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 Hölder’s condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm

$$\begin{aligned} \Vert e^{i\lambda \phi (\xi -\eta )}\Vert _{{\mathcal {M}}_{p,q,r}({{\mathbb {R}}}^n)}\rightarrow \infty ,~ \lambda \in {{\mathbb {R}}},~ |\lambda |\rightarrow \infty . \end{aligned}$$

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.