Lemma: Rapid decay for microcausal kernels

Let $v\in {\mathcal{E}}^{\prime }$ satisfy (eq:mc) with $n=3$, let ${\varphi }_0$ be smooth on $M$, and put $v_{{\varphi }_0}:=⟨v,\cdot \otimes \cdot \otimes {\varphi }_0⟩$, a compactly supported tensor-valued distribution on $M^2$. Let $k_1,k_2$ be nonzero causal covectors, both in ${\tilde{V}}_{+}$ or both in ${\tilde{V}}_{-}$. Then for all constant ${ϵ}_1,{ϵ}_2$ and all $N$,

uniformly for $(k_1,k_2)$ in a compact subset of ${\tilde{V}}_{+}^2\setminus 0$ or ${\tilde{V}}_{-}^2\setminus 0$.

Proof. By XI-F's smearing lemma Hormander (Theorem 8.2.12), $\mathrm{WF}(v_{{\varphi }_0})\subset \{(y_1,p_1;y_2,p_2):\exists x,(y_1,p_1;y_2,p_2;x,0)\in \mathrm{WF}(v)\}$, so $\mathrm{WF}(v_{{\varphi }_0})$ contains no nonzero point of ${\tilde{V}}_{+}^2\cup {\tilde{V}}_{-}^2$, componentwise in the finite tensor indices. Write $e^{i\lambda k\cdot y}=e^{i⟨y,\lambda k_{♭}⟩}$ with $(k_{♭}{)}_{\mu }={\eta }_{\mu \nu }{k}^{\nu }$ and the Euclidean pairing $⟨y,\zeta ⟩=y^{\mu }{\zeta }_{\mu }$. Then the pairing is the Fourier transform $\widehat{v_{{\varphi }_0}}(\zeta )=⟨v_{{\varphi }_0},e^{-i⟨\cdot ,\zeta ⟩}⟩$ at $\zeta =-\lambda (k_{1♭},k_{2♭})$, contracted with ${ϵ}_1\otimes {ϵ}_2$. By Hormander (Definition 8.1.2, Proposition 8.1.3), for a compactly supported distribution the closed cone $\Sigma (v_{{\varphi }_0})$, the projection of $\mathrm{WF}(v_{{\varphi }_0})$ to the covector, is the complement of the set of directions having a conic neighbourhood of rapid decrease of the Fourier transform. The direction $-(k_{1♭},k_{2♭})$ lies in ${\tilde{V}}_{-}^2\setminus 0$ or ${\tilde{V}}_{+}^2\setminus 0$, hence outside $\Sigma (v_{{\varphi }_0})$, and the compact set of such directions has a conic neighbourhood of rapid decrease.