Uniform Bounds for the Bilinear Hilbert Transforms, II

We continue the investigation initiated in (8) of uniform Lp bounds for the family of bilinear Hilbert transforms H; (f; g)(x) = p.v. Z R f(x t )g(x t) dt t . In this work we show that H; map Lp 1 (R) Lp 2 (R) into Lp(R) uniformly in the real parameters , sat- isfying j 1j c > 0 when 1 < p1; p2 < 2 and 2 3 < p = p 1p2 p1+p2 < 1. As a corollary we obtain Lp L1 ! Lp uniform bounds in the range 4=3 < p < 4 for the H1; 's when 2 (0; 1).