[proofplan] We prove the equivalence by moving between two ways of detecting the absence of singularities near $(x_0,\xi_0)$. Fourier decay gives an elliptic pseudodifferential cutoff by choosing a symbol supported inside the regular cone and applying it only to a spatial localization of $u$. Conversely, an elliptic operator gives a microlocal parametrix, which rewrites a localized copy of $u$ as a smooth term plus a microlocally smoothing error, forcing rapid Fourier decay. The same parametrix also shows that every operator microsupported in a sufficiently small elliptic neighbourhood factors through the elliptic cutoff modulo a smoothing operator, giving the universal condition in part 3. [/proofplan]

[step:Record the pseudodifferential calculus facts used in the proof] We use the following standard facts from the properly supported pseudodifferential calculus on $\mathbb{R}^n$, with the terminology fixed in the theorem statement. Let $\mathcal{L}^n$ denote Lebesgue measure on $\mathbb{R}^n$. For $a \in S^m(\mathbb{R}^n \times \mathbb{R}^n)$, the notation $\operatorname{Op}(a)$ denotes the standard properly supported quantization of $a$, and its operator microsupport is the closed conic set outside which the full symbol is of order $-\infty$ in the frequency variable. The space $\mathcal{E}'(\mathbb{R}^n)$ denotes compactly supported distributions. Throughout the proof, $I: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ denotes the identity operator. First, if $P \in \Psi^m(\mathbb{R}^n)$ is properly supported, then $P$ extends continuously by duality to \begin{align*} P: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n). \end{align*} Second, properly supported operators in $\Psi^m(\mathbb{R}^n)$ map $C^\infty(\mathbb{R}^n)$ to $C^\infty(\mathbb{R}^n)$. Third, a properly supported smoothing operator maps $\mathcal{D}'(\mathbb{R}^n)$ to $C^\infty(\mathbb{R}^n)$. Fourth, multiplication by a function in $C_c^\infty(\mathbb{R}^n)$ is a properly supported order-zero pseudodifferential operator whose principal symbol is that function, and the principal symbol of a composition is the product of the principal symbols. Fifth, microlocal smoothing is used in the following precise form. If $R \in \Psi^0(\mathbb{R}^n)$ is smoothing microlocally on $U_0 \times \Gamma_0$, where $U_0 \subset \mathbb{R}^n$ is open and $\Gamma_0 \subset \mathbb{R}^n \setminus \{0\}$ is open and conic, then for every $\chi \in C_c^\infty(U_0)$ and every open conic set $\Gamma$ whose closure in $\mathbb{R}^n \setminus \{0\}$ is contained in $\Gamma_0$, the compactly supported distribution $\chi Rv \in \mathcal{E}'(\mathbb{R}^n)$ has rapidly decreasing Fourier transform in $\Gamma$ for every $v \in \mathcal{D}'(\mathbb{R}^n)$. Sixth, if $A \in \Psi^0(\mathbb{R}^n)$ is properly supported and elliptic at $(x_0,\xi_0)$, then there exist an open neighbourhood $U_0 \subset \mathbb{R}^n$ of $x_0$, an open conic neighbourhood $\Gamma_0 \subset \mathbb{R}^n \setminus \{0\}$ of $\xi_0$, and a properly supported operator $Q \in \Psi^0(\mathbb{R}^n)$ such that $QA-I$ is smoothing microlocally on $U_0 \times \Gamma_0$. Seventh, after replacing $U_0$ and $\Gamma_0$ by smaller neighbourhoods if necessary, there are an open neighbourhood $U \subset U_0$ of $x_0$ and an open conic neighbourhood $\Gamma \subset \Gamma_0$ of $\xi_0$ such that every properly supported $B \in \Psi^0(\mathbb{R}^n)$ whose operator microsupport is compactly contained in $U \times \Gamma$ satisfies \begin{align*} B(QA-I) \in \Psi^{-\infty}(\mathbb{R}^n). \end{align*} Equivalently, \begin{align*} B = BQA + R \end{align*} for the properly supported smoothing operator $R:=-B(QA-I)$. These are exactly the standard microlocal parametrix, composition, Fourier-smoothing, and smoothing remainder facts for properly supported pseudodifferential operators used in the proof. [/step]

