Skip to main content
SpringerLink
Log in

A Weighted Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

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

Abstract

Two proofs of a weighted weak-type \(\left( 1,\ldots ,1;\frac{1}{m}\right) \) estimate for multilinear Calderón–Zygmund operators are given. The ideas are motivated by different proofs of the classical weak-type (1, 1) estimate for Calderón–Zygmund operators. One proof uses the Calderón–Zygmund decomposition, and the other proof is motivated by ideas of Nazarov, Treil, and Volberg.

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. Caldarelli, M., Rivera-Ríos, I.P.: A sparse approach to mixed weak type inequalities. Math. Z. 296(1–2), 787–812 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weighted weak-type inequalities and a conjecture of Sawyer. Int. Math. Res. Not. 30, 1849–1871 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Damián, W., Lerner, A.K., Pérez, C.: Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators. J. Fourier Anal. Appl. 21(1), 161–181 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Domingo Salazar, C., Lacey, M., Rey, G.: Borderline weak-type estimates for singular integrals and square functions. Bull. Lond. Math. Soc. 48(1), 63–73 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 3rd edn. Springer, New York (2014)

    MATH  Google Scholar 

  6. Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)

    MATH  Google Scholar 

  7. Grafakos, L., Torres, R.: Multilinear Calderón-Zygmund theory. Adv. Math. 165(1), 124–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_{\infty }\). Anal. PDE 6(4), 777–818 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hytönen, T., Liu, S., Yang, D., Yang, D.: Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Can. J. Math. 64(4), 892–923 (2012)

    Article  MATH  Google Scholar 

  10. Lerner, A.K., Ombrosi, S., Pérez, C.: \(A_1\) bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett. 16(1), 149–156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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(4), 1222–1264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, K., Ombrosi, S., Pérez, C.: Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates. Math. Ann. 374(1–2), 907–929 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, K., Ombrosi, S.J., Picardi, B.: Weighted mixed weak-type inequalities for multilinear operators. Studia Math. 244(2), 203–215 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, Y.: Endpoint estimates for multilinear singular integral operators. Georgian Math. J. 23(4), 559–570 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Maldonado, D., Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity. J. Fourier Anal. Appl. 15(2), 218–261 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 9(9), 463–487 (1998)

    Article  MATH  Google Scholar 

  17. Ombrosi, S., Pérez, C.: Mixed weak type estimates: examples and counterexamples related to a problem of E. Sawyer. Colloq. Math. 145(2), 259–272 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Ombrosi, S., Pérez, C., Recchi, J.: Quantitative weighted mixed weak-type inequalities for classical operators. Indiana Univ. Math. J. 65(2), 615–640 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pérez, C., Torres, R.H.: Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators. Proc. Am. Math. Soc. Ser. B 1, 1–13 (2014)

    Article  MATH  Google Scholar 

  20. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  21. Stein, E.M., Weiss, G.: Interpolation of operators with change of measure. Trans. Am. Math. Soc. 87, 159–172 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stockdale, C.B.: A different approach to endpoint weak-type estimates for Calderón-Zygmund operators. J. Math. Anal. Appl. 487(2), 124016 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stockdale, C.B., Wick, B.D.: An endpoint weak-type estimate for multilinear Calderón-Zygmund operators. J. Fourier Anal. Appl. 25(5), 2635–2652 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO, 63130, USA

    Cody B. Stockdale

Authors
  1. Cody B. Stockdale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cody B. Stockdale.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stockdale, C.B. A Weighted Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators. J Geom Anal 33, 68 (2023). https://doi.org/10.1007/s12220-022-01123-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-01123-7

Keywords

Mathematics Subject Classification

Search

Navigation