[proofplan] We prove the assertion locally in a coordinate chart around $x_0$, where the wave front set is defined by rapid decay of localized Fourier transforms in conic neighbourhoods of $\xi_0$. If $(x_0,\xi_0)$ is not in the wave front set, we build an order-zero pseudodifferential cutoff whose symbol is supported inside such a conic region and equals $1$ near $(x_0,\xi_0)$; the rapid decay estimate implies that applying this operator to $u$ gives a smooth function. Conversely, if an elliptic operator $A$ smooths $u$, the microlocal elliptic parametrix theorem gives a local inverse to $A$ modulo a smoothing remainder, so a pseudodifferential cutoff equal to the identity near $(x_0,\xi_0)$ also smooths $u$. Translating this cutoff statement back into the Fourier definition gives $(x_0,\xi_0) \notin WF(u)$, and coordinate invariance globalizes the argument. [/proofplan]

[step:Localize the problem in one coordinate chart]Choose a coordinate chart $(U,\kappa)$ on $X$ with $x_0 \in U$, where \begin{align*} \kappa: U \to \kappa(U) \subset \mathbb{R}^n \end{align*} is a diffeomorphism onto an open set. Let $\mathcal{L}^n$ denote $n$-dimensional Lebesgue measure on $\mathbb{R}^n$. Let $y_0 := \kappa(x_0)$ and let $\eta_0 \in \mathbb{R}^n \setminus \{0\}$ denote the coordinate representative of $\xi_0$ under the cotangent map $d\kappa_{x_0}^{-\top}: T_{x_0}^*X \to \mathbb{R}^n$. For a cutoff $\varphi \in C_c^\infty(U)$, define the localized distribution \begin{align*} v_\varphi := (\kappa^{-1})^*(\varphi u) \in \mathcal{D}'(\kappa(U)). \end{align*} The coordinate definition of the wave front set says that $(x_0,\xi_0) \notin WF(u)$ if and only if there are a cutoff $\varphi \in C_c^\infty(U)$ with $\varphi(x_0) \neq 0$ and an open conic neighbourhood $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ of $\eta_0$ such that, for every integer $N \geq 0$, there is a constant $C_N > 0$ satisfying \begin{align*} |\widehat{v_\varphi}(\eta)| \leq C_N(1+|\eta|)^{-N} \end{align*} for every $\eta \in \Gamma$. We shall prove the equivalence in these coordinates. The passage back to $X$ uses the standard coordinate invariance of pseudodifferential symbols and wave front sets, namely that ellipticity and rapid conic Fourier decay transform under the cotangent lift of a change of chart. This is a standard prerequisite not yet in the wiki: coordinate invariance of pseudodifferential symbols and wave front sets.[/step]

[guided]We first reduce the theorem to the Euclidean model used in the definition of the wave front set. Choose a coordinate chart $(U,\kappa)$ with $x_0 \in U$, where \begin{align*} \kappa: U \to \kappa(U) \subset \mathbb{R}^n \end{align*} is a diffeomorphism onto an open subset of $\mathbb{R}^n$. Let $\mathcal{L}^n$ denote $n$-dimensional Lebesgue measure on $\mathbb{R}^n$. Define $y_0 := \kappa(x_0)$, and let $\eta_0 \in \mathbb{R}^n \setminus \{0\}$ be the coordinate representative of $\xi_0$ under the cotangent map $d\kappa_{x_0}^{-\top}: T_{x_0}^*X \to \mathbb{R}^n$. For each cutoff $\varphi \in C_c^\infty(U)$, define the localized distribution \begin{align*} v_\varphi := (\kappa^{-1})^*(\varphi u) \in \mathcal{D}'(\kappa(U)). \end{align*} The local Fourier definition of the wave front set says that $(x_0,\xi_0) \notin WF(u)$ exactly when there exist such a cutoff $\varphi$ with $\varphi(x_0) \neq 0$ and an open conic neighbourhood $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ of $\eta_0$ such that, for every integer $N \geq 0$, there is a constant $C_N>0$ with \begin{align*} |\widehat{v_\varphi}(\eta)| \leq C_N(1+|\eta|)^{-N} \end{align*} for every $\eta \in \Gamma$. This is the only local condition we need to prove or recover. The reason this reduction is legitimate is the standard coordinate invariance of wave front sets and pseudodifferential principal symbols. Under a change of chart, conic Fourier decay transforms by the cotangent lift, and the homogeneous principal symbol of a pseudodifferential operator transforms by the same cotangent map. Thus ellipticity at $(x_0,\xi_0)$ and absence of wave front set at $(x_0,\xi_0)$ are independent of the chosen chart.[/guided]

[step:Construct an elliptic order-zero cutoff from rapid Fourier decay]Assume first that $(x_0,\xi_0) \notin WF(u)$. By the preceding step, choose $\varphi_0 \in C_c^\infty(U)$ with $\varphi_0(x_0) \neq 0$ and an open conic neighbourhood $\Gamma \subset \mathbb{R}^n \setminus \{0\}$ of $\eta_0$ on which $\widehat{v_{\varphi_0}}$ decays rapidly. After shrinking the chart neighbourhood, choose $\varphi \in C_c^\infty(U)$ such that $\varphi(x_0)=1$, $\operatorname{supp}\varphi \subset \{x \in U: \varphi_0(x) \neq 0\}$, and $\varphi_0^{-1}\varphi$ is smooth on a neighbourhood of $\operatorname{supp}\varphi$. Since $\varphi u=(\varphi_0^{-1}\varphi)(\varphi_0u)$ on this neighbourhood, the standard cutoff stability lemma for microlocal rapid decay, applied with a possible shrinking of $\Gamma$, shows that $\widehat{v_\varphi}$ decays rapidly on a conic neighbourhood still denoted by $\Gamma$. Choose open conic neighbourhoods $\Gamma_1$ and $\Gamma_0$ of $\eta_0$ with \begin{align*} \overline{\Gamma_1} \cap S^{n-1} \subset \Gamma_0 \cap S^{n-1} \subset \overline{\Gamma_0} \cap S^{n-1} \subset \Gamma \cap S^{n-1}. \end{align*} Let $S^m(\kappa(U) \times \mathbb{R}^n)$ denote the order-$m$ symbol class in the coordinate variables $(y,\eta)$. By the standard conic cutoff construction for symbols, choose a symbol \begin{align*} a: \kappa(U) \times \mathbb{R}^n \to \mathbb{C} \end{align*} with $a \in S^0(\kappa(U) \times \mathbb{R}^n)$, compact support in the first variable, conic support in $\Gamma_0$ for $|\eta| \geq 1$, and satisfying $a(y,\eta)=1$ for $y$ near $y_0$ and $\eta \in \Gamma_1$ with $|\eta| \geq 2$. This uses the standard prerequisite not yet in the wiki: existence of pseudodifferential cutoffs with prescribed conic symbol support. Choose $\rho,\chi \in C_c^\infty(U)$ with $\rho=1$ on a neighbourhood of $x_0$, $\chi=1$ on a neighbourhood of $\operatorname{supp}\rho$, and $\operatorname{supp}\chi \subset \{x \in U: \varphi(x)=1\}$. Define the local operator \begin{align*} P_a: \mathcal{D}'(\kappa(U)) \to \mathcal{D}'(\kappa(U)) \end{align*} by left quantization, \begin{align*} P_a w(y) := (2\pi)^{-n}\int_{\mathbb{R}^n} e^{iy\cdot \eta} a(y,\eta)\widehat{w}(\eta)\,d\mathcal{L}^n(\eta), \end{align*} first for compactly supported distributions $w$ for which the oscillatory integral is defined distributionally. Define $A \in \Psi^0(X)$ by transporting $P_a$ through $\kappa$, applying it only to $\chi u$, multiplying the output by $\rho$, and extending by zero outside $U$: \begin{align*} Au := \rho\,\kappa^*\bigl(P_a((\kappa^{-1})^*(\chi u))\bigr). \end{align*} The chosen cutoffs make $A$ properly supported. Since $\rho(x_0)=1$, $\chi(x_0)=1$, and $a(y,\eta)=1$ near $(y_0,\eta_0)$ for $|\eta|$ large, the principal symbol of $A$ is nonzero at $(x_0,\xi_0)$; hence $A$ is elliptic at $(x_0,\xi_0)$. We now verify that $Au$ is smooth. Because $\operatorname{supp}\chi \subset \{\varphi=1\}$, the localized input