Lemma: Necessary restriction compatibility

Let $N$ be a submanifold of an open set $U$, and let $t\in {\mathcal{D}}^{\prime }(U)$ extend $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus N)$. Then

Consequently a bound $\mathrm{WF}(t)\subset N^{\ast }N$ on all of $U$ would require $t^{\circ }$ to be smooth on $U\setminus N$.

Proof. Wavefront sets are local and commute with restriction to an open set. The conormal bundle $N^{\ast }N$ has base points only in $N$. Hence the proposed bound leaves no wavefront point outside $N$, which is equivalent to smoothness there.