Translation invariant test function operators are convolutions

Suppose $L$ commutes with translations and put $\check{\psi }(y)=\psi (-y)$. Reflection maps ${\mathcal{D}}_K$ to ${\mathcal{D}}_{-K}$ with unchanged derivative seminorms, so F2 makes it continuous on $\mathcal{D}$. Evaluation at zero is continuous on $C({\mathbb{R}}^n)$ for the compact-open topology. Consequently $u(\psi )=(L\check{\psi })(0)$ is a continuous linear test functional, hence a distribution.

For fixed $x$, reflection of the test $y\mapsto \phi (x-y)$ is $z\mapsto \phi (x+z)=T_{-x}\phi (z)$. Thus $u(\phi (x-\cdot ))=(L{T}_{-x}\phi )(0)=(T_{-x}L\phi )(0)=L\phi (x)$. F1 (or F3 for this parameter family) shows this is smooth. If another distribution gives the same operator, evaluating its convolution with $\check{\psi }$ at zero recovers its value on every $\psi$, so it equals $u$.

Conversely fix a distribution $u$, a source support $K$, a target compact $H$, and target derivative order $r$. All tests $({\mathrm{\partial }}^{\alpha }\phi )(x-\cdot )$ with $x\in H$, $\mathrm{\mid }\alpha \mathrm{\mid }\le r$, and $\phi \in {\mathcal{D}}_K$ are supported in the single compact $H-K$. F4 there gives $C,m$, and F1 gives ${\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}{\sup }_H\mathrm{\mid }{\mathrm{\partial }}^{\alpha }(u\ast \phi )\mathrm{\mid }\le C{p}_{m+r}(\phi )$. This proves continuity from each stage into the smooth-function topology, hence continuity on $\mathcal{D}$ by F2. Direct substitution gives $u\ast (T_h\phi )(x)=u(\phi (x-h-\cdot ))=(T_h(u\ast \phi ))(x)$, proving translation commutation. Empty source or target compact sets give zero seminorms; the zero distribution gives the zero operator. No closed-graph theorem or choice is used. $\mathrm{\square }$