Unimodular bilinear Fourier multipliers on $L^p$ spaces

Abstract

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(\xi ,\eta )=e^{i\varphi (\xi -\eta )}$. We prove that if $\varphi$ is a $C^1({\mathbb{R}}^n)$ real-valued non-linear function, then for all exponents $p,q,r$ lying outside the local $L^2-$range and satisfying the Hölder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm $$\Vert e^{i\lambda \varphi (\xi -\eta )}{\Vert }_{{\mathcal{M}}_{p,q,r}({\mathbb{R}}^n)}\to \infty ,\lambda \in \mathbb{R},|\lambda |\to \infty .$$ For exponents in the local $L^2-$range, we give examples of unimodular functions of the form $e^{i\varphi (\xi -\eta )}$, which do not give rise to bilinear multipliers. Further, we also discuss the essential continuity property of bilinear multipliers for exponents outside local $L^2-$ range.

Typos corrected. To appear in Monatsh. Math