Uniform bounds for the bilinear hilbert transforms, II

We continue the investigation initiated in [8] of uniform L^p bounds for the family of bilinear Hilbert transforms H _α,β(f,g)(x) = p.v. ∫_ℝ f(x - αt)g(x - βt)dt/t. In this work we show that H _α,β map L^p1(ℝ) × L ^p2(ℝ) into L^p(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 L^p × L^∞ → L^p uniform bounds in the range 4/3 < p < 4 for the H_1,α's when α ∈ [0, 1).