Let $X$ and $Y$ be smooth second-countable Hausdorff manifolds. Let $T^*X \setminus 0$ and $T^*Y \setminus 0$ denote the cotangent bundles with their zero sections removed. Let

\begin{align*} C \subset (T^*X \setminus 0) \times (T^*Y \setminus 0) \end{align*}

be a closed conic canonical relation, and let $A \in I^m(X,Y;C)$ be a properly supported Fourier integral operator of order $m \in \mathbb{R}$ with Schwartz kernel $K_A \in \mathcal{D}'(X \times Y)$. Use the twisted kernel wave front convention

\begin{align*} WF'(K_A) := \{(x,\xi;y,\eta) \in (T^*X \setminus 0) \times (T^*Y \setminus 0) : (x,y;\xi,-\eta) \in WF(K_A)\}. \end{align*}

Assume

\begin{align*} WF'(K_A) \subset C. \end{align*}

Assume also that $K_A$ has no one-sided kernel wave front components, meaning that no element of $WF(K_A)$ has the form $(x,y;\xi,0)$ with $\xi \neq 0$ or $(x,y;0,\eta)$ with $\eta \neq 0$. For every $u \in \mathcal{D}'(Y)$, define $Au \in \mathcal{D}'(X)$ by the properly supported kernel action

\begin{align*} (Au)(\phi) := u(A^t\phi), \qquad \phi \in C_c^\infty(X), \end{align*}

where $A^t:C_c^\infty(X) \to C_c^\infty(Y)$ is the transpose operator determined by $K_A$. Then

\begin{align*} WF(Au) \subset C \circ WF(u), \end{align*}

where

\begin{align*} C \circ WF(u) := \{(x,\xi) \in T^*X \setminus 0 : \text{there exists } (y,\eta) \in WF(u) \text{ such that } (x,\xi;y,\eta) \in C\}. \end{align*}

Equivalently, let $(x_0,\xi_0) \in T^*X \setminus 0$, and let $V \subset T^*X \setminus 0$ be an open conic neighbourhood of $(x_0,\xi_0)$ such that

\begin{align*} V \cap (C \circ WF(u)) = \varnothing. \end{align*}

Then, after replacing $V$ by a sufficiently small open conic neighbourhood of $(x_0,\xi_0)$ if necessary, every properly supported operator $P \in \Psi^0(X)$ whose essential support is contained in $V$ satisfies

\begin{align*} PAu \in C^\infty(X). \end{align*}

In particular, $Au$ is microlocally smooth at every covector in $V$.

Now assume the elliptic graph case. Let $(y_0,\eta_0) \in T^*Y \setminus 0$ and $(x_0,\xi_0) \in T^*X \setminus 0$. Suppose there are open conic neighbourhoods $U_0 \subset T^*Y \setminus 0$ of $(y_0,\eta_0)$ and $V_0 \subset T^*X \setminus 0$ of $(x_0,\xi_0)$, and a homogeneous canonical diffeomorphism $\kappa:U_0 \to V_0$ with $\kappa(y_0,\eta_0)=(x_0,\xi_0)$, such that

\begin{align*} C \cap (V_0 \times U_0) = \{(\kappa(y,\eta);y,\eta) : (y,\eta) \in U_0\}. \end{align*}

Assume that the principal symbol of $A$ is elliptic on this graph near $(x_0,\xi_0;y_0,\eta_0)$. Assume also that, after possibly shrinking $U_0$ and $V_0$,

\begin{align*} C \cap (V_0 \times (WF(u) \setminus U_0)) = \varnothing. \end{align*}

Then there exist smaller open conic neighbourhoods $U \subset U_0$ of $(y_0,\eta_0)$ and $V \subset V_0$ of $(x_0,\xi_0)$, an open conic neighbourhood $W \subset V_0$ of $\overline{V}$ satisfying $\kappa^{-1}(W) \subset U$, properly supported operators $Q_1,Q_0 \in \Psi^0(Y)$ and $P_1,P_0 \in \Psi^0(X)$, and an elliptic properly supported Fourier integral operator $B$ of order $-m$ associated with the inverse canonical graph $\kappa^{-1}:V_0 \to U_0$, such that $\overline{U} \subset U_0$, $Q_1$ is elliptic on $U$, $P_1$ is elliptic on $V$, $\operatorname{ess\,supp}(P_1) \subset W$, $\operatorname{ess\,supp}(P_0) \subset W$, $Q_0$ is microlocally equal to the identity on a conic neighbourhood of $\kappa^{-1}(W)$, $P_0$ is microlocally equal to the identity on a conic neighbourhood of $W$, the essential supports of $Q_0$ and $P_0$ are contained in $U_0$ and $V_0$, respectively, and

\begin{align*} Q_1BP_0AQ_0 = Q_1 + R \end{align*}

microlocally on $U$, where $R$ is smoothing microlocally over $U \times U$. Moreover,

\begin{align*} P_1A(I-Q_0)u \end{align*}

is microlocally smooth in $V$. Consequently,

\begin{align*} (y_0,\eta_0) \in WF(u) \quad \Longleftrightarrow \quad (x_0,\xi_0) \in WF(Au). \end{align*}