[step:Construct an elliptic cutoff from conic Fourier decay]Assume condition 1. Thus there are $\chi \in C_c^\infty(\mathbb{R}^n)$ with $\chi(x_0) \neq 0$ and an open conic neighbourhood $\Gamma_1 \subset \mathbb{R}^n \setminus \{0\}$ of $\xi_0$ such that $\widehat{\chi u}$ is rapidly decreasing in $\Gamma_1$. Choose $\rho \in C_c^\infty(\mathbb{R}^n)$ such that $\rho(x_0) \neq 0$ and $\operatorname{supp}\rho$ is contained in the open set where $\chi$ is nonzero. Define \begin{align*} \beta: \mathbb{R}^n \to \mathbb{C}, \qquad \beta(x) := \rho(x)/\chi(x) \end{align*} on $\operatorname{supp}\rho$ and extend $\beta$ smoothly by $0$ outside a compact subset of the set where $\chi$ is nonzero. Then $\beta \chi = \rho$ on $\mathbb{R}^n$. Choose a smooth conic cutoff $\psi \in C^\infty(\mathbb{R}^n \setminus \{0\})$ of degree $0$ for $|\xi| \geq 1$ such that $\psi(\xi_0) \neq 0$ and the conic support of $\psi$ is contained in $\Gamma_1$ for $|\xi| \geq 1$. Extending $\psi$ smoothly to all of $\mathbb{R}^n$, define a symbol map \begin{align*} a: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}, \qquad (x,\xi) \mapsto \beta(x)\psi(\xi). \end{align*} Here $a \in S^0(\mathbb{R}^n \times \mathbb{R}^n)$ and $\operatorname{Op}(a)$ denotes the standard properly supported quantization. Define the multiplication map \begin{align*} M_\chi: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{E}'(\mathbb{R}^n), \qquad v \mapsto \chi v \end{align*} to be multiplication by $\chi$, where $\mathcal{E}'(\mathbb{R}^n)$ is the space of compactly supported distributions. Set $A := \operatorname{Op}(a)\circ M_\chi$. By the multiplication and composition facts recorded above, $M_\chi$ is a properly supported order-zero pseudodifferential operator, $A \in \Psi^0(\mathbb{R}^n)$ is properly supported, and the principal symbol of $A$ at $(x_0,\xi_0)$ is the product of the principal symbols of $\operatorname{Op}(a)$ and $M_\chi$. Hence \begin{align*} \sigma_0(A)(x_0,\xi_0)=\beta(x_0)\psi(\xi_0)\chi(x_0)=\rho(x_0)\psi(\xi_0)\neq 0. \end{align*} Therefore $A$ is elliptic at $(x_0,\xi_0)$. The distribution $\chi u$ has compact support, so $\widehat{\chi u}$ is a smooth function of at most polynomial growth. By the standard properly supported quantization convention, $\operatorname{Op}(a)(\chi u)$ agrees, modulo a smooth function, with the Kohn-Nirenberg oscillatory expression determined by $a$ and $\widehat{\chi u}$. It is therefore enough to prove smoothness for that oscillatory expression. Split the frequency integral defining this representative of $Au$ into the region $|\xi|\leq 2$ and the region $|\xi|>2$. The low-frequency region is compact in $\xi$, so differentiating under the integral produces only compactly supported smooth amplitudes. On the high-frequency region where $\psi(\xi)$ is nonzero, the conic support condition places $\xi$ in $\Gamma_1$, and the rapid decay estimates from condition 1 hold. If $\alpha \in \mathbb{N}_0^n$ is a spatial multi-index with $|\alpha|=k$, differentiating $k$ times in $x$ introduces at most a polynomial factor bounded by $C_\alpha(1+|\xi|)^k$ for a constant $C_\alpha>0$. Choose $N>k+n$. Then the differentiated high-frequency integrand is bounded by a constant times $(1+|\xi|)^{k-N}$, which is integrable over $\mathbb{R}^n$ with respect to $\mathcal{L}^n$. Dominated convergence justifies differentiation under the integral and gives continuity of the derivative. Since $\alpha$ was arbitrary, $Au \in C^\infty(\mathbb{R}^n)$. This proves condition 2.[/step]

