[proofplan] The assertion is local, so we pass to a real-analytic coordinate chart near $x_0$ and use the FBI-transform definition of the analytic wavefront set. If $u$ is analytic near $x_0$, [Cauchy estimates](/theorems/2571) allow the FBI phase to be shifted into a complex neighbourhood, giving exponential decay in every nonzero cotangent direction over $x_0$. Conversely, if no nonzero covector over $x_0$ lies in $\operatorname{WF}_a(u)$, compactness of the unit cotangent sphere gives uniform exponential FBI decay over all directions after shrinking the base neighbourhood. The FBI inversion formula and the analytic Paley-Wiener estimate then give factorial derivative bounds, which imply convergence of the Taylor series and hence real-analyticity near $x_0$. [/proofplan] [step:Reduce the statement to a local FBI transform estimate in analytic coordinates] Choose a real-analytic coordinate chart $(U,\kappa)$ of $X$ with $x_0\in U$, where $\kappa:U\to V\subset\mathbb R^m$ is a real-analytic diffeomorphism onto an [open set](/page/Open%20Set) and $m=\dim X$. Write $a_0:=\kappa(x_0)$. Let $\kappa_*u\in\mathcal D'(V)$ denote the pushforward distribution, defined by \begin{align*} (\kappa_*u)(\varphi):=u\bigl((\varphi\circ\kappa)\,|\det J\kappa|\bigr) \end{align*} for every $\varphi\in C_c^\infty(V;\mathbb C)$, with the Jacobian factor understood with respect to the coordinate density. Real-analyticity of $u$ near $x_0$ is equivalent to real-analyticity of $\kappa_*u$ near $a_0$, because $\kappa$ and $\kappa^{-1}$ are real-analytic. The analytic wavefront set transforms under the cotangent lift of $\kappa$, so \begin{align*} \operatorname{WF}_a(u)\cap\bigl(T_{x_0}^*X\setminus\{0\}\bigr)=\varnothing \end{align*} is equivalent to the absence of all nonzero covectors over $a_0$ from $\operatorname{WF}_a(\kappa_*u)$. It is therefore enough to prove the result for a distribution on an open set $V\subset\mathbb R^m$ at the point $a_0$. Fix an open ball $B(a_0,r_0)\subset V$ with $r_0>0$, and choose $\chi\in C_c^\infty(V;[0,1])$ such that $\chi=1$ on $B(a_0,r_0/2)$. Define the compactly supported distribution $v:=\chi\,\kappa_*u\in\mathcal E'(\mathbb R^m)$ by extending by zero outside $V$. For $0<h\le 1$, define the FBI transform $\mathcal T_h v:\mathbb R^m\times(\mathbb R^m\setminus\{0\})\to\mathbb C$ by \begin{align*} \mathcal T_h v(y,\eta):=v(\phi_{y,\eta,h}), \end{align*} where $\phi_{y,\eta,h}:\mathbb R^m\to\mathbb C$ is the entire Gaussian phase \begin{align*} \phi_{y,\eta,h}(x):=(2\pi h)^{-3m/4}\exp\left(-\frac{|x-y|^2}{2h}-\frac{i\,\eta\cdot(x-y)}{h}\right). \end{align*} Since $v$ has compact support, this pairing is well-defined by the standard extension of compactly supported distributions to $C^\infty$ functions. [guided] The first point is that both properties in the theorem are local at $x_0$. Choose a real-analytic coordinate chart $(U,\kappa)$ with $x_0\in U$ and $\kappa:U\to V\subset\mathbb R^m$ a real-analytic diffeomorphism. Write $a_0:=\kappa(x_0)$. The distribution $u$ is transported to a distribution $\kappa_*u\in\mathcal D'(V)$ by \begin{align*} (\kappa_*u)(\varphi):=u\bigl((\varphi\circ\kappa)\,|\det J\kappa|\bigr) \end{align*} for every $\varphi\in C_c^\infty(V;\mathbb C)$. This coordinate change does not change the question. Real-analyticity is preserved under composition with real-analytic coordinate changes. Likewise, the analytic wavefront