Lemma: Smoothing estimates

For $k\in {\mathcal{S}}^{\prime }({\mathbb{R}}^4)$, a compact set $L$, and any multi-index $\alpha$, there are constants $C,N$ such that

If $L\cap \mathrm{supp}k=\varnothing$, then for every $M$,

Proof. A tempered distribution is bounded by finitely many Schwartz seminorms. Apply that bound to ${\partial }^{\alpha }{G}_{\tau }(r-\cdot )$. For $r$ in a fixed compact set, its seminorms grow at most as a fixed inverse power of $\tau$, proving the first assertion.

For the second, insert a smooth cutoff equal to one near $L$ and supported away from $\mathrm{supp}k$. The pairing with its near-$L$ part is zero. In the remaining pairing the Gaussian argument stays a positive distance from zero. Gaussian decay, including all derivatives and polynomial weights required by the tempered-distribution seminorm, dominates every power of $\tau$. This proves the uniform remainder estimate.