Limited smoothness conditions with mixed norms for bilinear Fourier multipliers

In this paper, the $L^2\times L^{\mathrm{\infty }}\to L^2$ 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.