Skip to main content
SpringerLink
Log in

Two weight estimates for \(L^{r}\)-Hörmander singular integral operators and rough singular integral operators with matrix weights

  • Original Paper
  • Published:
Annals of Functional Analysis Aims and scope Submit manuscript

Abstract

In this paper, we give new bump conditions for two matrix weight inequalities of \(L^{r}\)-Hörmander singular integral operators and rough singular integral operators, which are new even in the scalar cases. As applications, we obtain quantitative one weight inequalities for rough singular integral operators.

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Bickel, K., Petermichl, S., Wick, B.D.: Bounds for the Hilbert transform with matrix \(A_2\) weights. J. Funct. Anal. 270, 1719–1743 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bownik, M.: Inverse volume inequalities for matrix weights. Indiana Univ. Math. J. 50, 383–410 (2001)

    Article  MathSciNet  Google Scholar 

  3. Culiuc, A., Di Plinio, F., Ou, Y.: Uniform sparse domination of singular integrals via dyadic shifts. Math. Res. Lett. 25, 21–42 (2018)

    Article  MathSciNet  Google Scholar 

  4. Cruz-Uribe, D., Isralowitz, J., Moen, K.: Two weight bump conditions for matrix weights. Integral Equ. Oper. Theory 90, 36 (2018)

    Article  MathSciNet  Google Scholar 

  5. Cruz-Uribe, D., Isralowitz, J., Moen, K., Pott, S., Rivera-Ríos, I.P.: Weak endpoint bounds for matrix weights. Rev. Mat. Iberoam. 37, 1513–1538 (2021)

    Article  MathSciNet  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  MathSciNet  Google Scholar 

  7. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia. Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer Basel AG, Basel (2011). (ISBN:978-3-0348-0071-6)

    Google Scholar 

  8. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229, 408–441 (2012)

    Article  MathSciNet  Google Scholar 

  9. Cruz-Uribe, D., Pérez, C.: On the two-weight problem for singular integral operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(1), 821–849 (2002)

    MathSciNet  Google Scholar 

  10. Di Plinio, F., Hytönen, T.P., Li, K.: Sparse bounds for maximal rough singular integrals via the Fourier transform. Ann. Inst. Fourier (Grenoble) 70, 1871–1902 (2020)

    Article  MathSciNet  Google Scholar 

  11. Domelevo, K., Kakaroumpas, S., Petermichl, S., Gibert, O., Soler, I.: Oundedness of Journé operators with matrix weights. J. Math. Anal. Appl. 532, 127956 (2024)

    Article  Google Scholar 

  12. Duong, X.T., Li, J., Yang, D.: Variation of Calderón–Zygmund operators with matrix weight. Commun. Contemp. Math. 23, 2050062 (2021)

    Article  Google Scholar 

  13. Goldberg, M.: Matrix \(A_p\) weights via maximal functions. Pac. J. Math. 211, 201–220 (2003)

    Article  Google Scholar 

  14. Huo, Z., Wick, B.D.: Weighted estimates of the Bergman projection with matrix weights. Preprint arXiv:2012.13810v2 (2020)

  15. Isralowitz, J.J., Kwon, H.-K., Pott, S.: Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols. J. Lond. Math. Soc. 2(96), 243–270 (2017)

    Article  MathSciNet  Google Scholar 

  16. Isralowitz, J., Pott, S., Rivera-Ríos, I.P.: Sharp \(A_1\) weighted estimates for vector-valued operators. J. Geom. Anal. 31, 3085–3116 (2021)

    Article  MathSciNet  Google Scholar 

  17. Lerner, A.K.: On an estimate of Calderón–Zygmund operators by dyadic positive operators. J. Anal. Math. 121, 141–161 (2013)

    Article  MathSciNet  Google Scholar 

  18. Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. 37, 225–265 (2019)

    Article  MathSciNet  Google Scholar 

  19. Lerner, A.K.: On separated bump conditions for Calderón–Zygmund operators. Proc. Am. Math. Soc. 150, 1197–1208 (2022)