[guided]We begin from the Fourier condition and want to build a pseudodifferential operator that sees only the regular part of $u$ near the covector $\xi_0$. The spatial cutoff $\chi$ is already part of condition 1, and the hypothesis says that the compactly supported distribution $\chi u$ has a Fourier transform that decays faster than every power of $|\xi|$ in a cone around $\xi_0$. Choose $\rho \in C_c^\infty(\mathbb{R}^n)$ with $\rho(x_0)\neq 0$ and with support contained in the set where $\chi$ is nonzero. This support condition matters because it lets us divide by $\chi$ on the support of $\rho$. Define \begin{align*} \beta: \mathbb{R}^n \to \mathbb{C}, \qquad \beta(x) := \rho(x)/\chi(x) \end{align*} on $\operatorname{supp}\rho$ and extend $\beta$ smoothly by $0$ outside a compact subset of the set where $\chi$ is nonzero. Then $\beta\chi=\rho$ on $\mathbb{R}^n$. Choose also a conic frequency cutoff $\psi \in C^\infty(\mathbb{R}^n \setminus \{0\})$, homogeneous of degree $0$ for $|\xi|\geq 1$, such that $\psi(\xi_0)\neq 0$ and the conic support of $\psi$ for large $|\xi|$ lies inside the cone $\Gamma_1$ where $\widehat{\chi u}$ decays rapidly. After smoothing $\psi$ near $\xi=0$, define the symbol map \begin{align*} a: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}, \qquad (x,\xi) \mapsto \beta(x)\psi(\xi). \end{align*} The symbol $a$ belongs to $S^0(\mathbb{R}^n \times \mathbb{R}^n)$ because $\beta$ is compactly supported and, for every multi-index $\alpha \in \mathbb{N}_0^n$, the derivatives $\partial_\xi^\alpha \psi$ satisfy the order-zero symbol estimate. To make the operator act only on the localized distribution, define the multiplication map \begin{align*} M_\chi: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{E}'(\mathbb{R}^n), \qquad v \mapsto \chi v \end{align*} where $\mathcal{E}'(\mathbb{R}^n)$ is the space of compactly supported distributions, and set $A:=\operatorname{Op}(a)\circ M_\chi$. Since multiplication by a smooth compactly supported function is a properly supported order-zero pseudodifferential operator and principal symbols multiply under composition, $A$ remains in $\Psi^0(\mathbb{R}^n)$ and its principal symbol satisfies \begin{align*} \sigma_0(A)(x_0,\xi_0)=\beta(x_0)\psi(\xi_0)\chi(x_0)=\rho(x_0)\psi(\xi_0)\neq 0. \end{align*} Thus $A$ is elliptic at $(x_0,\xi_0)$. It remains to check smoothness of $Au$. The compact support of $\chi u$ implies that $\widehat{\chi u}$ is smooth and grows at most polynomially. The standard properly supported quantization agrees, on compact spatial supports and modulo a smoothing operator, with the corresponding Kohn-Nirenberg oscillatory formula. The smoothing error already sends distributions to smooth functions, so we only need to estimate the oscillatory representative. We split the frequency integral into two regions. On $|\xi|\leq 2$, the frequency domain is compact, so differentiating in $x$ produces a compactly supported smooth amplitude and hence a smooth contribution. On the large-frequency part of the support of $\psi$, the support condition for $\psi$ places $\xi$ inside $\Gamma_1$, and the hypothesis gives rapid decay: for every $N \in \mathbb{N}$ there exists $C_N>0$ such that \begin{align*} |\widehat{\chi u}(\xi)| \leq C_N(1+|\xi|)^{-N} \end{align*} for all such $\xi$. Let $\alpha\in\mathbb{N}_0^n$ be a spatial multi-index and write $k=|\alpha|$. After applying $\partial_x^\alpha$, the differentiated oscillatory integrand is bounded by a constant times $(1+|\xi|)^k|\widehat{\chi u}(\xi)|$ on the high-frequency region, because $a$ is an order-zero symbol and the exponential contributes at most $k$ powers of $\xi$. Choose $N>k+n$. Then the integrand is bounded by a constant times $(1+|\xi|)^{k-N}$, which is integrable over $\mathbb{R}^n$ with respect to $\mathcal{L}^n$. Dominated convergence therefore justifies differentiating under the integral and proves that $\partial_x^\alpha Au$ is continuous. Since $\alpha$ was arbitrary, $Au\in C^\infty(\mathbb{R}^n)$. Thus the Fourier decay condition produces an elliptic properly supported order-zero cutoff with smooth output.[/guided]

