An improved bilinear estimate for Benjamin-Ono type equations

A bilinear estimate in Fourier restriction norm spaces with applications to the Cauchy problem associated to u_t - |D|^{\alpha}u_x + uu_x =0 is proved, for 1 -{3/4}(\alpha-1) and \omega=1/\alpha-1/2. This extends to global well-posedness for all s \geq 0.