    Article  Google Scholar 

  20. Lerner, A.K., Li, K., Ombrosi S., Rivera-Ríos, I.P.: On the sharpness of some matrix weighted endpoint estimates. Preprint arXiv:2310.06718 (2023)

  21. Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: On pointwise and weighted estimates for commutators of Calderón–Zygmund operators. Adv. Math. 319, 153–181 (2017)

    Article  MathSciNet  Google Scholar 

  22. Liu, L., Luque, T.: A \(B_p\) condition for the strong maximal function. Trans. Am. Math. Soc. 366, 5707–5726 (2014)

    Article  Google Scholar 

  23. Muller, P.A., Rivera-Ríos, I.P.: Quantitative matrix weighted estimates for certain singular integral operators. J. Math. Anal. Appl. 509, 125939 (2022)

    Article  MathSciNet  Google Scholar 

  24. Nazarov, F., Petermichl, S., Treil, S., Volberg, A.: Convex body domination and weighted estimates with matrix weights. Adv. Math. 318, 279–306 (2017)

    Article  MathSciNet  Google Scholar 

  25. Nazarov, F., Reznikov, A., Treil, S., Volberg, A.: A Bellman function proof of the \(L^2\) bump conjecture. J. Anal. Math. 121, 255–277 (2013)

    Article  MathSciNet  Google Scholar 

  26. Neugebauer, C.J.: Inserting \(A_p\)-weights. Proc. Am. Math. Soc. 87, 644–648 (1983)

    Google Scholar 

  27. Pérez, C.: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc. 2(49), 296–368 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. 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  Google Scholar 

  30. Rahm, R., Spencer, S.: Entropy bumps and another sufficient condition for the two-weight boundedness of sparse operators. Isr. J. Math. 223, 197–204 (2018)

    Article  MathSciNet  Google Scholar 

  31. Roudenko, S.: Matrix-weighted Besov spaces. Trans. Am. Math. Soc. 355, 273–314 (2003)

    Article  MathSciNet  Google Scholar 

  32. Sawyer, E.T., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)

    Article  MathSciNet  Google Scholar 

  33. Treil, S., Volberg, A.: Wavelets and the angle between past and future. J. Funct. Anal. 143, 269–308 (1997)

    Article  MathSciNet  Google Scholar 

  34. Treil, S., Volberg, A.: Entropy conditions in two weight inequalities for singular integral operators. Adv. Math. 301, 499–548 (2016)

    Article  MathSciNet  Google Scholar 

  35. Volberg, A.: Matrix \(A_p\) weights via S-funcions. J. Am. Math. Soc. 10, 445–466 (1997)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee cordially for his/her valuable remarks and suggestions. Yongming Wen is supported by the National Natural Science Foundation of China (Grant No. 12301119), the Natural Science Foundation of Fujian Province (No. 2021J05188), President’s fund of Minnan Normal University (No. KJ2020020), Institute of Meteorological Big Data-Digital Fujian, Fujian Key Laboratory of Data Science and Statistics and Fujian Key Laboratory of Granular Computing and Applications (Minnan Normal University), China. Fuli Ku is supported by The Key project of Fujian Provincial Education Department (Grant No. JZ230054), the scientific research start-up fund project for introducing high-level talents from Sanming University (No. 23YG09).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, People’s Republic of China

    Yongming Wen

  2. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Wenting Hu

  3. School of Information Engineering, Sanming University, Sanming, 365004, People’s Republic of China

    Fuli Ku

Authors
  1. Yongming Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenting Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fuli Ku
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fuli Ku.

Additional information

Communicated by Klaus Guerlebeck.

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

Wen, Y., Hu, W. & Ku, F. Two weight estimates for \(L^{r}\)-Hörmander singular integral operators and rough singular integral operators with matrix weights. Ann. Funct. Anal. 15, 24 (2024). https://doi.org/10.1007/s43034-024-00326-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43034-024-00326-z

Keywords

Mathematics Subject Classification

Search

Navigation