set is invariant under real-analytic diffeomorphisms through the induced cotangent map. Thus the original condition \begin{align*} \operatorname{WF}_a(u)\cap\bigl(T_{x_0}^*X\setminus\{0\}\bigr)=\varnothing \end{align*} is exactly the local Euclidean condition that no nonzero covector over $a_0$ belongs to $\operatorname{WF}_a(\kappa_*u)$. Now choose an open ball $B(a_0,r_0)\subset V$ and a smooth cutoff $\chi\in C_c^\infty(V;[0,1])$ such that $\chi=1$ on $B(a_0,r_0/2)$. This cutoff is not analytic, and that is intentional: compactly supported analytic cutoffs do not exist except in the zero case. The analytic microlocal information is encoded in the FBI estimates, while $\chi$ is only used to make the distribution compactly supported. Define $v:=\chi\,\kappa_*u$ and extend it by zero to a compactly supported distribution on $\mathbb R^m$. For $0<h\le 1$, define \begin{align*} \mathcal T_h v(y,\eta):=v(\phi_{y,\eta,h}), \end{align*} where \begin{align*} \phi_{y,\eta,h}(x):=(2\pi h)^{-3m/4}\exp\left(-\frac{|x-y|^2}{2h}-\frac{i\,\eta\cdot(x-y)}{h}\right). \end{align*} Here $y\in\mathbb R^m$ is the base variable, $\eta\in\mathbb R^m\setminus\{0\}$ is the frequency variable, and $h\in(0,1]$ is the semiclassical parameter. Because $v$ has compact support, it can act on the smooth function $\phi_{y,\eta,h}$ even though this function is not compactly supported. This is the standard FBI transform used to define the analytic wavefront set. [/guided] [/step] [step:Derive exponential FBI decay from real-analyticity] Assume first that $u$ is real-analytic in a neighbourhood of $x_0$. After shrinking $r_0>0$, the distribution $\kappa_*u$ is represented on $B(a_0,r_0)$ by a real-[analytic function](/page/Analytic%20Function) $f:B(a_0,r_0)\to\mathbb C$. Hence there are constants $R>0$ and $A>0$ such that $f$ extends holomorphically to the complex polydisc \begin{align*} P_R(a_0):=\{z\in\mathbb C^m:|\operatorname{Re}z-a_0|<R,\ |\operatorname{Im}z|<R\} \end{align*} and satisfies $|f(z)|\le A$ on $P_R(a_0)$. We use the following precise external analytic microlocal result: the local FBI criterion for analytic regularity says that if a distribution $w\in\mathcal D'(V)$ is represented by a [holomorphic function](/page/Holomorphic%20Function) in a complex neighbourhood of $a_0\in V$, then for every $\eta_0\in\mathbb R^m\setminus\{0\}$ and every cutoff $\chi\in C_c^\infty(V)$ equal to $1$ near $a_0$, the FBI transform of $\chi w$, with the normalization fixed above, satisfies an estimate \begin{align*} |\mathcal T_h(\chi w)(y,\eta)|\le C_{\eta_0}\exp\left(-\frac{c_{\eta_0}}{h}\right) \end{align*} for $y$ in a neighbourhood of $a_0$, for $\eta$ in a conic neighbourhood of $\eta_0$ with $|\eta|$ in any fixed compact subinterval of $(0,\infty)$, and for $0<h\le h_{\eta_0}$. Its proof splits $\chi w$ into a near part supported where $w=f$ and a separated part. The near part is estimated by shifting the FBI phase inside the holomorphic neighbourhood $P_R(a_0)$; the constants depend on $A$, $R$, the chosen annulus in $\eta$, and the FBI normalization. The separated part has support a positive Euclidean distance from $a_0$, so the Gaussian factor contributes $\exp(-d^2/(4h))$ for $y$ sufficiently close to $a_0$, where $d>0$ is that support separation; this absorbs the distribution seminorms of the compactly supported separated part. Applying this criterion to $w=\kappa_*u$ and the