On some bilinear Fourier multipliers with oscillating factors, I

Bilinear Fourier multipliers of the form $e^{i(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\xi +\eta \mathrm{\mid })}\sigma (\xi ,\eta )$ are considered. It is proved that if $\sigma (\xi ,\eta )$ is in the Hörmander class $S_{1,0}^m({\mathbb{R}}^{2n})$ with $m=-(n+1)\mathrm{/}2$ then the corresponding bilinear operator is bounded in $L^{\mathrm{\infty }}\times L^{\mathrm{\infty }}\to bmo$, $h^1\times L^{\mathrm{\infty }}\to L^1$, and $L^{\mathrm{\infty }}\times h^1\to L^1$. This improves a result given by Rodríguez-López, Rule and Staubach.