satisfies $(\kappa^{-1})^*(\chi u)= (\kappa^{-1})^*(\chi\varphi u)$. Multiplication by the compactly supported smooth function $\chi\circ\kappa^{-1}$ preserves microlocal rapid decay after shrinking cones: since $\widehat{v_\varphi}$ decays rapidly on $\Gamma$ and $\overline{\Gamma_0}\cap S^{n-1}\subset \Gamma\cap S^{n-1}$, the Fourier transform of $(\kappa^{-1})^*(\chi u)$ decays rapidly on $\Gamma_0$. This is the standard cutoff stability lemma, obtained by writing multiplication as convolution with the Schwartz Fourier transform of $\chi\circ\kappa^{-1}$ and using the angular separation between $\Gamma_0$ and $\mathbb{R}^n\setminus\Gamma$. The factors $\rho$ and $\chi$ are smooth compactly supported cutoffs, and the part of the transported kernel where the two spatial supports are separated is smooth by the standard off-support smoothness of pseudodifferential kernels. Thus it remains to prove smoothness of the displayed local oscillatory term on a neighbourhood of $\operatorname{supp}\rho$; multiplication by $\rho$ then gives a globally smooth function after extension by zero outside $U$. For every multi-index $\alpha$, differentiating under the oscillatory integral gives a finite sum of terms of the form \begin{align*} (2\pi)^{-n}\int_{\mathbb{R}^n} e^{iy\cdot \eta} b_\alpha(y,\eta)\widehat{(\kappa^{-1})^*(\chi u)}(\eta)\,d\mathcal{L}^n(\eta), \end{align*} where $b_\alpha \in S^{|\alpha|}$ and the conic support of $b_\alpha$ is contained in $\Gamma_0$ for $|\eta|$ large, because differentiation in $y$ and multiplication by powers of $\eta$ do not enlarge the conic support of $a$. On the compact $y$-support of $b_\alpha$, the symbol seminorms give a constant $C_\alpha>0$ such that $|b_\alpha(y,\eta)| \leq C_\alpha(1+|\eta|)^{|\alpha|}$. The low-frequency region $\{\eta: |\eta|\leq 1\}$ is compact, and the Fourier transform of a compactly supported distribution has at most polynomial growth, so it contributes a smooth term. On the high-frequency conic support, $\widehat{(\kappa^{-1})^*(\chi u)}$ decays faster than every power on $\Gamma_0$; choose $N>n+|\alpha|+1$. Then \begin{align*} |b_\alpha(y,\eta)\widehat{(\kappa^{-1})^*(\chi u)}(\eta)| \leq C_\alpha C_N(1+|\eta|)^{|\alpha|-N} \end{align*} on the support of $b_\alpha$, and the right-hand side is integrable with respect to $\mathcal{L}^n(\eta)$ on $\mathbb{R}^n$. Therefore every derivative of the local term is represented by an absolutely convergent integral depending continuously on $y$. Thus the local term is smooth on $\kappa(U)$, and the cutoff construction above gives $Au \in C^\infty(X)$.[/step]

[guided]The forward direction asks us to turn Fourier decay into an operator. The definition of $(x_0,\xi_0) \notin WF(u)$ gives a spatial cutoff $\varphi \in C_c^\infty(U)$, with $\varphi(x_0)=1$, and a conic neighbourhood $\Gamma$ of the coordinate covector $\eta_0$ such that the Fourier transform of \begin{align*} v_\varphi := (\kappa^{-1})^*(\varphi u) \end{align*} decays rapidly on $\Gamma$. First we normalize the spatial cutoff. The definition may give only a cutoff $\varphi_0$ with $\varphi_0(x_0)\neq 0$. Since nonvanishing is an open condition, we shrink the coordinate neighbourhood and choose $\varphi \in C_c^\infty(U)$ with $\varphi(x_0)=1$ and $\operatorname{supp}\varphi \subset \{\varphi_0\neq 0\}$. On this support, $\varphi u=(\varphi_0^{-1}\varphi)(\varphi_0u)$, so the cutoff stability lemma for Fourier decay transfers rapid decay from $v_{\varphi_0}$ to $v_\varphi$ after possibly shrinking the cone. We then choose cones \begin{align*} \Gamma_1 \Subset \Gamma_0 \Subset \Gamma \end{align*} on the unit sphere, meaning $\overline{\Gamma_1}\cap S^{n-1}\subset\Gamma_0\cap S^{n-1}$ and $\overline{\Gamma_0}\cap S^{n-1}\subset\Gamma\cap S^{n-1}$. The point of the intermediate cone $\Gamma_0$ is that the symbol will be supported in $\Gamma_0$, while it will equal $1$ only on the smaller cone $\Gamma_1$. The standard conic symbol cutoff construction gives a symbol \begin{align*} a: \kappa(U) \times \mathbb{R}^n \to \mathbb{C} \end{align*} with $a \in S^0(\kappa(U) \times \mathbb{R}^n)$, compact support in $y$, conic support in $\Gamma_0$ for high frequencies, and $a(y,\eta)=1$ near $(y_0,\eta_0)$ for $\eta\in\Gamma_1$ with $|\eta|$ large. Choose $\rho,\chi \in C_c^\infty(U)$ with $\rho=1$ near $x_0$, $\chi=1$ near $\operatorname{supp}\rho$, and $\operatorname{supp}\chi \subset \{\varphi=1\}$. Define $A$ by transporting the local operator through $\kappa$, applying it to $\chi u$, multiplying the result by $\rho$, and extending by zero outside $U$. Because the cutoffs equal $1$ at $x_0$ and the leading symbol is nonzero at $(y_0,\eta_0)$, the resulting properly supported operator $A \in \Psi^0(X)$ is elliptic at $(x_0,\xi_0)$. Now we check smoothness rather than merely asserting it. In local coordinates the relevant operator is \begin{align*} P_a w(y) := (2\pi)^{-n}\int_{\mathbb{R}^n} e^{iy\cdot \eta}a(y,\eta)\widehat{w}(\eta)\,d\mathcal{L}^n(\eta). \end{align*} Apply this to the localized input $(\kappa^{-1})^*(\chi u)$. Since $\operatorname{supp}\chi \subset \{\varphi=1\}$, this input equals $(\kappa^{-1})^*(\chi\varphi u)$. Multiplication by $\chi\circ\kappa^{-1}$ becomes convolution with a Schwartz function on the Fourier side, so the rapid decay of $\widehat{v_\varphi}$ on $\Gamma$ implies rapid decay of $\widehat{(\kappa^{-1})^*(\chi u)}$ on the smaller cone $\Gamma_0$. The cone shrink is what controls the part of the convolution coming from frequencies outside $\Gamma$, and we need decay on $\Gamma_0$ because that is where the high-frequency support of the symbol lies. If $\alpha$ is a multi-index, differentiating in $y$ produces terms with symbols $b_\alpha \in S^{|\alpha|}$ whose high-frequency conic support remains in $\Gamma_0$, so each derivative is a finite sum of integrals \begin{align*} (2\pi)^{-n}\int_{\mathbb{R}^n} e^{iy\cdot \eta} b_\alpha(y,\eta)\widehat{(\kappa^{-1})^*(\chi u)}(\eta)\,d\mathcal{L}^n(\eta). \end{align*} On the compact $y$-support of $b_\alpha$, symbol seminorms give $|b_\alpha(y,\eta)| \leq C_\alpha(1+|\eta|)^{|\alpha|}$. The low-frequency region is compact in $\eta$, and the Fourier transform of a compactly supported distribution has polynomial growth there. On the high-frequency conic support, which is contained in $\Gamma_0$, $\widehat{(\kappa^{-1})^*(\chi u)}$ decays faster than every power. Choosing $N>n+|\alpha|+1$ gives \begin{align*} |b_\alpha(y,\eta)\widehat{(\kappa^{-1})^*(\chi u)}(\eta)| \leq C_\alpha C_N(1+|\eta|)^{|\alpha|-N}, \end{align*} and this bound is integrable over $\mathbb{R}^n$ with respect to $\mathcal{L}^n(\eta)$. Hence each derivative exists and is continuous by dominated convergence. This proves the localized term is smooth, while the off-support kernel terms are smooth by the pseudodifferential kernel theorem. Therefore the properly supported global operator $A$ satisfies $Au \in C^\infty(X)$.[/guided]

