Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 675–711 | Cite as

A modulation invariant Carleson embedding theorem outside local L2

  • Francesco di Plinio
  • Yumeng Ou


Yen Do and Christoph Thiele developed a theory of Carleson embeddings in outer Lp spaces for the wave packet transform
$${F_\phi }(f)(u,t,\eta ) = \int {f(x){e^{i\eta (u - x)}}\phi \left( {\frac{{u - x}}{t}} \right)} \frac{{dx}}{t},(u,t,\eta ) \in R \times (0,\infty ) \times R$$
of functions fLp(R) in the range 2 ≤ p ≤ ∞, referred to as local L2. In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Borwein and T. Erdélyi, Nikolskii-type inequalities for shift invariant function spaces, Proc. Amer. Math. Soc. 134 (2006), 3243–3246.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    F. Di Plinio, Weak-Lp bounds for the Carleson and Walsh-Carleson operators, C. R. Math. Acad. Sci. Paris 352 (2014), 327–331.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    F. Di Plinio and Y. Ou, Banach-valued multilinear singular integrals, Indiana Univ. Math. J., to appear. arXiv:1506.05827[math.CA].Google Scholar
  5. [5]
    F. Di Plinio and C. Thiele, Endpoint bounds for the bilinear Hilbert transform, Trans. Amer. Math. Soc. 368 (2016), 3931–3972.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Y. Do and C. Thiele, Lp theory for outer measures and two themes of Lennart Carleson united, Bull. Amer. Math. Soc. (N.S.) 52 (2015), 249–296.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    L. Grafakos and X. Li, Uniform bounds for the bilinear Hilbert transforms. I, Ann. of Math. (2) 159 (2004), 889–933.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    T. P. Hytönen M. T. Lacey, Pointwise convergence of vector-valued Fourier series, Math. Ann. 357 (2013), 1329–1361.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    M. Lacey and C. Thiele, Lp estimates on the bilinear Hilbert transform for 2 < p < ∞, Ann. of Math. (2) 146 (1997), 693–724.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Lacey and C. Thiele, On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), 475–496.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    X Li, Uniform bounds for the bilinear Hilbert transforms. II, Rev. Mat. Iberoam. 22 (2006), 1069–1126.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    C. Muscalu, J. Pipher, T. Tao, and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam. 22 (2006), 963–976.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis. Vol. II, Cambridge University Press, Cambridge, 2013.MATHGoogle Scholar
  14. [14]
    C. Muscalu, T. Tao, and C. Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), 469–496.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    F. Nazarov, R. Oberlin, and C. Thiele, A Calderón-Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain, Math. Res. Lett. 17 (2010), 529–545.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    R. Oberlin and C. Thiele, New uniform bounds for a Walsh model of the bilinear Hilbert transform, Indiana Univ. Math. J. 60 (2011), 1693–1712.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    C. Thiele, A uniform estimate, Ann. of Math. (2) 156 (2002), 519–563.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    C. Thiele, Wave Packet Analysis, American Mathematical Society, Providence RI, 2006.CrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations