Let $M$ be a smooth closed manifold, $E$ a Hermitian vector bundle over $M$ and $P$ a pseudodifferential operator. Let $u\in D’(M,E)$ such that $Pu=0$. I want show that $$ WF(u) \subset T_0^{*}M \setminus \ell(P),$$ with $\ell(P)$ the set of elliptic points of $P$.
The thing is that I know that if $P$ is an elliptic pseudodifferential operator, and that $u \in \ker(P)$, then $u$ is smooth (so $u$ has no wavefront set). So my question is : can I use this result point by point, that is, if $(x,\xi) \in \ell(P)$, then as $Pu = 0$, $u$ is smooth at $x$, so $(x,\xi) \not\in WF(u)$ ? That would give the result.
Any ideas ? Thanks.