On some bilinear Fourier multipliers with oscillating factors, I

TL;DR

This paper studies bilinear Fourier multipliers with oscillating exponential factors, proving boundedness results in various function spaces for specific symbol classes, thus extending previous work in harmonic analysis.

Contribution

It establishes new boundedness results for bilinear multipliers with oscillating factors in the H"ormander class, improving prior results by Rodrf3guez-Lf3pez, Rule, and Staubach.

Findings

Boundedness in $L^{ } imes L^{ } o bmo$

Boundedness in $h^{1} imes L^{ } o L^{1}$

Extension of previous boundedness results

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 H\"ormander class $S^{m}_{1,0} (\mathbb{R}^{2n})$ 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 Rodr\'iguez-L\'opez, Rule and Staubach.