Uniform bounds for the bilinear Hilbert transforms, I


It is shown that the bilinear Hilbert transforms \[ H_{\alpha,\beta} (f,g)(x) = \text{p.v.}\int_{\mathbf{R}} f(x-\alpha t)g(x-\beta t)\, \frac{dt}{t} \] map $L^{p_1}(\mathbf{R})\times L^{p_2}(\mathbf{R})\to L^p(\mathbf{R})$ uniformly in the real parameters $\alpha,\beta $ when $2 < p_1, p_2 < \infty$ and $1\lt p= \frac{p_1p_2}{p_1+p_2}<2$. Combining this result with the main result in [9], we deduce that the operators $H_{1, \alpha}$ map $L^2(\mathbf{R})\times L^\infty(\mathbf{R})\to L^2(\mathbf{R})$ uniformly in the real parameter $\alpha\in [0,1]$. This completes a program initiated by A. Calderón.


Loukas Grafakos

Department of Mathematics
University of Missouri
Columbia, MO 65211
United States

Xiaochun Li

Department of Mathematics
University of California, Los Angeles
Los Angeles, CA 90095
United States