Let $M$ be closed manifold and $\Psi^0(M)$ denote the space of pseudodifferential operators of order $0$. For $Q\in \Psi^0(M)$ one can define the operator wave-front set $\mathrm{WF}'(Q)\subset T^*M\backslash 0$, which consists of those points in phase-space, where $Q$ is microlocally non-trivial (see e.g. Definition 6.5 here). Suppose now that $U\subset T^*M\backslash 0$ is an open conic set and let $\Psi^0_U(M)$ consist of those operators with $\mathrm{WF}'(Q)\subset U$.
Question. Given $f\in \mathscr{D}'(M)$ is there a way to tell whether $f$ is fixed by some $Q\in \Psi^0_U(M)?$ In other words, can we characterise the space $F_U= \bigcup_{Q\in \Psi^0_U(M)}\ker (\mathrm{id}-Q)$ in terms of $U$?
Denoting $\pi:T^*M\rightarrow M$ the base-point projection, if $\pi^{-1}(\mathrm{supp}(f))\subset U$, then one can simply take $Q$ to be multiplication by $q\in C^\infty(M)$ with $\mathrm{supp}(q)\subset \pi(U)$ and $q\equiv 1$ on $\mathrm{supp} f$.
The more interesting case is when the complement of $U$ intersects every fibre of $T^*M$, which gives a true microlocal constraint. E.g. if $Qf=f$ for $Q\in \Psi^0_U(M)$, microlocality yields $\mathrm{WF}(f)\subset U$.
What I'd be most keen to know is, how 'large' $F_U\cap C^\infty(M)$ is inside of $C^\infty(M)$.