[step:Use ellipticity to recover a microlocal identity near the cotangent point]Conversely, suppose there exists $A \in \Psi^0(X)$ elliptic at $(x_0,\xi_0)$ with $Au \in C^\infty(X)$. By ellipticity, the principal symbol of $A$ is nonzero in some conic neighbourhood $\mathcal{V} \subset T^*X \setminus 0$ of $(x_0,\xi_0)$. Let $\Psi^{-\infty}(X)$ denote the class of properly supported smoothing pseudodifferential operators on $X$, meaning operators whose Schwartz kernels are smooth on $X \times X$. Choose $Q \in \Psi^0(X)$ whose full symbol is supported in the conic neighbourhood $\mathcal{V}$ and equals $1$ on a smaller conic neighbourhood of $(x_0,\xi_0)$. Since $A$ is elliptic on $\mathcal{V}$ and the microsupport of $Q$ is contained in $\mathcal{V}$, the microlocal elliptic parametrix theorem gives $B \in \Psi^0(X)$ and $R \in \Psi^{-\infty}(X)$ such that the genuine operator identity \begin{align*} Q = BA + R \end{align*} holds on distributions. This invokes a standard prerequisite not yet in the wiki: microlocal elliptic parametrix theorem with a prescribed microlocal cutoff. Since $Au \in C^\infty(X)$ and pseudodifferential operators preserve smoothness, $B(Au) \in C^\infty(X)$. Since $R$ is smoothing, $Ru \in C^\infty(X)$; this uses the standard prerequisite not yet in the wiki: smoothing pseudodifferential operators have smooth output on distributions. Therefore \begin{align*} Qu = B(Au) + Ru \end{align*} is smooth on $X$. After multiplying by a spatial cutoff supported in the coordinate chart and equal to $1$ near $x_0$, we may regard $Qu$ as a smooth local function whose symbol is equal to $1$ in a conic neighbourhood of $\eta_0$.[/step]

[guided]The reverse direction uses ellipticity as microlocal invertibility. The hypothesis says that $A$ has a nonvanishing principal symbol at $(x_0,\xi_0)$. Therefore, after shrinking to a conic neighbourhood of that point in $T^*X \setminus 0$, the symbol of $A$ can be inverted in the pseudodifferential calculus. Choose $Q \in \Psi^0(X)$ whose full symbol is supported where the principal symbol of $A$ is nonzero and whose full symbol equals $1$ in a smaller conic neighbourhood of $(x_0,\xi_0)$. The microlocal elliptic parametrix theorem with this prescribed cutoff gives an operator $B \in \Psi^0(X)$ and a smoothing operator $R \in \Psi^{-\infty}(X)$ such that the actual operator identity \begin{align*} Q = BA + R \end{align*} holds on distributions. The cutoff $Q$ is the device that turns the phrase "near $(x_0,\xi_0)$" into an ordinary equality: outside the elliptic region, $Q$ has no microlocal support, so the inverse construction is not being asserted there. Now apply this identity to $u$. Since $Au \in C^\infty(X)$ by hypothesis and pseudodifferential operators map smooth functions to smooth functions, we have \begin{align*} B(Au) \in C^\infty(X). \end{align*} The operator $R$ is smoothing, so \begin{align*} Ru \in C^\infty(X). \end{align*} Thus the identity \begin{align*} Qu = B(Au) + Ru \end{align*} shows that $Qu$ is smooth on $X$, in particular microlocally near $(x_0,\xi_0)$. This is the decisive consequence of ellipticity: although $A$ may not be globally invertible, it is invertible after inserting the phase-space cutoff $Q$ in the region relevant to the wave front question.[/guided]

