Revista Matemática Iberoamericana


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Volume 22, Issue 3, 2006, pp. 1069–1126
DOI: 10.4171/RMI/483

Uniform Bounds for the Bilinear Hilbert Transforms, II

Xiaochun Li (1)

(1) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Street, IL 61801-2975, URBANA, UNITED STATES

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 $\alpha\in [0,1)$.

Keywords: Time-frequency analysis, bilinear Hilbert transform, uniform bounds