On some bilinear Fourier multipliers with oscillating factors, II

For $s>0$, $s\mathrm{\ne }1$, bilinear Fourier multipliers of the form $e^{i(\mathrm{\mid }\xi {\mathrm{\mid }}^s+\mathrm{\mid }\eta {\mathrm{\mid }}^s+\mathrm{\mid }\xi +\eta {\mathrm{\mid }}^s)}\sigma (\xi ,\eta )$ are considered, where $\sigma (\xi ,\eta )$ belongs to the Hörmander class $S_{1,0}^m({\mathbb{R}}^{2n})$. A criterion for $m$ to ensure the $L^{\mathrm{\infty }}\times L^{\mathrm{\infty }}\to L^{\mathrm{\infty }}$, $L^1\times L^{\mathrm{\infty }}\to L^1$, and $L^{\mathrm{\infty }}\times L^1\to L^1$ boundedness of the corresponding bilinear operators is given.