Manuscript XVIII-A@2 · XVIII-A-v2.tex · lemma · 0f4336ec

Lemma: Nonvanishing smooth multiplication preserves local wavefront

Microlocal operations and their boundaries · lines 683–697 · proof included or linked

XVIII-A@2#lem:switchWF

For $u \in D^{'} (X)$ and $a \in C^{\infty} (X)$ , let $U = {a \neq = 0}$ . Then

WF (a u) ∣_{U} = WF (u) ∣_{U}, WF (a u) \subseteq WF (u) .

Proof. Smooth multiplication does not enlarge wavefront. On $U$ , $a^{- 1}$ is smooth and $u = a^{- 1} (a u)$ , giving the reverse local inclusion.

Displayed formulas in this passage

unlabeled · line 687
$WF (a u) ∣_{U} = WF (u) ∣_{U}, WF (a u) \subseteq WF (u) .$