Unimodular bilinear Fourier multipliers on \(L^p\) spaces

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.