cutoff $\chi$ fixed above gives the displayed exponential estimate for $v=\chi\kappa_*u$ near every nonzero $\eta_0$. The estimate is exactly the FBI-transform exclusion criterion for $(a_0,\eta_0)$ not to belong to $\operatorname{WF}_a(\kappa_*u)$, because $\chi=1$ near $a_0$. Since $\eta_0$ was arbitrary, no nonzero covector over $a_0$ belongs to $\operatorname{WF}_a(\kappa_*u)$. Returning through the coordinate chart gives \begin{align*} \operatorname{WF}_a(u)\cap\bigl(T_{x_0}^*X\setminus\{0\}\bigr)=\varnothing. \end{align*} [/step] [step:Convert absence of analytic wavefront directions into uniform exponential decay] Assume conversely that \begin{align*} \operatorname{WF}_a(u)\cap\bigl(T_{x_0}^*X\setminus\{0\}\bigr)=\varnothing. \end{align*} In the chosen coordinates this means that $(a_0,\eta_0)\notin\operatorname{WF}_a(\kappa_*u)$ for every $\eta_0\in\mathbb R^m\setminus\{0\}$. By the FBI definition of $\operatorname{WF}_a$, for each unit covector $\omega\in S^{m-1}$ there are an open neighbourhood $Y_\omega\subset\mathbb R^m$ of $a_0$, an open conic neighbourhood $\Gamma_\omega\subset\mathbb R^m\setminus\{0\}$ of $\omega$, and constants $C_\omega>0$, $c_\omega>0$, and $h_\omega\in(0,1]$ such that \begin{align*} |\mathcal T_h v(y,\eta)|\le C_\omega\exp\left(-\frac{c_\omega}{h}\right) \end{align*} for $y\in Y_\omega$, $\eta\in\Gamma_\omega$ with $1/2\le|\eta|\le 2$, and $0<h\le h_\omega$. The sphere $S^{m-1}:=\{\omega\in\mathbb R^m:|\omega|=1\}$ is compact, so choose finitely many unit covectors $\omega_1,\dots,\omega_N\in S^{m-1}$ such that \begin{align*} S^{m-1}\subset\bigcup_{j=1}^N\Gamma_{\omega_j}. \end{align*} Define \begin{align*} Y:=\bigcap_{j=1}^N Y_{\omega_j}. \end{align*} Choose $\rho\in(0,r_0/2)$ with $B(a_0,\rho)\subset Y$, and define \begin{align*} C:=\max_{1\le j\le N}C_{\omega_j},\qquad c:=\min_{1\le j\le N}c_{\omega_j},\qquad h_0:=\min_{1\le j\le N}h_{\omega_j}. \end{align*} Then \begin{align*} |\mathcal T_h v(y,\eta)|\le C\exp\left(-\frac{c}{h}\right) \end{align*} for every $y\in B(a_0,\rho)$, every $\eta\in\mathbb R^m$ with $1/2\le|\eta|\le 2$, and every $0<h\le h_0$. By homogeneity in the frequency variable in the FBI characterization, the same estimate holds after rescaling on every compact annulus in $\mathbb R^m\setminus\{0\}$, with constants depending only on the annulus. This is the uniform all-direction exponential FBI decay over the base point $a_0$. [/step] [step:Apply the FBI analytic regularity theorem to get factorial derivative bounds] We now use the converse part of the same local FBI analytic [regularity theorem](/theorems/2750). In the normalization fixed above, it states: if $v\in\mathcal E'(\mathbb R^m)$ and there are an open set $Y\subset\mathbb R^m$ containing $a_0$, constants $C>0$, $c>0$, and $h_0\in(0,1]$ such that \begin{align*} |\mathcal T_h v(y,\eta)|\le C\exp\left(-\frac{c}{h}\right) \end{align*} for every $y\in Y$, every $\eta\in\mathbb R^m$ with $1/2\le |\eta|\le 2$, and every $0<h\le h_0$, then $v$ is represented by a real-analytic function in a neighbourhood of $a_0$. Quantitatively, after shrinking to a ball $B(a_0,\rho/4)\subset Y$, there are constants $M>0$ and $B>0$, depending only on $m$, the FBI normalization, $C$, $c$, $h_0$, $\rho$, and finitely many compact-support seminorms of $v$, such that, for every multi-index $\alpha\in\mathbb N_0^m$ and every $x\in B(a_0,\rho/4)$, \begin{align*} |\partial_x^\alpha v(x)|\le M\,B^{|\alpha|+1}\,\alpha!. \end{align*} Here $\alpha!:=\alpha_1!