[step:Convert the microlocal cutoff smoothness into Fourier decay]Return to the coordinate chart $(U,\kappa)$. Choose $\psi \in C_c^\infty(U)$ with $\psi(x_0)=1$ and support sufficiently small that the coordinate symbol of $Q$ equals $1$ for $x$ near $x_0$ and $\eta$ in an open conic neighbourhood $\Gamma_2$ of $\eta_0$ with $|\eta|$ large. Let \begin{align*} w_\psi := (\kappa^{-1})^*(\psi u) \in \mathcal{D}'(\kappa(U)). \end{align*} Let \begin{align*} q: \kappa(U) \times \mathbb{R}^n &\to \mathbb{C} \end{align*} denote the coordinate full symbol of $Q$ on the chart, and define \begin{align*} r: \kappa(U) \times \mathbb{R}^n &\to \mathbb{C} \end{align*} \begin{align*} (y,\eta) &\mapsto 1-q(y,\eta). \end{align*} Shrink the cone to an open conic neighbourhood $\Gamma_3$ of $\eta_0$ with $\overline{\Gamma_3}\cap S^{n-1}\subset \Gamma_2\cap S^{n-1}$. Since $q(y,\eta)=1$ on $\operatorname{supp}\psi \times \Gamma_2$ for $|\eta|$ sufficiently large, every $\eta$-derivative of $r$ vanishes on $\operatorname{supp}\psi \times \Gamma_3$ for high frequency. The local identity is \begin{align*} \psi u = \psi Q u + \psi(I-Q)u. \end{align*} The distribution $\psi u$ is compactly supported in the coordinate chart, so the local Fourier transform of $\psi(I-Q)u$ is well-defined as the Fourier transform of a compactly supported distribution. The high-frequency symbol of the localized operator $\psi(I-Q)$ vanishes on $\operatorname{supp}\psi\times\Gamma_3$ and is separated on the unit sphere from the output cone $\Gamma_3$. The standard non-stationary phase estimate for pseudodifferential kernels applied to this localized operator then gives, for every integer $N\geq 0$, a constant $C_N'>0$ such that \begin{align*} |\widehat{\psi(I-Q)u}(\eta)| \leq C_N'(1+|\eta|)^{-N} \end{align*} for every $\eta \in \Gamma_3$. This estimate is obtained by repeatedly integrating by parts in the oscillatory variables using that angular separation; no commutator with $\psi$ is being ignored, because multiplication by $\psi$ is part of the error operator being estimated. Since $\psi Qu$ is smooth and compactly supported in the chart, its Fourier transform is rapidly decreasing in every direction. Combining the decomposition $\psi u=\psi Qu+\psi(I-Q)u$ with the two rapid-decay estimates gives, for every integer $N \geq 0$, a constant $C_N > 0$ such that \begin{align*} |\widehat{w_\psi}(\eta)| \leq C_N(1+|\eta|)^{-N} \end{align*} for every $\eta \in \Gamma_3$. By the localized Fourier definition of the wave front set in the first step, this rapid decay estimate implies \begin{align*} (x_0,\xi_0) \notin WF(u). \end{align*}[/step]

[guided]We now convert the microlocal identity into the Fourier estimate that appears in the definition of $WF(u)$. Choose $\psi \in C_c^\infty(U)$ with $\psi(x_0)=1$ and define \begin{align*} w_\psi := (\kappa^{-1})^*(\psi u) \in \mathcal{D}'(\kappa(U)). \end{align*} The operator $Q$ has coordinate full symbol $q(y,\eta)$ equal to $1$ for $y$ near $y_0$ and $\eta$ in a cone $\Gamma_2$ around $\eta_0$ at high frequency. Therefore the symbol $r(y,\eta):=1-q(y,\eta)$ of the error $I-Q$ vanishes on $\operatorname{supp}\psi \times \Gamma_3$ after shrinking to a smaller cone $\Gamma_3\Subset\Gamma_2$ on the unit sphere. In local coordinates we use the exact decomposition \begin{align*} \psi u = \psi Q u + \psi(I-Q)u. \end{align*} This is the correct place to put the spatial cutoff: replacing $\psi(I-Q)u$ by $(I-Q)(\psi u)$ would introduce a commutator, and that commutator is not zero in general. Why does the error have rapid decay? The distribution $\psi u$ is compactly supported in the coordinate chart, so its local Fourier transform is meaningful as the Fourier transform of a compactly supported distribution. Also, the high-frequency symbol of the localized operator $\psi(I-Q)$ vanishes on $\operatorname{supp}\psi\times\Gamma_3$ and is separated from the output cone $\Gamma_3$ on the unit sphere. Therefore, in the Fourier transform of $\psi(I-Q)u$, the phase has no stationary interaction between output frequencies $\eta \in \Gamma_3$ and the high-frequency support of the symbol of $\psi(I-Q)$. Repeated integration by parts in the oscillatory kernel gives, for each integer $N\geq 0$, a constant $C_N'>0$ such that \begin{align*} |\widehat{\psi(I-Q)u}(\eta)| \leq C_N'(1+|\eta|)^{-N} \end{align*} for all $\eta \in \Gamma_3$. This is precisely the pseudodifferential cutoff estimate: an operator whose symbol vanishes in a conic region cannot create non-rapid Fourier growth in that region. The other term is easier. Since $Qu$ is smooth microlocally near $(x_0,\xi_0)$ and $\psi$ is supported inside the chosen coordinate neighbourhood, $\psi Qu$ is a compactly supported smooth function in the chart. The Fourier transform of a compactly supported smooth function is rapidly decreasing in every direction. Combining the identity $\psi u=\psi Qu+\psi(I-Q)u$ with the rapid decay of both terms gives, for every integer $N\geq 0$, a constant $C_N>0$ such that \begin{align*} |\widehat{w_\psi}(\eta)| \leq C_N(1+|\eta|)^{-N} \end{align*} for all $\eta\in\Gamma_3$. Since $\psi(x_0)=1$ and $\Gamma_3$ is an open conic neighbourhood of $\eta_0$, the localized Fourier definition of the wave front set gives \begin{align*} (x_0,\xi_0)\notin WF(u). \end{align*}[/guided]

[step:Globalize the local equivalence on the manifold]The preceding arguments were made in a coordinate chart, but each ingredient is invariant under change of coordinates: the Fourier decay cone transforms by the cotangent lift of the coordinate change, and the principal symbol of a pseudodifferential operator transforms by the same cotangent map. Therefore ellipticity at $(x_0,\xi_0)$ and absence of wave front set at $(x_0,\xi_0)$ are coordinate-independent statements. The forward direction constructs an operator $A \in \Psi^0(X)$ elliptic at $(x_0,\xi_0)$ with $Au \in C^\infty(X)$. The reverse direction shows that any such elliptic smoothing operator forces the localized Fourier transform of $u$ to decay rapidly in a conic neighbourhood of $\xi_0$. Thus \begin{align*} (x_0,\xi_0) \notin WF(u) \end{align*} if and only if there exists $A \in \Psi^0(X)$ elliptic at $(x_0,\xi_0)$ such that $Au \in C^\infty(X)$.[/step]

[guided]The argument has been written in a chart only because the definition of $WF(u)$ is expressed by Fourier decay after choosing coordinates. This does not make the conclusion coordinate-dependent. If we change charts, covectors transform by the cotangent lift of the coordinate change, the conic decay region transforms by the same map, and the homogeneous principal symbol of a pseudodifferential operator transforms by that cotangent map as well. Hence the two relevant properties, namely $(x_0,\xi_0)\notin WF(u)$ and ellipticity of $A$ at $(x_0,\xi_0)$, are intrinsic statements on $T^*X\setminus 0$. The forward implication used local rapid Fourier decay to build a properly supported order-zero operator $A\in\Psi^0(X)$ whose principal symbol is nonzero at $(x_0,\xi_0)$ and whose application to $u$ is smooth. The reverse implication used ellipticity to construct a microlocal parametrix and a cutoff $Q$ equal to the identity near $(x_0,\xi_0)$; from $Au\in C^\infty(X)$ it followed that $Qu\in C^\infty(X)$, and the pseudodifferential cutoff estimate converted this into rapid decay of a localized Fourier transform in a conic neighbourhood of the coordinate representative of $\xi_0$. By the local Fourier definition of the wave front set, this is exactly $(x_0,\xi_0)\notin WF(u)$.[/guided]