[proofplan] We prove the equality fiberwise over an arbitrary point $x_0 \in V$ and nonzero covector $\xi_0 \in T_{x_0}^*V$. Since $V$ is open, we may choose a coordinate chart of $M$ whose domain is contained in $V$; in that chart, the restriction $u|_V$ and the original distribution $u$ give exactly the same localized distribution whenever the cutoff is supported in the chart domain. The Euclidean cutoff definition of the wave front set therefore gives the same rapid-decay condition for $u$ and for $u|_V$. This proves both inclusions after translating covectors by the chart differential. [/proofplan]

[step:Identify cotangent vectors over $V$ through the open inclusion] Let $\iota: V \to M$ be the inclusion map. Since $V$ is open in $M$, for each $x \in V$ the differential \begin{align*} d\iota_x: T_xV \to T_xM \end{align*} is a linear isomorphism. Its dual map \begin{align*} (d\iota_x)^*: T_x^*M \to T_x^*V \end{align*} is therefore also a linear isomorphism. Throughout the proof, when a covector $\xi \in T_x^*V$ is regarded as a covector on $M$, it means the unique covector $\widetilde{\xi} \in T_x^*M$ satisfying \begin{align*} (d\iota_x)^*\widetilde{\xi}=\xi. \end{align*} Thus the asserted identity is equivalent to the following pointwise statement: for every $x_0 \in V$ and every $\xi_0 \in T_{x_0}^*V \setminus \{0\}$, if $\widetilde{\xi}_0 \in T_{x_0}^*M$ denotes the corresponding covector on $M$, then \begin{align*} (x_0,\xi_0)\in \operatorname{WF}(u|_V) \iff (x_0,\widetilde{\xi}_0)\in \operatorname{WF}(u). \end{align*} [/step]

[step:Choose one chart contained in the open set] Fix $x_0 \in V$ and $\xi_0 \in T_{x_0}^*V \setminus \{0\}$. Since $V$ is open in $M$, there exists a coordinate chart $(U,\kappa)$ of $M$ such that \begin{align*} x_0 \in U \subset V. \end{align*} Write $\Omega := \kappa(U) \subset \mathbb{R}^n$, where $n=\dim M$, and let $y_0 := \kappa(x_0)$. Define the coordinate representative of $\xi_0$ by \begin{align*} \eta_0 := (d\kappa_{x_0}^{-1})^*\xi_0 \in T_{y_0}^*\mathbb{R}^n \setminus \{0\}. \end{align*} Here \begin{align*} d\kappa_{x_0}^{-1}: T_{y_0}\mathbb{R}^n \to T_{x_0}V \end{align*} is the inverse of the chart differential, and $(d\kappa_{x_0}^{-1})^*$ is its dual map. Because $U \subset V$, the restricted chart $(U,\kappa)$ is also a coordinate chart for $V$. Moreover the coordinate representative of $\widetilde{\xi}_0 \in T_{x_0}^*M$ in the chart $(U,\kappa)$ is the same covector $\eta_0$, because $d\iota_x$ identifies the tangent spaces of the open subset and the ambient manifold. [/step]

[step:Show that all localized distributions in the chosen chart agree]Let $\varphi \in C_c^\infty(U)$ be a compactly supported smooth cutoff on $U$. Define the localized distribution for $u$ in the chart $(U,\kappa)$ as the distribution $w_\varphi \in \mathcal{D}'(\Omega)$ given by \begin{align*} w_\varphi(\psi) := u(\varphi \cdot (\psi \circ \kappa)) \end{align*} for every [test function](/page/Test%20Function) $\psi \in C_c^\infty(\Omega)$. Define the localized distribution for $u|_V$ in the same chart as $\widetilde{w}_\varphi \in \mathcal{D}'(\Omega)$ by \begin{align*} \widetilde{w}_\varphi(\psi) := (u|_V)(\varphi \cdot (\psi \circ \kappa)) \end{align*} for every $\psi \in C_c^\infty(\Omega)$. Since $\operatorname{supp}\varphi \subset U \subset V$, the test function $\varphi \cdot (\psi \circ \kappa)$ is a compactly supported smooth function on $V$, and also a compactly supported smooth function on $M$ after extension by $0$ outside $V$. By the definition of restriction of distributions to open subsets, \begin{align*} (u|_V)(\varphi \cdot (\psi \circ \kappa))=u(\varphi \cdot (\psi \circ \kappa)). \end{align*} Therefore $\widetilde{w}_\varphi(\psi)=w_\varphi(\psi)$ for every $\psi \in C_c^\infty(\Omega)$, hence \begin{align*} \widetilde{w}_\varphi=w_\varphi \end{align*} as distributions on $\Omega$.[/step]