[step:Recover conic Fourier decay from an elliptic cutoff] Assume condition 2. Let $A \in \Psi^0(\mathbb{R}^n)$ be properly supported, elliptic at $(x_0,\xi_0)$, and satisfy $Au \in C^\infty(\mathbb{R}^n)$. By the microlocal parametrix fact recorded above, choose open neighbourhoods $U_0$ of $x_0$ and $\Gamma_0$ of $\xi_0$, with $\Gamma_0$ conic, and choose $Q \in \Psi^0(\mathbb{R}^n)$ properly supported such that $QA-I$ is smoothing microlocally on $U_0 \times \Gamma_0$. Choose $\chi \in C_c^\infty(U_0)$ with $\chi(x_0)\neq 0$ and choose a smaller open conic neighbourhood $\Gamma \subset \Gamma_0$ of $\xi_0$ whose closure in $\mathbb{R}^n\setminus\{0\}$ is contained in $\Gamma_0$. Define the microlocal remainder operator \begin{align*} R: \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n), \qquad v \mapsto (QA-I)v. \end{align*} Then \begin{align*} \chi u = \chi QAu - \chi Ru. \end{align*} The first term $\chi QAu$ is compactly supported and smooth because $Au$ is smooth and $Q$ maps smooth functions to smooth functions. Its Fourier transform is rapidly decreasing in every direction. The second term $\chi Ru$ has rapid Fourier decay in $\Gamma$ by the Fourier-smoothing consequence recorded in the first step: $R$ is microlocally smoothing on $U_0 \times \Gamma_0$, $\operatorname{supp}\chi \subset U_0$, and the closure of $\Gamma$ in $\mathbb{R}^n\setminus\{0\}$ is contained in $\Gamma_0$. Therefore $\widehat{\chi u}$ is rapidly decreasing in $\Gamma$, proving condition 1. [/step]

[step:Factor every smaller microlocal cutoff through the elliptic operator]Assume condition 2 again, with the same operator $A$. Choose $U_0$, $\Gamma_0$, and $Q$ as in the preceding step. Shrink to an open neighbourhood $U \subset U_0$ of $x_0$ and an open conic neighbourhood $\Gamma \subset \Gamma_0$ of $\xi_0$ such that the operator identity \begin{align*} B = BQA + R_B \end{align*} holds for every properly supported $B \in \Psi^0(\mathbb{R}^n)$ with operator microsupport compactly contained in $U\times\Gamma$, where $R_B$ is a properly supported smoothing operator. Let such a $B$ be fixed. Define \begin{align*} C_B: C_c^\infty(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n), \qquad C_B := BQ. \end{align*} Then $C_B \in \Psi^0(\mathbb{R}^n)$ is properly supported by the composition calculus. Applying the factorization to $u$ gives \begin{align*} Bu = C_B(Au) + R_Bu. \end{align*} The term $C_B(Au)$ is smooth because $Au\in C^\infty(\mathbb{R}^n)$ and properly supported pseudodifferential operators preserve smoothness. The term $R_Bu$ is smooth because $R_B$ is smoothing. Hence $Bu \in C^\infty(\mathbb{R}^n)$ for every such $B$, proving condition 3.[/step]

[guided]The point of ellipticity is that $A$ has an inverse microlocally near $(x_0,\xi_0)$, even if it has no global inverse. The parametrix statement gives a properly supported operator $Q\in\Psi^0(\mathbb{R}^n)$ such that $QA$ agrees with the identity modulo an operator that is smoothing microlocally in a neighbourhood $U_0\times\Gamma_0$ of the point. More explicitly, the operator $QA-I$ is microlocally smoothing on $U_0\times\Gamma_0$, where $U_0$ is open in $\mathbb{R}^n$ and $\Gamma_0$ is open and conic in $\mathbb{R}^n\setminus\{0\}$. We now shrink to a smaller neighbourhood $U\times\Gamma$ inside this elliptic region with compact closure in the microlocal sense: $U$ has closure contained in $U_0$ near