Lemma: Kernel of the first coefficient

Let $G_{\mathrm{R}}$ be the retarded fundamental solution of $\square$ on $M$. Then

with $P^{\alpha \mu ,\rho \sigma }=\frac{1}{2}{\eta }^{\alpha \mu }{\eta }^{\rho \sigma }-\frac{1}{2}({\eta }^{\alpha \rho }{\eta }^{\mu \sigma }+{\eta }^{\alpha \sigma }{\eta }^{\mu \rho })$. As a distribution in $(x,z)$, $\mathrm{WF}({\Lambda }_{\epsilon })\subset \mathrm{WF}(G_{\mathrm{R}}(x-z))$.

Proof. The source $-{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[h])$ vanishes on $\{t\le 0\}$, so ${\mathsf{E}}_0$ acts on it as the retarded operator. Integrating by parts in $z$ over the compact set $J^{-}(x)\cap \{z^0\ge 0\}$, with no boundary contribution because ${\sigma }_{\epsilon }(z)=0$ at $z^0=0$, gives (eq:Lambda). Multiplication by the smooth function ${\sigma }_{\epsilon }(z)$ and differentiation do not enlarge wavefront sets Hormander (Section 8.1).