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On a hybrid of bilinear Hilbert transform and paraproduct

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Abstract

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range \(D = \left\{ {\left( {{p_1},{p_2},{p_3}} \right) \in {\mathbb{R}^3}:1 < {p_1},{p_2} < \infty ,\frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} < \frac{3}{2},1 < {p_3} < \infty } \right\}\).

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References

  1. Bernicot, F.: Uniform estimates for paraproducts and related multilinear multipliers. Rev. Mat. Iberoam, 25, 1055–1088 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernicot, F.: L p estimates for non-smooth bilinear Littlewood–Paley square functions on R. Math. Ann., 351, 1–49 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Grafakos, L., Li, X.: Uniform bounds for the bilinear Hilbert transforms. I. Ann. of Math., 159, 889–933 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lacey, M., Thiele, C.: L p estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. of Math., 146, 693–724 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lacey, M., Thiele, C.: On Calder´on’s conjecture. Ann. of Math., 149, 475–496 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, X.: Uniform bounds for the bilinear Hilbert transforms. II. Rev. Mat. Iberoam., 22, 1069–1126 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, X.: Uniform estimates for some paraproducts. New York J. Math., 14, 145–192 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Li, X.: Bilinear Hilbert transforms along curves I: The monomial case. Anal. PDE, 6, 197–220 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, X., Xiao, L.: Uniform estimates for bilinear Hilbert transform and bilinear maximal functions associated to polynomials. Amer. J. Math., 138, 907–962 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Muscalu, C., Schlag, W.: Classical and Multilinear Harmonic Analysis. Vol. II, Cambridge University Press, Cambridge, 2013

    MATH  Google Scholar 

  11. Muscalu, C., Tao, T., Thiele, C.: Uniform estimates on paraproducts. J. Anal. Math., 87, 369–384 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Thiele, C.: A uniform estimate. Ann. of Math., 156, 519–563 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Dong Dong & Xiao Chun Li

Authors
  1. Dong Dong
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  2. Xiao Chun Li
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Correspondence to Dong Dong.

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Dong, D., Li, X.C. On a hybrid of bilinear Hilbert transform and paraproduct. Acta. Math. Sin.-English Ser. 34, 29–41 (2018). https://doi.org/10.1007/s10114-017-6415-9

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  • DOI: https://doi.org/10.1007/s10114-017-6415-9

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