Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to $Z$, i.e. $K \in I^m(X\times Y,Z)$ for some $m$. It defines the F.I.O. $A_K \colon C^\infty_c(X) \to \mathscr D'(Y)$ such that for any $u \in C^\infty_c(X)$, $v \in C^\infty_c(Y)$ the equality $\langle A_K u, v\rangle = \langle K, u \otimes v \rangle$ holds. If $\dot N^\ast Z \subset \dot T^\ast X \times \dot T^\ast Y$ (dot means the zero section removed) then $A_K \colon C^\infty_c(X) \to C^\infty(|\Lambda|Y)$ and we can extend $A_K$ to $A_K \colon \mathscr E'(X) \to \mathscr D'(Y)$.

I would like to estimate the analytical wavefront set $WF_A(u)$ of $u \in \mathscr E'(X)$ given $WF_A(A_Ku)$. Please tell me, are there some related results in literature? If I'm not mistaken, if $\dot N^\ast Z$ is a graph of some bijective symplectomorphism $\pi$ from $\dot T^\ast X$ to $\dot T^\ast Y$ and $A_K$ is elliptic then $\dot T^\ast_y Y \cap WF_A(A_K u) = \varnothing$ must imply $\pi^{-1}(\dot T^\ast Y) \cap WF_A(u) = \varnothing$ but is it possible to say something in general, e.g. when $\dot N^\ast Z$ is only a local canonical graph?