论文标题
$ l^p $空间上的单型双线性傅立叶乘数
Unimodular bilinear Fourier multipliers on $L^p$ spaces
论文作者
论文摘要
在本文中，我们研究了与形式$ m（ξ，η）= e^{i ϕ（ξ-η）} $相关的双线性乘数运算符的有界性能。我们证明，如果$ ϕ $是$ c^1（\ mathbb r^n）$ reameal valued非线性功能，则对于所有指数$ \ frac {1} {p}+\ frac {1} {q} = \ frac {1} {1} {r} $，biinear乘数norm $ $ $ \ | e^{iλcation（iλcation） \ infty，〜λ \ in \ mathbb r，〜|λ| \ rightarrow \ infty。此外，我们还讨论了本地$ l^2- $范围以外的指数的双线性乘数的基本连续性属性。
In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(ξ,η)=e^{i ϕ(ξ-η)}$. We prove that if $ϕ$ 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 $$\|e^{iλϕ(ξ-η)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ λ\in \mathbb R,~ |λ|\rightarrow \infty.$$ For exponents in the local $L^2-$range, we give examples of unimodular functions of the form $e^{iϕ(ξ-η)}$, 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.
