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!


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 $\xi_0$".


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.

  • $\begingroup$ Thanks, I should have realised that as stated, the characterisation was nonsensical. I'll add an edit to the question. $\endgroup$ – onamoonlessnight May 23 '17 at 14:34

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