The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator

Article, Preprint English OPEN
Lie, Victor (2007)
  • Publisher: eScholarship, University of California
  • Journal: 457-497 (issn: 1016-443X)
  • Related identifiers: doi: 10.1007/s00039-009-0010-x
  • Subject: Geometry and Topology | Mathematics | Time-frequency analysis | 42A99 | Mathematics - Classical Analysis and ODEs | quadratic phase | Analysis | Carleson’s theorem

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase.
  • References (12)
    12 references, page 1 of 2

    Now, for x ∈ Ij fixed, we conclude that 1 B(x) sup I⊃Ij |I| I

    1 + |2ξk| −n−2 (δ1/3dj,l)n|ξ|ndξ 1 + |2ξk| −n−2 dξ 2k(2kδ1/3dj,l)n .

    [C] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157.

    [F] C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551-571.

    [L1] M. Lacey, Carleson's theorem: proof, complements, variations, Publ. Mat. 48:2 (2004), 251-307.

    [L2] M. Lacey, Carleson's theorem with quadratic phase functions, Studia Math. 153 (2002), 249-267.

    [LT1] M. Lacey, C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7:4 (2000), 361-370.

    [LT2] M. Lacey, C. Thiele, On Calderon's conjecture, Ann. of Math. 149 (1999), 475- 496.

    [LT3] M. Lacey, C. Thiele, Lp bounds for the bilinear Hilbert transform, p > 2, Ann. of Math. 146 (1997), 693-724.

    [S1] E.M. Stein, On limits of sequences of operators, Ann. of Math. 74 (1961), 140-170.

  • Metrics
    No metrics available
Share - Bookmark