This question is related to my previous one: "Distributions valued in Fréchet spaces".
Consider a function $u = u(t,x)$ such that
- $u \in C^{\infty}([0,T] \times \mathbb{R}^n )$
- $u \in D'(0,T; S(\mathbb{R}^n))$
where $S(\mathbb{R}^n)$ is the space of rapidly decreasing functions.
Q1) Can I conclude that $u \in C^{\infty}(0,T; S(\mathbb{R}^n))$ ?
It seems quite reasonable but I couldn't find a neat argument.
Further thoughts (added later)
As a first step I would like to answer the following:
Q2) Given 1. and 2., is it possible to conclude that $u(t) \in S(\mathbb{R}^n)$ for any $t \in [0,T]$?
Note that Q2) means:
$$\forall \varphi \in D(0,T) \quad \int_0^T u(t,x) \varphi(t) d t \in S(\mathbb{R}^n) \implies u(t,x) \in S(\mathbb{R}^n) \quad\forall t \in [0,T] $$
This can be informally phrased as:
Q2') When true in the average is true pointwise?