$\dot{T}{\mathcal{M}}$ etc) is the bundle $T^*{\mathbb{R}}^n$ ($T{\mathbb{R}}^n$, $T{\mathcal{M}}$, etc) with the zero section removed.

A distribution $u \in {\mathcal{D}}^\prime ({\mathbb{R}}^n)$ is said to be microlocally $H^s$ at $(x,\xi)\in \dot{T}^*{\mathbb{R}}^n$ if there exists an open cone $\Gamma \subset {\mathbb{R}}^n\setminus 0$ about $\xi$ and a smooth function $\varphi \in C^\infty_0 ({\mathbb{R}}^n)$, $\varphi(x)\not=0$, such that $$\int_\Gamma {\mathrm{d}}^n\zeta \, (1+|\zeta|^2)^s |[\varphi u]^\wedge(\zeta)|^2 < \infty \, .$$ The Sobolev wave-front set $WF^s(u)$ of a distribution $u \in {\mathcal{D}}^\prime ({\mathbb{R}}^n)$ is the complement, in $\dot{T}^*{\mathbb{R}}^n$, of the set of all pairs $(x,\xi)$ at which $u$ is microlocally $H^s$.

To define the Sobolev wave-front set of a distribution on a manifold one works locally. Let ${\mathcal{U}}$ be an open patch of a manifold ${\mathcal{M}}$ with associated coordinate map $\kappa:{\mathcal{U}}\mapsto {\mathbb{R}}^n$. If $(\kappa(x),\xi)\in WF^s(u\circ\kappa^{-1})$ then $(x,\kappa^{-1}_*(\xi))\in WF^s(u)\subset \dot{T}^*{\mathcal{M}}$. We shall occasionally use the notation $u\in H^s_{\mathrm{loc}}({\mathcal{M}})$ if $WF^s(u)=\emptyset$ for a distribution $u\in{\mathcal{D}}'({\mathcal{M}})$ and direct the reader to the remarks following definition 8.2.5 of [@hormander; @hyper] to justify this notation.

The Sobolev wave-front set is a closed cone in $\dot{T}^*{\mathcal{M}}$. Furthermore, we have the following properties[^6] of $WF^s$:\

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      i) The smooth wave-front set is related to the Sobolev wave-front set

         via $WF(u)=\overline{\bigcup_{s\in{\mathbb{R}}} WF^s(u)}$.

     ii) If $\varphi \in 