A Bourgain type bilinear estimate for a class of water-wave models

The bilinear estimtate in proposition 7.15 [J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Parts II, Geometric Funct. Anal. 3(3) (1993) 209-262.] plays an essential role in the study of the nonlinear term of KdV equation. In this paper, this estimate is extended to the a more general water-vave equations. We hope this result could shed some light on the estimates of nonlinear terms of water-vave equations.