Skip to main content
SpringerLink
Log in

Quantitative weighted bounds for singular integrals and fractional differentiations

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we study the weighted bounds for the singular integrals with variable kernels. Let \(T_\Omega \) be the singular integral operator with variable kernel and \(T_\Omega ^\star \) be the associated maximal singular integral operator. More precisely, we first obtain the quantitative weighted bounds that depend on some variants of the \(A_p\) constant of w for the maximal singular integral \(T_\Omega ^{\star }\) and it’s commutator. Let \(T_\Omega ^{*}\) be the adjoint of \(T_\Omega \) and let \(T_\Omega ^{\sharp }\) be the pseudo-adjoint of \(T_\Omega \). Secondly, we get some quantitative weighted bounds for these singular integral operators with the fractional differentiation operator \(D^{\gamma }(0<\gamma \le 1)\).

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 statement

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

References

  1. Aguilera, N.E., Harboure, E.O.: Some inequalities for maximal operators. Indiana Univ. Math. J. 29, 559–576 (1980)

    Article  MathSciNet  Google Scholar 

  2. Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc. 340, 253–272 (1993)

    Article  MathSciNet  Google Scholar 

  3. Calderón, A.P., Zygmund, A.: On a problem of Mihlin. Trans. Amer. Math. Soc. 78, 209–224 (1955)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Calderón, A.P., Zygmund, A.: Singular integral operators and differential equations. Amer. J. Math. 79, 901–921 (1957)

    Article  MathSciNet  Google Scholar 

  6. Calderón, A.P.: Commutators of singular integrals. Proc. Nat. Acad. Sci. USA 53, 1092–1099 (1965)

    Article  MathSciNet  Google Scholar 

  7. Calderón, A. P., Zygmund, A.: On singular integrals with variable kernels, Appl. Anal. 7(1977/78), 221-238

  8. Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2, p}\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ric. Mat. 40, 149–168 (1991)

    MATH  Google Scholar 

  9. Christ, M., Duoandikoetxea, J., Rubio De Francia, J.: Maximal operators related to the radon transform and the Calderón-Zygmund method of rotations. Duke Math. J. 53, 189–209 (1986)

    Article  MathSciNet  Google Scholar 

  10. Chung, D., Pereyra, C., Pérez, C.: Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Amer. Math. Soc. 364, 1163–1177 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Conde-Alonso, J.M., Culiuc, A., Di Plinio, F., Ou, Y.: A sparse domination principle for rough singular integrals. Anal. PDE 10, 1255–1284 (2017)

    Article  MathSciNet  Google Scholar 

  13. Di Fazio, G., Palagachev, D., Ragusa, M.: Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, 179–196 (1999)

    Article  MathSciNet  Google Scholar 

  14. Di Fazio, G., Ragusa, M.A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112, 241–256 (1993)

    Article  MathSciNet  Google Scholar 

  15. Di Plinio, F., Hytönen, T. P., Li, K.: Sparse bounds for maximal rough singular integrals via the Fourier transform, Annales de l’institut Fourier, to appear. Available at arXiv:1706.09064

  16. Duong, X. T., Li, J., Yang, D.: Variation of Calderón-Zygmund operators with Matrix Weight, to appear in Communications in Contemporary Mathematics

  17. Fujii, N.: Weighted bounded mean oscillation and singular integrals, Math. Japon. 22 (1977/1978), 529-534

  18. Hu, G.: Quantitative weighted bounds for the composition of Calderón-Zygmund operators. Banach J. Math. Anal. 13, 133–150 (2019)

    Article  MathSciNet  Google Scholar 

  19. Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. 175, 1473–1506 (2012)

    Article  MathSciNet  Google Scholar 

  20. Hytönen, T., Lacey, M.T.: The \(A_p\)-\(A_\infty \) inequality for general Calderón-Zygmund operators. Indiana Univ. Math. J. 61, 2041–2092 (2012)

    Article  MathSciNet  Google Scholar 

  21. Hytönen, T., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218, 133–164 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lacey, M.T.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217, 181–195 (2017)

    Article  MathSciNet  Google Scholar 

  23. Lerner, A.K.: On pointwise estimates involving sparse operators. New York J. Math. 22, 341–349 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Lerner, A. K.: A note on weighted bounds for rough singular integrals, C. R. Acad. Sci. Paris, Ser. I. 356 (2018), 77-80

  25. Lu, S., Ding, Y., Yan, D.: Singular integrals and related topics, World Scientific, (2007)

  26. Murray, M.A.M.: Commutators with fractional differentiation and BMO Sobolev spaces. Indiana Univ. Math. J. 34, 205–215 (1985)

    Article  MathSciNet  Google Scholar 

  27. Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) -characteristic. Amer. J. Math. 129, 1355–1375 (2007)

    Article  MathSciNet  Google Scholar 

  28. Petermichl, S.: The sharp weighted bound for the Riesz transforms. Proc. Amer. Math. Soc. 136, 1237–1249 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N. J. (1971)

    MATH  Google Scholar 

  31. Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ (1993)

  32. Strichartz, R.S.: Bounded mean oscillation and Sobolev spaces. Indiana Univ. Math. J. 29, 539–558 (1980)

    Article  MathSciNet  Google Scholar 

  33. Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic \(A_\infty \). Duke Math. J. 55, 19–50 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their deep gratitude to the referees for giving many valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Teng Wang & Yanping Chen

Authors
  1. Teng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanping Chen.

Additional information

Publisher's Note

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

The project was in part supported by: Yanping Chen’s National Natural Science Foundation of China (# 11871096, # 11471033).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, T., Chen, Y. Quantitative weighted bounds for singular integrals and fractional differentiations. Anal.Math.Phys. 12, 58 (2022). https://doi.org/10.1007/s13324-022-00672-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-022-00672-y

Keywords

Mathematics Subject Classification

Search

Navigation