[guided]The point of choosing $U \subset V$ is that every cutoff supported in $U$ is automatically supported inside the [open set](/page/Open%20Set) on which the restriction $u|_V$ is defined. Let $\varphi \in C_c^\infty(U)$ be such a cutoff. For each test function $\psi \in C_c^\infty(\Omega)$, the composition $\psi \circ \kappa$ is smooth on $U$, and the product $\varphi \cdot (\psi \circ \kappa)$ has compact support in $U$. Since $U \subset V$, this product is a legitimate test function on $V$. Now define two distributions on the coordinate domain $\Omega$. The first comes from localizing $u$: \begin{align*} w_\varphi(\psi) := u(\varphi \cdot (\psi \circ \kappa)). \end{align*} The second comes from localizing the restricted distribution $u|_V$: \begin{align*} \widetilde{w}_\varphi(\psi) := (u|_V)(\varphi \cdot (\psi \circ \kappa)). \end{align*} The definition of restriction of a distribution to an open subset says exactly that, on test functions supported in $V$, the restricted distribution acts by the original distribution. Since $\operatorname{supp}(\varphi \cdot (\psi \circ \kappa)) \subset \operatorname{supp}\varphi \subset U \subset V$, we obtain \begin{align*} \widetilde{w}_\varphi(\psi)=w_\varphi(\psi). \end{align*} This equality holds for every $\psi \in C_c^\infty(\Omega)$, so the two localized distributions are identical: \begin{align*} \widetilde{w}_\varphi=w_\varphi. \end{align*} Thus every Fourier-transform decay test appearing in the coordinate definition of $\operatorname{WF}(u|_V)$ is literally the same test as the one appearing for $\operatorname{WF}(u)$, provided the cutoff is supported in $U$.[/guided]

[step:Compare microlocal smoothness in the shared chart] The coordinate definition of the wave front set is local in the base point: microlocal smoothness at a covector may be tested in any coordinate chart whose domain contains the base point, using cutoffs supported in that chart domain. Since $U$ is a chart domain for both $M$ and $V$ and $x_0 \in U$, $(x_0,\xi_0)$ is not in $\operatorname{WF}(u|_V)$ exactly when there exist a cutoff $\varphi \in C_c^\infty(U)$ with $\varphi(x_0)\neq 0$ and an open conic neighbourhood $\Gamma \subset T_{y_0}^*\mathbb{R}^n\setminus \{0\}$ of $\eta_0$ such that, for every integer $N \ge 1$, there is a constant $C_N>0$ satisfying \begin{align*} |\widehat{\widetilde{w}_\varphi}(\eta)| \le C_N(1+|\eta|)^{-N} \end{align*} for every $\eta \in \Gamma$. Here $\widehat{\widetilde{w}_\varphi}$ denotes the [Fourier transform](/page/Fourier%20Transform) of the compactly supported distribution $\widetilde{w}_\varphi$ on $\Omega$, evaluated after extending it by $0$ to $\mathbb{R}^n$. The same coordinate definition says that $(x_0,\widetilde{\xi}_0)$ is not in $\operatorname{WF}(u)$ exactly when the identical rapid-decay condition holds for $w_\varphi$ in the same chart and at the same coordinate covector $\eta_0$. From the previous step, \begin{align*} \widetilde{w}_\varphi=w_\varphi. \end{align*} Therefore \begin{align*} \widehat{\widetilde{w}_\varphi}=\widehat{w_\varphi}. \end{align*} The rapid-decay condition in every conic neighbourhood of $\eta_0$ is consequently identical for $u|_V$ and for $u$; in particular, the same constants $C_N$ work on both sides because the compactly supported distributions whose Fourier transforms are being estimated are equal. Hence \begin{align*} (x_0,\xi_0)\notin \operatorname{WF}(u|_V) \iff (x_0,\widetilde{\xi}_0)\notin \operatorname{WF}(u). \end{align*} [/step]

[step:Conclude the equality of the two wave front sets] Taking complements inside $T^*V\setminus 0$ in the equivalence just proved gives, for every $x_0 \in V$ and every $\xi_0 \in T_{x_0}^*V\setminus \{0\}$, \begin{align*} (x_0,\xi_0)\in \operatorname{WF}(u|_V) \iff (x_0,\widetilde{\xi}_0)\in \operatorname{WF}(u). \end{align*} Under the identification of $T^*V$ with $T^*M|_V$ induced by $(d\iota_x)^*$, this is precisely \begin{align*} \operatorname{WF}(u|_V)=\operatorname{WF}(u)\cap (T^*V\setminus 0). \end{align*} This proves the theorem. [/step]