On the bilinear Hilbert transform (Q1126744)

The author and \textit{C. Thiele} in their highly acclaimed paper ``On Calderón's conjecture'' [Ann. Math. (2) 149, No. 2, 475-496 (1999; Zbl 0934.42012)] have proved that the bilinear Hilbert transform \[ Hfg(x):= \lim_{\varepsilon \rightarrow 0}\int_{|y|>\varepsilon}{f(x+y)g(x-y) {\frac{dy}{y}}} \] extends to a bounded operator on \(L^p\times L^p\) into \(L^r\) if \(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\), \(1<p,q\leq{\infty}\) and \(\frac{2}{3}<r< \infty\). This proves, in particular, a conjecture of Calderón from 1964 that \(H\) maps \(L^2 \times L^2\) into \(L^1\). The current highly readable article illustrates the method of proof of the general result by giving a complete proof of the weaker result that the bilinear Hilbert transform maps \(L^2 \times L^2\) into weak \(L^1\).