论文标题
本地和多线性非交易性de leeuw定理
Local and multilinear noncommutative de Leeuw theorems
论文作者
论文摘要
令$γ<g $为本地紧凑的单模型$ g $的离散子组。令$ m \ in C_b（g）$为$ g $上的$ p $ -Multiplier，$ 1 \ leq p <\ infty $，让$ t_ {m}：l_p（\ wideHat {g} {g}）\ rightArrow l_p（\ wideHat {g}）$是相应的fouriperier。同样，令$ t_ {m \vert_γ}：l_p（\wideHatγ）\ rightArrow l_p（\wideHatγ）$是与限制$ m |_γ$ $ m $ of $ m $至$γ$相关的傅立叶乘数。我们表明 \ [ c（{\ rm supp}（m \vert_γ））\ vert t_ {m \vert_γ}：l_p（\wideHatγ）\ rightarrow l_p（\wideHatγ）\ vert \ vert \ leq \ leq \ leq \ vert t_ {m} l_p（\ wideHat {g}）\ vert，\]对于特定常数$ 0 \ leq c（u）\ leq 1 $，该\ leq 1 $是为每个$ u \ subseteqγ$定义的。 功能$ c $量化了$ g $的失败，以接纳小$γ$ invariant的社区，并且可以在混凝土案例中明确确定。尤其是，$ c（γ）= 1 $当$ g $具有小$γ$ - invariant社区时。因此，我们的结果从[CPPR15]以及De Leeuw的经典定理[Lee65]扩展了De Leeuw限制定理。 对于真正的还原性谎言组$ g $，我们为$ c $提供明确的下限，以$ c $的最大尺寸$ d $ d $在伴随表示中。我们表明$ c（b_ρ^g）\ geqρ^{ - d/4} $其中$b_ρ^g $是$ g \ in g $ in g $的球，$ \ vert {\ rm ad} _g \ \ \ \ \ vert <ρ$。 我们进一步证明了多线性傅立叶乘数的几个结果。最重要的是，我们证明了对$ c（γ）= 1 $的$γ<g $的多线性de leeuw限制定理。我们还获得了晶格近似定理，紧凑定理和周期定理的多线性版本。因此，我们能够在非亚伯群体上提供双线性乘数的第一个例子。
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.
