Uniform bounds for the bilinear hilbert transforms, II

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We continue the investigation initiated in [8] of uniform Lp bounds for the family of bilinear Hilbert transforms H α,β(f,g)(x) = p.v. ∫ f(x - αt)g(x - βt)dt/t. In this work we show that H α,β map Lp1(ℝ) × L p2(ℝ) into Lp(ℝ) uniformly in the real parameters α, β satisfying |α/β-1|≥c>0 when 1 < P1,P2 < 2 and 2/3 < p = p1p 2/p1+p2 < ∞. As a corollary we obtain Lp × L → Lp uniform bounds in the range 4/3 < p < 4 for the H1,α's when α ∈ [0, 1).

Original languageEnglish (US)
Pages (from-to)1069-1126
Number of pages58
JournalRevista Matematica Iberoamericana
Issue number3
StatePublished - 2006


  • Bilinear Hilbert transform
  • Time-frequency analysis
  • Uniform bounds

ASJC Scopus subject areas

  • Mathematics(all)


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