On some bilinear Fourier multipliers with oscillating factors, II

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