[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 $\Omega$.
[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]
[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]
