I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.

For instance,

Theorem: Any $A \in \Psi^{m}$ has the pseudolocal property $$ \begin{aligned} \text { sing } \sup (A u) & \subset \operatorname{sing} \operatorname{supp}(u) \\ \operatorname{sing} \operatorname{supp}^{\alpha-m}(A u) & \subset \operatorname{sing} \operatorname{supp}^{\alpha}(u) \end{aligned} $$ and the microlocal property $$ \begin{aligned} W F(A u) & \subset W F(u) \\ W F^{\alpha-m}(A u) & \subset W F^{\alpha}(u) \end{aligned} $$

As $\Psi^m$ is distribution whose domain is tempered distribution. But Author proves only for compactly supported distribution.

In functional analysis, with suitable norm using density property one can prove result for small set then it is also valid for larger set tho which that set is dense.

Is there any property between tempered and compactly supported distribution, that holds that similarly?

Any help or hint will be appreciated.



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