Uniform bounds for the bilinear Hilbert transforms, I

Abstract

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.