Theorem: Direct compact-current smoothing

Let $C$ be a compact symmetric contravariant distribution with $K^{\ast }C=0$. Suppose

is smooth. For a smooth temporal cutoff $\chi$, zero in the remote past and one in the remote future, define

Then $J_C\in \mathcal{E}$ and

Proof. Conservation of $C$ implies

Consequently $G^{(1)}(\chi q)$ is supported in the transition slab of $\chi$. Spacelike compactness makes this support compact. The Bianchi identity proves conservation.

First take $Dh=0$. Set

Both $F,d$ are compact, and

The second term pairs to zero with $h$, because ${\partial }^{\mu }(\mathsf{r}h{)}_{\mu \nu }=0$.

Retarded and advanced uniqueness give

Hence $E_{2}F=-E_2{C}^{♭♭}$. By (eq:compactprimitive), $F+C^{♭♭}=\square A_2$ for compact $A_2$. Since $\square h=0$, it follows that $⟨-F,h⟩=⟨C,h⟩$.

For general $h\in \mathcal{S}$, solve $\square w_0=Dh$ with zero data at $0$. Then $h-K{w}_0\in {\mathcal{S}}_{\mathrm{dD}}$. Both compact conserved currents annihilate $K{w}_0$, so the equality extends to all $h$.