关于黎曼对称空间的施瓦兹空间同构定理
On the Schwartz space isomorphism theorem for the Riemannian symmetric spaces
摘要：
给出了黎曼对称空间$X$上$L^p$ -Schwartz类函数$(0< p \leq 2) $的闭子空间$\mc S^p_Γ(X)$的同构定理的证明，其中$Γ$是$\what{K}_M$的有限子集.所考虑的傅里叶变换是Helgason傅里叶变换。我们的证明只依赖于相应的函数类的Paley-Wiener定理，因此它不使用复杂的高阶渐近的初等球函数。
英文摘要：
We deduce a proof of the isomorphism theorem for certain closed subspace $\mc S^p_Γ(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric space $X$ where $Γ$ is a finite subset of $\what{K}_M$. The Fourier transform considered is the Helgason Fourier transform. Our proof relies only on the Paley-Wiener theorem for the corresponding class of functions and hence it does not use the complicated higher asymptotics of the elementary spherical functions.
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