the spatial support under consideration, and the closure of $\Gamma$ in $\mathbb{R}^n\setminus\{0\}$ is contained in $\Gamma_0$. This is the hypothesis needed for the standard smoothing-remainder composition statement. It ensures that any operator $B$ whose operator microsupport is compactly contained in $U\times\Gamma$ only probes the part of phase space where $QA-I$ is microlocally smoothing. Fix such an operator $B\in\Psi^0(\mathbb{R}^n)$. Its operator microsupport is compactly contained in the chosen smaller set $U\times\Gamma$, and $QA-I$ is microlocally smoothing on the larger set $U_0\times\Gamma_0$. Therefore the microlocal composition fact applies and gives \begin{align*} B(QA-I) \in \Psi^{-\infty}(\mathbb{R}^n). \end{align*} Define $R_B:=-B(QA-I)$. Then $R_B$ is a properly supported smoothing operator and \begin{align*} B = BQA + R_B. \end{align*} This identity is the precise meaning of “$B$ factors through $A$ microlocally”: first apply $A$, then apply the parametrix $Q$, then apply $B$, and the only error is smoothing. Define the composed operator \begin{align*} C_B: C_c^\infty(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n), \qquad C_B := BQ. \end{align*} By the composition theorem for properly supported pseudodifferential operators, $C_B$ is again a properly supported order-zero pseudodifferential operator. Applying the factorization to the distribution $u$ gives \begin{align*} Bu = C_B(Au) + R_Bu. \end{align*} Now each term on the right is smooth for a different reason. The term $Au$ is smooth by condition 2, and $C_B$ preserves smoothness, so $C_B(Au)\in C^\infty(\mathbb{R}^n)$. The operator $R_B$ is smoothing, so $R_Bu\in C^\infty(\mathbb{R}^n)$ for every distribution $u\in\mathcal{D}'(\mathbb{R}^n)$. Therefore $Bu$ is the sum of two smooth functions, hence $Bu\in C^\infty(\mathbb{R}^n)$. Since $B$ was arbitrary subject to its microsupport being contained in $U\times\Gamma$, condition 3 follows.[/guided]

[step:Choose an elliptic test operator from the universal smoothness condition] Assume condition 3. Let $U \subset \mathbb{R}^n$ be an open neighbourhood of $x_0$ and let $\Gamma \subset \mathbb{R}^n\setminus\{0\}$ be an open conic neighbourhood of $\xi_0$ such that every properly supported $B\in\Psi^0(\mathbb{R}^n)$ with operator microsupport compactly contained in $U\times\Gamma$ satisfies $Bu\in C^\infty(\mathbb{R}^n)$. Choose $\rho \in C_c^\infty(U)$ with $\rho(x_0)\neq 0$. Choose an order-zero conic cutoff $\psi \in C^\infty(\mathbb{R}^n\setminus\{0\})$, smoothly extended near $\xi=0$, such that $\psi(\xi_0)\neq 0$ and the conic support of $\psi$ for large $|\xi|$ is contained in $\Gamma$. Define the symbol map \begin{align*} b: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}, \qquad (x,\xi) \mapsto \rho(x)\psi(\xi). \end{align*} Let $B:=\operatorname{Op}(b)$ be a properly supported quantization. By choosing $\rho$ with support compactly contained in $U$ and choosing $\psi$ with conic support compactly contained in $\Gamma$ for large $|\xi|$, we have $B\in\Psi^0(\mathbb{R}^n)$ and the operator microsupport of $B$ is compactly contained in $U\times\Gamma$. Moreover \begin{align*} b(x_0,\xi_0)=\rho(x_0)\psi(\xi_0)\neq 0. \end{align*} Thus $B$ is elliptic at $(x_0,\xi_0)$. By condition 3, $Bu\in C^\infty(\mathbb{R}^n)$. Taking $A:=B$ proves condition 2. [/step]

[step:Conclude the equivalence] We have proved condition 1 implies condition 2, condition 2 implies condition 1, condition 2 implies condition 3, and condition 3 implies condition 2. Therefore all three stated conditions are equivalent. This establishes the equivalence of the Fourier decay and pseudodifferential operator definitions of microlocal regularity at $(x_0,\xi_0)$. [/step]