Advertisement

Journal of Fourier Analysis and Applications

, Volume 20, Issue 4, pp 751–765 | Cite as

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

  • Kangwei Li
  • Kabe MoenEmail author
  • Wenchang Sun
Article
First Online:

Abstract

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).

Keywords

Multiple weights Multilinear maximal function Multilinear Calderón–Zygmund operators Weighted estimates 

Mathematics Subject Classification

42B20 42B25 47H60 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The authors thank Carlos Pérez for helping improve the quality of this article.

References

  1. 1.
    Buckley, S.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc. 340, 253–272 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Christ, M., Journé, J.-L.: Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159, 51–80 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212, 315–331 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229, 408–441 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Damián, W. Lerner A.K., Pérez C.: Sharp weighted bounds for multilinear maximal functions and Calderón–Zygmund operators, available at http://arxiv.org/abs/1211.5115
  7. 7.
    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York (2008)Google Scholar
  8. 8.
    Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. 175, 1473–1506 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hytönen T. P.: The \(A_2\) theorem: Remarks and complements, preprint, available at http://arxiv.org/abs/1212.3840
  11. 11.
    Hytönen T., Lacey M.: The \(A_p\)-\(A_\infty \) inequality for general Calderón–Zygmund operators. Indiana Univ. Math. J. 61, 2041–2052 (2012)Google Scholar
  12. 12.
    Hytönen T., Lacey M., Pérez C.: Sharp weighted bounds for \(q\)-variation of singular integrals. Bull. Lond. Math. Soc. 45, 529–540 (2013)Google Scholar
  13. 13.
    Hytönen T., Pérez C.: Sharp weighted bounds involving \(A_\infty \). J. Anal. & P.D.E. 6, 777–818 (2013)Google Scholar
  14. 14.
    Kenig, C.E., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lacey, M., Petermichl, S., Reguera, M.: Sharp \(A_2\) inequality for Haar shift operators. Math. Ann. 348, 127–141 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lerner, A.K.: An elementary approach to several results on the Hardy–Littlewood maximal operator. Proc. Am. Math. Soc. 136, 2829–2833 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lerner, A.K.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)MathSciNetGoogle Scholar
  18. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Moen, K.: Sharp weighted bounds without testing or extrapolation. Arch. Math. 99, 457–466 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjin China
  2. 2.Department of MathematicsUniversity of AlabamaTuscaloosaUSA

Personalised recommendations

Cite article

Buy options