    Toru Nogayama

Authors
  1. Toru Nogayama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toru Nogayama.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nogayama, T. Mixed Morrey spaces. Positivity 23, 961–1000 (2019). https://doi.org/10.1007/s11117-019-00646-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-019-00646-8

Keywords

Mathematics Subject Classification

Search

Navigation