Skip to main content
SpringerLink
Log in

Weighted oscillation and variation inequalities for singular integrals and commutators satisfying Hörmander type conditions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper is devoted to investigating the weighted L p-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type (p, p) estimates for 1 < p < ∞ and the weighted weak type (1, 1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hörmander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini’s conditions as model examples. Meanwhile, we also obtain the weighted L p-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(Rd)-functions.

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. Akcoglu, M., Jones, R. L., Schwartz, P.: Variation in probability, ergodic theory and analysis. Illions J. Math., 42(1), 154–177 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Betancor, J. J., Fariña, J. C., Harbour, E., et al.: L p-boundedness properits of variation operators in the Schrödinger setting. Rev. Mat. Complut., 26(2), 485–534 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bougain, J.: Pointwise ergodic theorem for arithmatic sets. Inst. Hautes Études Sci. Publ. Math., 69, 5–45 (1989)

    Article  Google Scholar 

  4. Campbell, J. T., Jones, R. L., Reinhold, K., et al.: Oscillation and variation for Hilbert transform. Duke Math. J., 105, 59–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Campbell, J. T., Jones, R. L., Reinhold, K., et al.: Oscillation and variation for singular integrals in higher dimension. Trans. Amer. Math. Soc., 355, 2115–2137 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crescimbeni, R., Macías, R. A., Menárguez, T., et al.: The p-variation as an operator between maximal operators and singular integrals. J. Evol. Equ., 9, 81–102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crescimbeni, R., Martin-Reyes, F. J., Torre, A. L., et al.: The p-variation of the Hermitian Riesz transform. Acta Math. Sin., Engl. Ser., 26, 1827–1838 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fefferman, C., Stein, E. M.: H p spaces of severable variables. Acta Math., 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gillespie, T. A., Torrea, J. L.: Dimension free estimates for the oscillation of Riesz transforms. Israel J. Math., 141, 125–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harboure, E., Macías, R. A., Menárguez, M. T., et al.: Oscillation and variation for the Gaussian Riesz transforms and Poisson integral. Proc. Roy. Soc. Edinburgh Set. A, 135, 85–104 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jones, R. L.: Ergodic theory and connections with analysis and probability. New York J. Math., 3A, 31–67 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Jones, R. L.: Variation inequalities for singular integrals and related operators. Contemp. Math., 411, 89–121 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jones, R. L., Kaufman, R., Rosenblatt, J., Wierdl, M.: Oscillation in ergodic theory. Ergodic Theory Dynam. Systems, 18, 889–935 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jones, R. L., Ostrovskii, I., Rosenblatt, J.: Square functions in ergodic theory. Ergodic Theory Dynam. Systems, 16, 267–305 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jones, R. L., Reinhold, K.: Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam. Systems, 21, 1809–1829 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jones, R. L., Rosenblatt, J. M., Wierdl, M.: Oscillation in ergodic theory: higher dimensional results. Israel J. Math., 135(1), 1–27 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jones, R. L., Seeger, A., Wright, J.: Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc., 360, 6711–6742 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, F., Wu, H.: A criterion on osillation and variation for the commutators of singular integral operators. Forum Math., 27, 77–97 (2015)

    MathSciNet  Google Scholar 

  20. Liu, H.: Operators related to truncated Hilbert transforms on H 1. Acta Math. Sin., Engl. Ser., 33(7), 1011–1020 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, T., Torrea, J. L., Xu, Q.: Weighted variation inequalities for differential operators and singular integrals. J. Funct. Anal., 268(2), 376–416 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, T., Torrea, J. L., Xu, Q.: Weighted variation inequalities for differential operators and singular integrals in higher dimensions. arXiv:1511.05129

  23. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Peréz, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commutators. J. London Math. Soc., 65(3), 672–692 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rubio de Francia, J. L., Ruiz, F. J., Torrea, J. L.: Calderón–Zygmund theory for vector-valued functions. Adv. Math., 62, 7–48 (1986)

    Article  MATH  Google Scholar 

  26. Watson, D. K.: Weighted estimates for singular integrals via Fourier transform estimates. Duke Math. J., 60(2), 389–399 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Yili Normal College, Yining, 835000, P. R. China

    Jing Zhang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Huo Xiong Wu

Authors
  1. Jing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huo Xiong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huo Xiong Wu.

Additional information

Supported by the NNSF of China (Grant Nos. 11371295 and 11471041), the NSF of Fujian Province of China (Grant No. 2015J01025) and Foundation for Doctors of Yili Normal College (Grant No. 2017YSBS09)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Wu, H.X. Weighted oscillation and variation inequalities for singular integrals and commutators satisfying Hörmander type conditions. Acta. Math. Sin.-English Ser. 33, 1397–1420 (2017). https://doi.org/10.1007/s10114-017-6379-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-017-6379-9

Keywords

MR(2010) Subject Classification

Search

Navigation