Uniform bounds for the bilinear Hilbert transforms, I

It is shown that the bilinear Hilbert transforms H _α,β(f,g)(x) = p.v. ∫_R f(x - αt)g(x - βt) dt/t map L^p1(R) × L^p2(R) → L ^p2(R) uniformly in the real parameters α,β when 2 < p₁,p₂ < ∞ and 1 < p = p₁p ₂/p₁p₂ < 2. Combining this result with the main result in [9], we deduce that the operators H_1,α map L²(R) × L^∞(R) → L²(R) uniformly in the real parameter α ∈ [0, 1]. This completes a program initiated by A. Calderón.