Skip to main content
SpringerLink
Log in

An Endpoint Estimate for the Commutators of Singular Integral Operators with Rough Kernels

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

Let Ω be homogeneous of degree zero and have mean value zero on the unit sphere Sd− 1, TΩ be the homogeneous singular integral operator with kernel \(\frac {\Omega (x)}{|x|^{d}}\) and TΩ,b be the commutator of TΩ with symbol b. In this paper, we prove that if \({\Omega }\in L(\log L)^{2}(S^{d-1})\), then for \(b\in \text {BMO}(\mathbb {R}^{d})\), TΩ,b satisfies an endpoint estimate of \(L\log L\) type.

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. Alvarez, J., Babgy, R. J., Kurtz, D., Pérez, C.: Weighted estimates for commutators of linear operators. Studia Math. 104, 195–209 (1993)

    Article  MATH  Google Scholar 

  2. Calderón, A. P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MATH  Google Scholar 

  3. Calderón, A. P., Zygmund, A.: On singular integrals. Amer. J. Math. 78, 289–309 (1956)

    Article  MATH  Google Scholar 

  4. Christ, M., Rubio de Francia, J. L.: Weak type (1,1) bounds for rough operators, II. Invent Math. 93, 225–237 (1988)

    Article  MATH  Google Scholar 

  5. Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103, 611–635 (1976)

    Article  MATH  Google Scholar 

  6. Ding, Y., Lai, X.: Weak type (1,1) bounded criterion for singular integral with rough kernel and its applications. Trans. Amer. Math. Soc. 371, 1649–1675 (2019)

    Article  MATH  Google Scholar 

  7. Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Amer. Math. Soc. 336, 869–880 (1993)

    Article  MATH  Google Scholar 

  8. Duoandikoetxea, J., Rubio de Francia, J. L.: Maximal and singular integrals via Fourier transform estimates. Invent Math. 84, 541–561 (1986)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  10. Hu, G.: \(L^{p}(\mathbb {R}^{n})\) boundedness for the commutator of a homogeneous singular integral. Studia Math. 154, 13–47 (2003)

    Article  Google Scholar 

  11. Hu, G., Sun, Q., Wang, X.: \(L^{p}(\mathbb {R}^{n})\) bounds for commutators of convolution operators. Colloquium Math. 93, 11–20 (2002)

    Article  Google Scholar 

  12. Hytönen, T., Pérez, C.: Sharp weighted bounds involving A1. Anal. PDE. 6, 777–818 (2013)

    Article  MATH  Google Scholar 

  13. Kurtz, D., Wheeden, R. L.: Results on weighted norm inequalities for multiplier. Trans. Amer. Math. Soc. 255, 343–362 (1979)

    Article  MATH  Google Scholar 

  14. Lerner, A. K.: A weak type estimates for rough singular integrals. Rev. Mat. Iberoam. 35, 1583–1602 (2019)

    Article  MATH  Google Scholar 

  15. Lan, J., Tao, X., Hu, G.: Weak type endpoint estimates for the commutators of rough singular integral operators. Math. Inequal. Appl. 23, 1179–1195 (2020)

    MATH  Google Scholar 

  16. Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for singular and fractional integrals. Trans. Amer. Math. Soc. 161, 249–258 (1971)

    Article  MATH  Google Scholar 

  17. O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Amer. Math. Soc. 115, 300–328 (1965)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Rao, M., Ren, Z.: Theory of Orlicz spaces Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc., New York (1991)

    Google Scholar 

  20. Ricci, F., Weiss, G.: A characterization of H1(Sn− 1). In: Wainger, S., Weiss, G. (eds.) Proc. Sympos. Pure Math. of Amer. Math. Soc., vol. 35 I, pp 289–294 (1979)

  21. Seeger, A.: Singular integral operators with rough convolution kernels. J. Amer. Math. Soc. 9, 95–105 (1996)

    Article  MATH  Google Scholar 

  22. Stein, E. M., Weiss, G.: Interpolation of operators with change of measures. Trans. Amer. Math. Soc. 87, 159–172 (1958)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincerely thanks to the referee for his/her valuable remarks and suggestions, which made this paper more readable. Also, The authors would like to thank Dr. Israel. P. Rivera-Ríos for helpful comments.

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Beijing Normal University, Zhuhai, 519087, People’s Republic of China

    Guoen Hu

  2. Department of Mathematics, School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    Xiangxing Tao

Authors
  1. Guoen Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiangxing Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangxing Tao.

Additional information

Publisher’s Note

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

The research of the first author was supported by the NNSF of China (Nos. 11871108, 11971295), and the research of the second (corresponding) author was supported by the NNSF of China (No. 11771399).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, G., Tao, X. An Endpoint Estimate for the Commutators of Singular Integral Operators with Rough Kernels. Potential Anal 58, 241–262 (2023). https://doi.org/10.1007/s11118-021-09939-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-021-09939-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation