拟内函数与局部因子
Quasi-inner functions and local factors
摘要：
我们引入了{ \it准内}函数的概念，并证明了在有理数域的位置的有限集合$F$上，局部{ $L$ -}因子{ $ρ_v(z)=γ_v(z)/γ_v(1-z)$ }的$m+1$比的乘积$u=ρ_\infty\prod ρ_v$在以下意义上是临界线$\Re(z)= \frac 12$左边的{准内}.平方可积函数在临界线上的Hilbert空间$L^2$到Hardy空间$H^2$及其正交补的正交分解中，与$u$相乘的矩阵的非对角部分$u_{21}$是紧算子.当在单位圆盘上解释时，准内部条件意味着相关的Haenkel矩阵是紧的。我们表明，没有个人的非阿基米德比$ρ_v$是准内部的，为了证明我们的主要结果，我们使用高斯乘法定理的因素阿基米德比$ρ_\infty$到$m$准内部功能的产品，其产品与每个$ρ_v$保留的属性是准内部的。最后证明了Sonin空间是拟内函数$u=ρ_\infty$的对角部分$u_{22}$的核，当$u(F)=\prod_{v\in F} ρ_v$的核构成无限维空间的归纳系统时，$u(F)_{22}$是Sonin空间的半局部类似物.
英文摘要：
We introduce the notion of {\it quasi-inner} function and show that the product $u=ρ_\infty\prod ρ_v$ of $m+1$ ratios of local {$L$-}factors {$ρ_v(z)=γ_v(z)/γ_v(1-z)$} over a finite set $F$ of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line $\Re(z)= \frac 12$ in the following sense. The off diagonal part $u_{21}$ of the matrix of the multiplication by $u$ in the orthogonal decomposition of the Hilbert space $L^2$ of square integrable functions on the critical line into the Hardy space $H^2$ and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios $ρ_v$ is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio $ρ_\infty$ into a product of $m$ quasi-inner functions whose product with each $ρ_v$ retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part $u_{22}$ for the quasi-inner function $u=ρ_\infty$, and when $u(F)=\prod_{v\in F} ρ_v$ the kernels of the $u(F)_{22}$ form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.
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