Characterisation of the wavefront set

Question

I am totally new to microlocal analysis, and have been studying Jared Wunsch's notes. I have been puzzling over the properties of the wavefront set.

Let $X$ be a compact Riemannian manifold, and $\Psi^m(X)$ denote the space of pseudodifferential operators of order $m \in \mathbb{R}$ on $X$, as defined there. The definition of the wavefront set given in the notes (see Definition 4.3, p. 29) is the set $\mathrm{WF}(u) \subset S^* X $ defined by $$ (x_0, \xi_0) \notin \mathrm{WF}(u) \iff \exists P \in \Psi^0(X), \text{ elliptic at } (x_0, \xi_0), \text{ such that } Pu \in C^{\infty}(X). $$

Having done some of the exercises he sets in the notes, I seem to have arrived at the following intuitive characterisation of the wavefront set: $$ (x_0, \xi_0) \notin \mathrm{WF}(u) \longleftrightarrow u \text{ is smooth in a neighbourhood of } (x_0, \xi_0). $$

Of course, it is true that $\mathrm{WF}(u) = \emptyset \iff u \in C^{\infty}(X) $. It seems to me that this should also be true locally, in the sense that if $\mathrm{WF}(u) \cap K = \emptyset$ for some compact set $K$, then $Bu \in C^{\infty}(X)$, where $B$ is a microlocal partition of unity on $K$: $B \in \Psi^0(X)$, $\mathrm{WF}'(\mathbb{1} - B) \cap K = \emptyset$, and $\mathrm{WF}'(B) \subset U$ for an open set $U$ containing $K$ (see Lemma 4.1, p. 29). My questions are

1. Is this true?, and if so,
2. Does this local statement imply anything about the distribution $u$ itself on $K$ (or a smaller open set)?

I am fairly confident that 1. should be true, and that 2. should be false, else the notion of a wavefront set would seem to be somewhat redundant. Nonetheless, it would be great to hear from people who actually understand this!

$\textbf{Edit:}$

As per Bazin's answer, here's an elaboration of my intuitive characterisation. It can be shown that $(x_0, \xi_0) \notin \mathrm{WF}(u)$ if and only if there exist cutoff functions $\phi(x)$, $\gamma(\xi)$, where $\gamma(\xi)$ is a cutoff in a cone of directions near $\xi_0$, smoothed out at the origin, such that $$ \gamma(\xi) \mathcal{F}(\phi u )(\xi) \in \mathcal{S}(X). $$ It is in this sense that $u$ is smooth "in a neighbourhood of $x_0$, and a conic neighbourhood of $\xi_0$".

Answer 1

Your intuitive characterization does not make sense: the function $u$ is defined on some neighborhood of $x_0$ and $(x_0,\xi_0)$ belongs to the sphere bundle. On the other hand, you may salvage part of your statements by DEFINING smoothness at $(x_0,\xi_0)$ by your intuitive hunch.

In particular, you have $π_1(WF u)=ss(u)$, where $ss(u)$ is the singular support of $u$ (whose complement is the largest open set where $u$ is $C^\infty$) and $π_1$ is the canonical projection of $T^*(X)$ onto $X$ ($(x,\xi)\mapsto x$). Then of course if $WF u=\emptyset$ it is true also for the singular support. Also if $K$ is a compact subset of $X$ such that $K\cap π_1(WF u)=\emptyset$, then $u$ is smooth on a neighborhood of $K$.

An addendum after the new edition of the question: in the definition of the wave-front-set that you give, you can replace the left-hand-side $(\exists P\dots)$ by $ \exists $ a neighborhood $W$ of $(x_0, \xi_0)$ in $S^*(X)$ such that for all $P\in \Psi^0(X)$ with a symbol supported in $W$, $Pu\in C^\infty$. It is obviously stronger but also implied by your condition using the invertibility due to your ellipticity assumption. Then your intuitive definition follows.