I am currently studying microlocal analysis, and I am working on understanding the proof of the Theorem of Propagation of Singularities as presented by Peter Hintz in his notes (link to notes, Theorem 8.5).
The theorem states:
Suppose $P \in \Psi^m(M)$ has a real-valued homogeneous principal symbol. Let $u \in D'(M)$ and $f := Pu \in D'(M)$. For $s \in \mathbb{R}$, we have: $$ WF^s(u) \subset \text{Char}(P) \cup WF^{s-m}(f). \tag{8.8} $$ Moreover, within $\text{Char}(P) \backslash WF^{s-m+1}(f)$, $$ WF^s(u) \backslash WF^{s-m+1}(f) \tag{8.9} $$ is a union of maximally extended null-bicharacteristics of $P$.
I wanted to ask the following:
- Is there a more straightforward or alternative proof for this theorem compared to the one provided in the notes?
- Does Hörmander’s work (in particular, his books on the analysis of linear PDEs) contain a version of this theorem or its proof? If so, what are the key differences between the approaches?
I would greatly appreciate any insights, references, or suggestions.
Many thanks in advance!