Multiplying tempered distributions by smooth functions

Question

This is roughly exercise V.23 of Reed, Simon - Methods of Modern Mathematical Physics, vol. 1.

Let $\psi$ be a $\mathcal{C}^\infty$ function such that for every multi-index $\alpha$ there exist $c_\alpha> 0$ constant and a $N_\alpha$ such that

$$ \|\partial^\alpha \psi(x)\| \leq c_\alpha(1 + \|x\|^2)^{N_\alpha} $$

for every $x \in \mathbb{R}^d$.

(i) If $\varphi \in \mathcal{D}(\mathbb{R}^d)$, then $\varphi\psi \in \mathscr{S}(\mathbb{R}^d)$.

This is clear from Leibniz' Rule. What I'm having trouble to show is parts (ii) and (iii):

(ii) If $h$ is a measurable function such that $h\psi \in \mathscr{S}$ for all $\psi \in \mathscr{S}$, then $h$ has to be smooth.

(iii) Given $u \in \mathscr{S}'$, putting $(\psi u)(\varphi) = u(\psi\varphi)$ (as expected) for every $\varphi\in\mathscr{S}$, then $\psi u$ is a tempered distribution, i.e., $\psi u \in \mathscr{S}'$.

I'm actually not sure on how (ii) helps me at all on showing (iii) and how to show (ii).

Answer 1

Are you sure about the statement (i)? As stated, it is trivial because for $\varphi\in\mathscr D$ and smooth $\psi$ one has $\varphi\psi\in\mathscr D \subseteq \mathscr S$. Perhaps you mean $\varphi\psi\in\mathscr S$ for all $\varphi\in \mathscr S$? Then the map $\varphi\mapsto \varphi\psi$ is continuous on $\mathscr S$ by the closed graph theorem and $\psi u$ is continuous as a composition. This solves (3).

I don't see, why (ii) should be useful. For the proof of (ii) just consider test functions $\psi\in\mathscr D$ which are constant on a neighbourhood of the point where you want to show smoothness.