Let $(M,g)$ be a time-oriented smooth Lorentzian manifold of dimension $n \geq 2$, let $\pi_E:E \to M$ be a finite-rank complex smooth vector bundle, and let $d\mu$ be a fixed smooth positive density on $M$. Let

\begin{align*} P:C^\infty(M;E)\to C^\infty(M;E) \end{align*}

be a second-order differential operator whose principal symbol satisfies

\begin{align*} \sigma_2(P)(x,\xi)=g_x^{-1}(\xi,\xi)\operatorname{id}_{E_x} \end{align*}

for every $(x,\xi)\in T^*M$. Let $O\subset M$ be an open causally convex geodesically convex neighbourhood. Define

\begin{align*} p:T^*O\to \mathbb{R},\qquad p(x,\xi):=g_x^{-1}(\xi,\xi) \end{align*}

and

\begin{align*} \Sigma_O:=\{(x,\xi)\in T^*O\setminus 0:p(x,\xi)=0\}. \end{align*}

Let $\Phi_t$ denote the Hamilton flow of $H_p$ on its domain of definition in $T^*O\setminus 0$. For each $\varepsilon\in\{+,-\}$, define

\begin{align*} \Lambda_\varepsilon:=\overline{\{((x,\xi),(y,-\eta))\in T^*(O\times O)\setminus 0:(x,\xi)=\Phi_t(y,\eta),\ (y,\eta)\in\Sigma_O,\ \varepsilon t>0,\text{ and the projected bicharacteristic segment from }y\text{ to }x\text{ lies in }O\}}. \end{align*}

Assume that, for each $\varepsilon\in\{+,-\}$,

\begin{align*} \Lambda_\varepsilon\cap N^*\Delta_O=\{((x,\xi),(x,-\xi)):(x,\xi)\in\Sigma_O\}, \end{align*}

where $\Delta_O\subset O\times O$ is the diagonal. Assume also that, for each $\varepsilon\in\{+,-\}$, there is a conic [open set](/page/Open%20Set)

\begin{align*} \Gamma_\varepsilon\subset T^*(O\times O)\setminus 0 \end{align*}

containing this characteristic diagonal and the part of $\Lambda_\varepsilon$ under consideration such that, inside $\Gamma_\varepsilon$, $\Lambda_\varepsilon$ is an embedded conic Lagrangian submanifold, the pair $(N^*\Delta_O,\Lambda_\varepsilon)$ intersects cleanly, and $\Lambda_\varepsilon\setminus N^*\Delta_O$ is locally parametrised by nondegenerate positively homogeneous phase functions.

Work in the standard real-principal-type paired-Lagrangian calculus for vector-bundle valued half-density kernels associated with the clean pair $(N^*\Delta_O,\Lambda_\varepsilon)$. Kernels are sections of $E\boxtimes E^*$ tensored with the half-density bundle determined by $d\mu(x)d\mu(y)$; left action of $P$ means action in the first $O$ variable, and right action means the transpose-kernel action determined by $d\mu$ and the fibre pairing $E^*\times E\to\mathbb{C}$. Assume this calculus includes the diagonal principal-symbol exact sequence, the standard $\varepsilon$-side boundary-value normalization of $p^{-1}\operatorname{id}_E$ at the characteristic diagonal, local solvability and Borel summation for the left and right transport equations determined by the complete symbol of $P$, and the residual correction theorem which turns a left microlocal parametrix with the corresponding right transport normalization into a two-sided microlocal parametrix modulo kernels microlocally smooth in $\Gamma_\varepsilon$.

Then, for each $\varepsilon\in\{+,-\}$, there exists a properly supported continuous linear operator

\begin{align*} G_O^\varepsilon:C_c^\infty(O;E)\to \mathcal{D}'(O;E) \end{align*}

with Schwartz kernel

\begin{align*} K_O^\varepsilon\in \mathcal{D}'(O\times O;E\boxtimes E^*) \end{align*}

where $E^*$ is paired with $E$ using the density $d\mu$, such that

\begin{align*} \operatorname{WF}'(PG_O^\varepsilon-I)\cap\Gamma_\varepsilon=\varnothing \end{align*}

and

\begin{align*} \operatorname{WF}'(G_O^\varepsilon P-I)\cap\Gamma_\varepsilon=\varnothing. \end{align*}

Microlocally in $\Gamma_\varepsilon$, the kernel $K_O^\varepsilon$ is a paired Lagrangian distribution associated with the clean pair $(N^*\Delta_O,\Lambda_\varepsilon)$ in the standard half-density convention for real-principal-type parametrices. Equivalently, away from $N^*\Delta_O$ it is a Fourier integral distribution associated with $\Lambda_\varepsilon$, and away from $\Lambda_\varepsilon$ it is a pseudodifferential kernel on the diagonal whose principal symbol is normalized so that applying $P$ gives the identity kernel modulo a kernel microlocally smooth in $\Gamma_\varepsilon$. In every phase chart for $\Lambda_\varepsilon\setminus N^*\Delta_O$, the phase satisfies

\begin{align*} p(x,\partial_x\phi)=0 \end{align*}

on its critical set, and the full $E\boxtimes E^*$-valued amplitude is determined modulo a smoothing amplitude by the transport equations obtained from the complete symbol of $P$ together with the diagonal identity normalization. No causal support property is asserted beyond proper support.