Skip to main content
SpringerLink
Log in

Weak (1,1) Boundedness of Oscillatory Singular Integral with Variable Phase

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

Abstract

In this paper, the weak (1,1) boundedness of oscillatory singular integral with variable phase P(x)γ(y) for any x, y ∈ ℝ,

$$Tf(x)\;:=\;{\rm{p}}.\>{\rm{v}}.\>\int_{- \infty}^\infty {{{\rm{e}}^{{\rm{i}}P(x)\gamma (y)}}f(x\;-\;y)\frac{dy}{y}}$$

is studied, where P is a real monic polynomial on ℝ.

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. Arias-de-Reyna, J.: Pointwise convergence of Fourier series. J. London Math. Soc., 65, 139–153 (2002)

    Article  MathSciNet  Google Scholar 

  2. Bennett, J. M.: Oscillatory singular integrals with variable flat phases and related operators. Ph.D. thesis, Edinburgh University, 1998

  3. Bennett, J. M.: Hilbert transforms and maximal functions along variable flat curves. Trans. Amer. Math. Soc., 354, 4871–4892 (2002)

    Article  MathSciNet  Google Scholar 

  4. Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math., 116, 135–157 (1966)

    Article  MathSciNet  Google Scholar 

  5. Chanillo, S., Christ, M.: Weak (1, 1) bounds for oscillatory singular integrals. Duke Math. J., 55, 141–155 (1987)

    Article  MathSciNet  Google Scholar 

  6. Chen, J., Zhu, X.: L 2-boundedness of Hilbert transforms along variable curves. J. Math. Anal. Appl., 395, 515–522 (2012)

    Article  MathSciNet  Google Scholar 

  7. Fan, D., Pan, Y.: Boundedness of certain oscillatory singular integrals. Studia Math., 114, 105–116 (1995)

    Article  MathSciNet  Google Scholar 

  8. Fefferman, C.: Pointwise convergence of Fourier series. Ann. of Math., 98, 551–571 (1973)

    Article  MathSciNet  Google Scholar 

  9. Folch-Gabayet, M., Wright, J.: Weak-type (1, 1) bounds for oscillatory singular integrals with rational phases. Studia Math., 210, 57–76 (2012)

    Article  MathSciNet  Google Scholar 

  10. Guo, S.: Oscillatory integrals related to Carleson’s theorem: fractional monomials. Commun. Pure Appl. Anal., 15, 929–946 (2016)

    Article  MathSciNet  Google Scholar 

  11. Hunt, R. A.: On the convergence of Fourier series. in: Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 235–255, Southern Illinois Univ. Press, Carbondale, Ill. 1968

    Google Scholar 

  12. Lacey, M., Thiele, C.: A proof of boundedness of the Carleson operator. Math. Res. Lett., 7, 361–370 (2000)

    Article  MathSciNet  Google Scholar 

  13. Lie, V.: The (weak-L 2) boundedness of the quadratic Carleson operator. Geom. Funct. Anal., 19, 457–497 (2009)

    Article  MathSciNet  Google Scholar 

  14. Lie, V.: The polynomial Carleson operator. arXiv:1105.4504

  15. Lie, V.: On the boundedness of the Carleson operator near L 1. Rev. Mat. Iberoam., 29, 1239–1262 (2013)

    Article  MathSciNet  Google Scholar 

  16. Li, J., Xiao, J., Yu, H.: L p boundedness of Carleson and Hilbert transforms associated with variable plane curves. Preprint

  17. Pan, Y.: L 2 boundedness of oscillatory integral operators. Duke Math. J., 62, 157–178 (1991)

    Article  MathSciNet  Google Scholar 

  18. Pan, Y.: Uniform estimates for oscillatory integral operators. J. Funct. Anal., 100, 207–220 (1991)

    Article  MathSciNet  Google Scholar 

  19. Pan, Y.: On the weak (1, 1) boundedness of a class of oscillatory singular integrals. Studia Math., 106, 279–287 (1993)

    Article  MathSciNet  Google Scholar 

  20. Pan, Y.: Weak (1, 1) estimate for oscillatory singular integrals with real-analytic phases. Proc. Amer. Math. Soc., 120, 789–802 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Pan, Y.: Oscillatory singular integrals on L p and Hardy spaces. Proc. Amer. Math. Soc., 124, 2821–2825 (1996)

    Article  MathSciNet  Google Scholar 

  22. Phong, D. H., Stein, E. M.: Hilbert integrals, singular integrals, and Radon transforms. I. Acta Math., 157, 99–157 (1986)

    Article  MathSciNet  Google Scholar 

  23. Sjölin, P.: An inequality of Paley and convergence a.e. of Walsh-Fourier series. Ark. Mat., 7, 551–570 (1969)

    Article  MathSciNet  Google Scholar 

  24. Soria, F.: Note on differentiation of integrals and the halo conjecture. Studia Math., 81, 29–36 (1985)

    Article  MathSciNet  Google Scholar 

  25. Soria, F.: On an extrapolation theorem of Carleson-Sjölin with applications to a.e. convergence of Fourier series. Studia Math., 94, 235–244 (1989)

    Article  MathSciNet  Google Scholar 

  26. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton, 1970

    MATH  Google Scholar 

  27. Stein, E. M., Wainger, S.: Oscillatory integrals related to Carleson’s theorem. Math. Res. Lett., 8, 789–800 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China

    Hai Xia Yu

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P.R. China

    Jun Feng Li

Authors
  1. Hai Xia Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Feng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Feng Li.

Additional information

Supported by NSFC-DFG (Grant No. 11761131002)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, H.X., Li, J.F. Weak (1,1) Boundedness of Oscillatory Singular Integral with Variable Phase. Acta. Math. Sin.-English Ser. 35, 1741–1759 (2019). https://doi.org/10.1007/s10114-019-8270-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-019-8270-3

Keywords

MR(2010) Subject Classification

Search

Navigation