On some bilinear Fourier multipliers with oscillating factors, II

Abstract

For $s>0$, $s\ne 1$, 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 $S_{1,0}^m({\mathbb{R}}^{2n})$. 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.

17 pages