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 Holder'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.

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$$L^p$$Lp 空间上的单模双线性傅立叶乘法器

在本文中，我们研究了与 $m(\xi,\eta)=e^{i\phi(\xi-\eta)}$ 形式的单模函数相关的双线性乘法算子的有界性质。我们证明如果 $\phi$ 是一个 $C^1(\mathbb R^n)$ 实值非线性函数，那么对于所有位于局部 $L^2 之外的指数 $p,q,r$- $range 并满足 Holder 条件 $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$，双线性乘子范数 $$\|e^{i\lambda \phi(\xi-\eta)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ \lambda \in \mathbb R,~ |\lambda |\rightarrow \infty.$$ 对于局部 $L^2-$ 范围内的指数，我们给出了 $e^{i\phi(\xi-\eta)}$ 形式的单模函数的例子，它不产生双线性乘法器。更多，