Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ is given by:


$WF\_h(u)=(\mathrm{supp}(u)\times\{0\})\cup WF(u)$.


Would someone please explain this relationship? Thanks!



C^\infty versus semiclassical wavefront sets
Heuristically, this result follows because the semiclassical wavefront set measures oscillations at frequency $\frac{\xi}{h}$. To understand this, observe that since $u$ does not depend on $h$, if $u$ is smooth, it has small high frequency oscillations, and hence, for $\xi\neq 0$, and hence points where $u$ is smooth and $\xi\neq 0$ do not appear in $WF\_h(u)$. However, where $u$ is not smooth, $u$ has high frequency oscillations and so these points do appear.


Below you will find a proof of the statement.


To see that $WF\_h(u)\cap \mathbb{R}^d\times \{0\}=$ supp $(u)\times\{0\}$ observe that 
$$\mathcal{F}\_h(\chi u)(0)=\langle u, \chi\rangle \neq 0$$
for $\chi$ supported near $x\_0\in $ supp $(u)$ and
$$\mathcal{F}\_h(\chi u)(0)=0 $$
for $\chi$ supported near $x\_0\notin $ supp $(u)$. 


To see the second part of $WF\_h(u)$, simply use the characterization of $WF(u)$ that for $\xi\neq 0$, $(x\_0,\xi\_0)\notin WF(u)$ if and only if for $\chi$ supported sufficiently close to $x\_0$, all $N>0$ and $\xi$ in a conic neighborhood of $\xi\_0$, 
$$|\mathcal{F}(\chi u)(\xi)|\leq C\_N\langle \xi \rangle ^{-N}.$$


Thus, $(x\_0,\xi\_0)\notin WF(u)$ if and only if for all $N>0$ and all $\xi$ in a neighborhood of $\xi\_0$,
$$|\mathcal{F}\_h(\chi u)(\xi)|\leq C\_N\langle \xi/h\rangle^{-N}.$$
But, since $\xi\_0\neq 0$, this gives $(x\_0,\xi\_0)\notin WF(u)$ if and only if for all $N>0$ and $\xi$ in a neighborhood of $\xi\_0$,
$$|\mathcal{F}\_h(\chi u)(\xi)|\leq C\_Nh^N$$


and hence if and only if $(x\_0,\xi\_0)\notin WF\_h(u)$. 


