Wave front set

Remarks

Since $\varphi u$ has compact support, $\widehat{\varphi u}$ is a smooth function, so the decay condition makes sense pointwise. By construction $\operatorname{WF}(u)$ is closed, and it is conic: $(x, \xi) \in \operatorname{WF}(u)$ implies $(x, t\xi) \in \operatorname{WF}(u)$ for $t > 0$. Its projection to $\Omega$ is the singular support of $u$, the complement of the largest open set on which $u$ is smooth. In particular $\operatorname{WF}(u) = \emptyset$ if and only if $u$ is smooth.

Example: for the Dirac delta $\delta$ at $0$ in $\R^n$, one has $\operatorname{WF}(\delta) = \set{0} \times (\R^n \setminus \set{0})$. Check: at a point $x_0 \ne 0$ one may take $\varphi$ vanishing near $0$, so $\varphi \delta = 0$. At $x_0 = 0$, $\widehat{\varphi \delta}(\xi) = \varphi(0)$ is a nonzero constant, which decays in no direction.

Non-example: let $u(x) = H(x_1)$ on $\R^2$, where $H$ is the Heaviside step function, and look at the point $(0, (0, 1))$. Take $\varphi(x) = \chi(x_1) \chi(x_2)$ with $\chi \in C_c^\infty(\R)$ and $\chi(0) \ne 0$. Then $\widehat{\varphi u}(\xi) = a(\xi_1)\, \hat\chi(\xi_2)$, where $a$ is the Fourier transform of $\chi H$ and is bounded. In the cone $\abs{\xi_1} < \abs{\xi_2}$ around $(0, 1)$ one has $\abs{\xi} < 2 \abs{\xi_2}$, and $\hat\chi(\xi_2)$ decays faster than any power of $\abs{\xi_2}$. So the point is microlocally regular, and it is not in $\operatorname{WF}(u)$. In fact $\operatorname{WF}(u) = \set{x_1 = 0} \times \set{(\xi_1, 0) : \xi_1 \ne 0}$.

The wave front set transforms well under changes of coordinates, as a subset of the cotangent bundle minus its zero section, so the notion extends to distributions on manifolds.