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 $m(\xi,\eta)=e^{i \phi(\xi-\eta)}$. We prove that if $\phi$ 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 H\"{o}lder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm $$\|e^{i\lambda \phi(\xi-\eta)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ \lambda \in \mathbb R,~ |\lambda|\rightarrow \infty.$$ For exponents in the local $L^2-$range, we give examples of unimodular functions of the form $e^{i\phi(\xi-\eta)}$, 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.