局部和多线性非交换de Leeuw定理
Local and multilinear noncommutative de Leeuw theorems
摘要：
设$Γ< G$是局部紧幺模群$G$的离散子群.令$m\in C_b(G)$是具有$1 \leq p < \infty$的$G$上的$p$ -乘法器，并令$T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$是相应的傅立叶乘法器。类似地，令$T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ)$为与$m$到$Γ$的限制$m|_Γ$相关联的傅立叶乘数。我们表明，\[ c( {\rm supp}( m\vert_Γ ) ) \Vert T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \]为一个特定的常数$0 \leq c(U) \leq 1$，这是为每个$U \subseteq Γ$定义。函数$c$量化$G$的失败，承认小几乎$Γ$不变的邻域，并可以明确地确定在具体情况下。特别地，当$G$具有小的几乎$Γ$不变的邻域时，$c(Γ) =1$。因此，我们的结果推广了De Leeuw限制定理[CPPR 15]以及De Leeuw的经典定理[Lee 65]。对于实约化李群$G$，我们给出了$c$在伴随表示中幂零轨道的最大维数$d$的一个显式下界。我们证明了$c(B_ρ^G) \geq ρ^{-d/4}$，其中$B_ρ^G$是具有$\Vert {\rm Ad}_g \Vert < ρ$的$g\in G$的球。我们进一步证明了多线性傅立叶乘法器的几个结果。最重要的是，我们证明了一个多线性De Leeuw限制定理对$Γ<G$与$c(Γ) = 1$。我们还得到了格逼近定理，紧化定理和周期化定理的多线性版本。因此，我们能够提供的第一个例子，双线性乘子的nonabel群。
英文摘要：
Let $Γ< G$ be a discrete subgroup of a locally compact unimodular group $G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p < \infty$ and let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding Fourier multiplier. Similarly, let $T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ)$ be the Fourier multiplier associated to the restriction $m|_Γ$ of $m$ to $Γ$. We show that \[ c( {\rm supp}( m\vert_Γ ) ) \Vert T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \] for a specific constant $0 \leq c(U) \leq 1$ that is defined for every $U \subseteq Γ$. The function $c$ quantifies the failure of $G$ to admit small almost $Γ$-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, $c(Γ) =1$ when $G$ has small almost $Γ$-invariant neighbourhoods. Our result thus extends the De Leeuw restriction theorem from [CPPR15] as well as De Leeuw's classical theorem [Lee65]. For real reductive Lie groups $G$ we provide an explicit lower bound for $c$ in terms of the maximal dimension $d$ of a nilpotent orbit in the adjoint representation. We show that $c(B_ρ^G) \geq ρ^{-d/4}$ where $B_ρ^G$ is the ball of $g\in G$ with $\Vert {\rm Ad}_g \Vert < ρ$. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs $Γ<G$ with $c(Γ) = 1$. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.
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