Skip to main content
SpringerLink
Log in

The Multilinear Strong Maximal Function

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved.

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. Bagby, R.: Maximal functions and rearrangements: some new proofs. Indiana Univ. Math. J. 32, 879–891 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. Coifman, R.R., Meyer, Y.: Commutateurs d’ intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)

    MATH  MathSciNet  Google Scholar 

  3. Coifman, R.R., Meyer, Y.: Au délà des opérateurs pseudo-différentiels. Astérisque, No. 57, Societé Mathématique de France (1979)

  4. Córdoba, A.: On the Vitali covering properties of a differentiation basis. Studia Math. 57, 91–95 (1976)

    MATH  MathSciNet  Google Scholar 

  5. Córboda, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. 102, 95–100 (1975)

    Article  Google Scholar 

  6. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture. Adv. Math. 216, 647–676 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fefferman, R.: Multiparameter Fourier analysis. In: Beijing Lectures in Harmonic Analysis. Annals of Math. Studies, vol. 112, pp. 47–130. Princeton University Press, Princeton (1986)

    Google Scholar 

  8. García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North Holland, Amsterdam (1985)

    MATH  Google Scholar 

  9. Grafakos, L.: Modern Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 250. Springer, New York (2009)

    MATH  Google Scholar 

  10. Grafakos, L., Kalton, N.: Some remarks on multilinear maps and interpolation. Math. Ann. 319, 151–180 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51, 1261–1276 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hagelstein, P., Stokolos, A.: An extension of the Córdoba–Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators. C. R. Acad. Sci. Paris, Ser. I 346, 1063–1065 (2008)

    MATH  MathSciNet  Google Scholar 

  14. Hagelstein, P., Stokolos, A.: Tauberian conditions for geometric maximal operators. Trans. Am. Math. Soc. 361, 3031–3040 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jawerth, B.: Weighted inequalities for maximal operators: linearization, localization, and factorization. Am. J. Math. 108, 361–414 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jawerth, B., Torchinsky, A.: The strong maximal function with respect to measures. Studia Math. 80, 261–285 (1984)

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo–González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220, 1222–1264 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lerner, A.K.: An elementary approach to several results on the Hardy–Littlewood maximal operator. Proc. Am. Math. Soc. 136, 2829–2833 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60, 213–238 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Neugebauer, C.J.: Inserting A p -weights. Proc. Am. Math. Soc. 87, 644–648 (1983)

    MATH  MathSciNet  Google Scholar 

  23. Pérez, C.: Weighted norm inequalities for general maximal operators. Publ. Mat. 35, 169–186 (1991)

    MATH  MathSciNet  Google Scholar 

  24. Pérez, C.: A remark on weighted inequalities for general maximal operators. Proc. Am. Math. Soc. 119, 1121–1126 (1993)

    Article  MATH  Google Scholar 

  25. Pérez, C.: On sufficient conditions for the boundedness of the Hardy–Littlewood maximal operator between weighted L p-spaces with different weights. Proc. Lond. Math. Soc. (3) 71, 135–157 (1995)

    Article  MATH  Google Scholar 

  26. Pérez, C.: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc. 49, 296–308 (1994)

    MATH  Google Scholar 

  27. Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43, 663–683 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-point estimates for iterated commutators of multilinear singular integrals. arXiv:1004.4976

  30. Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

    Google Scholar 

  31. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

  32. Sawyer, E.T.: A characterization of a two-weight norm weight inequality for maximal operators. Studia Math. 75, 1–11 (1982)

    MATH  MathSciNet  Google Scholar 

  33. Tao, T.: A converse extrapolation theorem for translation-invariant operators. J. Funct. Anal. 180, 1–10 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wilson, M.: Littlewood–Paley Theory and Exponential-Square Integrability. Lectures Notes in Math., vol. 1924. Springer, Berlin (2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

  2. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, China

    Liguang Liu

  3. Departamento De Análisis Matemático, Facultad de Matemáticas, Universidad De Sevilla, 41080, Sevilla, Spain

    Carlos Pérez

  4. Department of Mathematics, University of Kansas, 405 Snow Hall 1460 Jayhawk Blvd, Lawrence, KS, 66045-7523, USA

    Rodolfo H. Torres

Authors
  1. Loukas Grafakos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liguang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rodolfo H. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Loukas Grafakos.

Additional information

Communicated by Marco Peloso.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grafakos, L., Liu, L., Pérez, C. et al. The Multilinear Strong Maximal Function. J Geom Anal 21, 118–149 (2011). https://doi.org/10.1007/s12220-010-9174-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-010-9174-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation