Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

Question

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\Sigma(\phi v) \subset \Sigma(v)$. I am wondering if this is still true if $v \in \mathcal{D^{\prime}}$ or $v \in \mathcal{S^{\prime}}$. He seems to switch between distributions in two different space $\mathcal{E^{\prime}}$ and $\mathcal{D^{\prime}}$ without explanations in the chapter. Would appreciate if someone can explain the reasons behind these.

Answer 1

Hörmander defines first the wave front set for a distribution with compact support in $\mathcal{E}'$. If $v\in\mathcal{E}'$, then $\hat{v}$ is the polynomialy bounded growth function $\langle v,e^{-i\xi\cdot}\rangle$. But for a general distribution $u\in\mathcal{D}'(\mathbb{R}^n)$, for all $\varphi\in\mathcal{D}(\mathbb{R}^n)$, $u\phi\in \mathcal{E}'(\mathbb{R}^n)$, so we can define $ \Sigma_x(u)=\bigcap_\varphi \Sigma(\varphi u) $ taking the intersection over all the $\varphi\in\mathcal{D}(\mathbb{R}^n)$ such that $\varphi(x)\neq0$. Then we can define $WF(u)=\{(x,\xi)\in\mathbb{R}^n\times(\mathbb{R}^n\setminus \{0\}), \xi\in\Sigma_x(u)\}$ for an arbitrary distribution.

Answer 2

The definition of $\Sigma(v)$ assumes that the Fourier transform of $v$ is a function. This is true if $v\in\mathcal{E}'$ but not, in general, for (tempered) distributions.