On some bilinear Fourier multipliers with oscillating factors, I

Abstract

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

24 pages