Differentiation and polynomial multiplication preserve tempered distributions

Each displayed functional is the composition of $u$ with a continuous Schwartz endomorphism, followed in the derivative case by a scalar sign. It is therefore continuous and complex-linear on $\mathcal{S}$, hence belongs to ${\mathcal{S}}^{\mathrm{\prime }}$.

For a single test $\phi$, the absolute value after either operation is a source weak seminorm evaluated at the transformed test. Thus each operation is weakly continuous.

A continuous linear Schwartz endomorphism sends bounded sets to bounded sets. For a bounded $B$, the target strong seminorm is therefore the source strong seminorm on ${\mathrm{\partial }}^{\alpha }B$ or $PB$ (the derivative sign disappears under absolute values). This proves strong continuity.

If $\psi \in \mathcal{D}$, then ${\mathrm{\partial }}^{\alpha }\psi$ and $P\psi$ are again compactly supported tests. Evaluating the two displayed definitions on $\psi$ gives exactly the formulas in [F3]. Hence restriction to ${\mathcal{D}}^{\mathrm{\prime }}$ commutes with both operations. The cases $\alpha =0$, constant $P$, $P=0$, and $u=0$ follow from the same formulas. No choice axiom is used.