The bilinear Hilbert transform is pointwise finite (Q1382014)

For the bilinear Hilbert transform \[ H(fg)(x)=\text{p.v. }\int^\infty_{-\infty} f(x+ y) g(x- y){dy\over y}, \] the author proved that if \(f\in L^\infty\) and \(g\in L^2\) are supported on \([0,1]\), then \[ \| H(fg)\|_{L^1(\mathbb{R})}\leq C\| f\|_{L^\infty(\mathbb{R})}\| g\|_{L^2(\mathbb{R})}. \] As a consequence, the bilinear Hilbert transform is pointwise finite for \(f\in L^\infty\) and \(g\in L^2\). The paper gives some concrete indication that the conjecture, by A. Calderón, that \(H\) maps \(L^2\times L^\infty\) into \(L^2\) could be true. A further development of the paper can be found in [\textit{M. Lacey} and \textit{C. Thiele}, Proc. Natl. Acad. Sci. USA 94, No. 1, 33-35 (1997)].