Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Question

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?

UPD. I have realized that I don't have a good answer even in the case of the ordinary $C^\infty$ wavefront set $WF$. Suppose that $A_K$ is proper (hence it maps $\mathscr E'(X) \to \mathscr E'(Y))$ and suppose it has a left parametrix $B$, so that $BA - I_X$ has a $C^\infty$ kernel. Then we can write $$ WF(u) = WF(BA_Ku) \subset C^{-1} \circ WF(A_K u), \quad C = (\dot N^\ast Z)'. $$ This is the desired estimate in the case of $C^\infty$ wavefront set but according to the proof it holds for proper FIOs with left parametrixes. It is possible to say something in the case of nonproper but elliptic FIOs? Does this inclusion still hold?

Answer 1

When $A_K$ is a (pseudo)differential operator with analytic coefficients, a good answer to your question is given by Theorem 9.5.1 on page 353 of the first volume of Hörmander's ALPDO (Springer Grund. 256): $$WF_A (Pu)\subset WF_A(u)\subset \text{Char} P\cup WF_A (Pu), $$ where the characteristic set of $P$, denoted by $\text{Char} P$ is defined as the subset of the cosphere bundle where the principal symbol of $P$ vanishes. In particular, if $P$ is elliptic (i.e. with empty characteristic set), we get $WF_A(u)=WF_A (Pu)$: the singularities of $u$ such that $Pu=f$ are located exactly at the singular set of $f$, and it is not necessary to solve the equation to get that piece of information.