We continue the investigation initiated in [Grafakos and Li:
Uniform bounds for the bilinear Hilbert transforms, I. Ann. of
Math. (2) 159 (2004), 889-933] of uniform $L^{p}$ bounds for
the family of bilinear Hilbert transforms
$$
H_{\alpha,\beta} (f,g)(x) = \text{p.v.}
\displaystyle\int_{\mathbb{R}} f(x-\alpha t) g(x-\beta t)
\frac{dt}{t} \,.
$$
In this work we show that $H_{\alpha,\beta}$ map $L^{p_1}(\mathbb R)\times
L^{p_2}(\mathbb R)$ into $L^p(\mathbb R)$ uniformly in the real parameters $\alpha$,
$\beta$ satisfying $|\frac{\alpha}{\beta}-1|\ge c > 0$ when $1 < p_1, p_2 < 2$ and
$\frac{2}{3} < p= \frac{p_1p_2}{p_1+p_2} < \infty$. As a corollary we obtain
$L^p \times L^\infty \to L^p$ uniform bounds in the range $4/3 < p < 4 $ for the
$H_{1,\alpha}$'s when $\alpha\in [0,1)$.