# Propagation of Singularities

I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the Propagation of Singularities Theorem.

Since $WF(u)$ is a closed set and a bicharacteristic curve $\gamma$ is a connected set, the only fact to prove is that $WF(u) \cap \gamma$ is an open subset of $\gamma$. Thus, given a $(x_0,\xi_0) \in WF(u) \subset \operatorname{Char} a$, we want to prove that the bicharacteristic curve $(x(t),\xi(t))$starting at this point is locally contained in $WF(u)$. We will proceed by contradiction, assuming that this is not true.

I could not see where the contradiction hypothesis -there is a point $(x(s),\xi(s)) \not\in WF(u)$ for an $s$ in the domain of definition of $\gamma$ - is used in the proof (Raymond, p. 83).

Can anybody help me?

Update: I think I understood, it remains to show two things: $\operatorname{supp} c(s) \subset \operatorname{supp}c_0 \subset e^{sH_q}\operatorname{supp}c_0$ and $w(s)=c(s,c,D)v=\chi \in C_0^{\infty}$.

Then, by a reversion in the time we will have that $\tilde{w}(t)=w(t+s)$ is solution of the problem $\partial \tilde{w}(t)-ib(x,D)\tilde{w}(t)=g(t+s)$ with $\tilde{w}(0)=\chi \in C_{0}^{\infty}$. Hence, $\tilde{w} \in \bigcap_{k} C^{0}(I:H^{k})$. In particular, $\tilde{w}(-s)=w(0)=c(x,D)v \in C^\infty$, which proves that $(x_0,\xi_0) \not\in WF(u),$ since $c_0(x_0,\xi_0)\neq 0.$

P.S: I have posted this in Math Stack Exchange but didn't get a reply and so I have posted it here. The page mentioned is in the link below.

https://math.stackexchange.com/questions/2688446/propagation-of-singularities-for-differential-operators

• I could not find the page in the lecture notes you linked. The proof you presented seems to be okay, but I am not familiar with the symbols used in Joshi's notes. – Bombyx mori Mar 19 '18 at 3:03
• I'm using the book "Raymond, X. S. Elementary Introduction To The Theory Of Pseudodifferential Operators", to complete the details the proof, since they are very similiar. In the Joshi's notes is the Theorem 10.2, page 35. It remains to prove those statements. – Victor Hugo Mar 19 '18 at 3:11
• I take a look, and unless I am missing something subtle, it seems this is just a routine localizing argument. I did not see any proof of contradiction or something. I do not have the printed book with me (I learned PsiDO from Melrose's notes and Shubin's book). So there may be some discrepenacies. – Bombyx mori Mar 19 '18 at 3:18
• Isnt this proved by proving that the complement of the wavefront set is invariant under the null bicharacteristc flow? – Deane Yang Mar 19 '18 at 3:47
• I was trying to build a conic neighborhood $U$ such that $WF(u) \cap U=\emptyset$ and $\mu-\operatorname{supp} c(s,x,D) \subset U$. Noting that $WF(c(s,x,D)v) \subset WF(v)\cap \mu-\operatorname{supp}c(s,x,D) \subset WF(v)\cap U=\emptyset$, the result follows, since $WF(u)=WF(v)$. – Victor Hugo Mar 20 '18 at 21:07