Uniform Bounds for the Bilinear Hilbert Transforms, II

Abstract

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

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 $\frac{2}{3}<p=\frac{p_1{p}_2}{p_1+p_2}<\infty$. 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 $\alpha \in [0,1)$.