Lemma: Smearing

Let $u$ be a compactly supported tensor-valued distribution on $M^3$ and $v,\psi$ smooth on $M$. Put $\ell :=⟨u,\cdot \otimes v\otimes \psi ⟩$, a compactly supported distribution on $M$. Then

In particular, if $u$ satisfies (eq:mc) with $n=3$, then $\mathrm{WF}(\ell )$ contains no nonzero covector in ${\tilde{V}}_{+}\cup {\tilde{V}}_{-}$.

Proof. Smooth multiplication by $v\otimes \psi$ does not enlarge the wavefront set, and integration over $(y_2,y_3)$ of a compactly supported distribution is the pushforward estimate of Hormander (Theorem 8.2.12). If $(y_1,p;y_2,0;y_3,0)\in \mathrm{WF}(u)$ with $0\ne p\in {\tilde{V}}_{+}$, then the triple lies in ${\tilde{V}}_{+}^3$, contradicting (eq:mc); similarly for ${\tilde{V}}_{-}$.