Question

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!

Answer 1

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 $(x_0,\xi_0)\notin WF_h(u)$.