  In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to
L^1$ boundedness of bilinear Fourier multiplier operators is discussed under
weak smoothness conditions on multipliers. As an application, we prove the $L^2
\times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear
operators with multipliers of limited smoothness satisfying vanishing
conditions.
