  We prove $L^p$ estimates for the Bi-Carleson operator, which is a natural
hybrid of the Carleson maximal operator and the bilinear Hilbert transform. The
methods used are essentially based on the treatment of the Walsh analogue of
the operator in the prequel of this paper, but with additional technicalities
due to the fact that in the Fourier model one cannot obtain perfect
localization in both space and frequency.