\cdots\alpha_m!$, $|\alpha|:=\alpha_1+\cdots+\alpha_m$, and $\partial_x^\alpha:=\partial_{x_1}^{\alpha_1}\cdots\partial_{x_m}^{\alpha_m}$. The hypotheses of this theorem match the preceding step: $v\in\mathcal E'(\mathbb R^m)$ by construction, $B(a_0,\rho)\subset Y$, and the finite conic cover gives the required uniform exponential estimate on the annulus $1/2\le |\eta|\le 2$ for $0<h\le h_0$. Therefore the stated constants $M$ and $B$ exist. [guided] The remaining implication is not a separate ad hoc inversion calculation; it is the quantitative converse built into the local FBI analytic regularity theorem. That theorem is the precise external result being used here. It assumes a compactly supported distribution $v\in\mathcal E'(\mathbb R^m)$ and uniform exponential decay of its FBI transform over a base neighbourhood and one fixed frequency annulus. Its conclusion is that $v$ is analytic in the base neighbourhood, with factorial derivative bounds. We verify the hypotheses one by one. First, $v=\chi\kappa_*u$ was extended by zero outside $V$, so $v\in\mathcal E'(\mathbb R^m)$. Second, the preceding finite-cover argument produced constants $C>0$, $c>0$, and $h_0\in(0,1]$ and a ball $B(a_0,\rho)$ such that \begin{align*} |\mathcal T_h v(y,\eta)|\le C\exp\left(-\frac{c}{h}\right) \end{align*} whenever $y\in B(a_0,\rho)$, $1/2\le |\eta|\le 2$, and $0<h\le h_0$. This is exactly the annular FBI estimate required by the theorem. No additional low-frequency estimate is being smuggled in: the theorem's proof treats the low-frequency part inside the fixed-parameter FBI inversion formula and uses only compact-support seminorms of $v$ there, while the annular exponential estimate controls the analytic high-frequency directions. Applying the theorem on the smaller ball $B(a_0,\rho/4)$ gives constants $M>0$ and $B>0$, depending only on the dimension, the chosen FBI normalization, the decay constants, the radius, and finitely many seminorms controlling the compactly supported distribution $v$, such that \begin{align*} |\partial_x^\alpha v(x)|\le M\,B^{|\alpha|+1}\,\alpha! \end{align*} for every $x\in B(a_0,\rho/4)$ and every multi-index $\alpha\in\mathbb N_0^m$. [/guided] [/step] [step:Conclude analyticity from the factorial bounds] Fix $x_1\in B(a_0,\rho/8)$. The factorial estimate gives, for every multi-index $\alpha\in\mathbb N_0^m$, \begin{align*} \left|\frac{\partial_x^\alpha v(x_1)}{\alpha!}(x-x_1)^\alpha\right|\le M\,B^{|\alpha|+1}|x-x_1|^{|\alpha|}. \end{align*} Choose $r_1>0$ such that $B(x_1,r_1)\subset B(a_0,\rho/4)$ and $mBr_1<1$. Then the Taylor series \begin{align*} \sum_{\alpha\in\mathbb N_0^m}\frac{\partial_x^\alpha v(x_1)}{\alpha!}(x-x_1)^\alpha \end{align*} converges absolutely and uniformly on $B(x_1,r_1)$ by comparison with a convergent geometric series. Standard Taylor remainder estimates using the same factorial bound show that this series represents $v$ on $B(x_1,r_1)$. Hence $v$ is real-analytic on $B(a_0,\rho/8)$. Because $\chi=1$ on $B(a_0,r_0/2)$ and $\rho<r_0/2$, the equality $v=\kappa_*u$ holds on $B(a_0,\rho/8)$. Therefore $\kappa_*u$ is real-analytic near $a_0$. Pulling back by the real-analytic chart $\kappa$ shows that $u$ is real-analytic in a neighbourhood of $x_0$. Combining this implication with the forward implication proves the equivalence. [/step]