[proofplan] The proof is a microlocal identification of the Hilbert-space Szego projection with the Boutet de Monvel-Guillemin Hermite projector constructed from the strictly pseudoconvex CR geometry. The deep input is the Boutet de Monvel-Sjostrand and Boutet de Monvel-Guillemin Szego kernel theorem, which gives a local oscillatory-integral parametrix whose phase parametrizes the positive contact cone. The Hermite calculus then converts the projection identity $\Pi^2=\Pi$ into an idempotent equation for the principal symbol, while the closed range theory for $\bar\partial_b$ identifies the microlocal parametrix with the actual [orthogonal projection](/theorems/437) modulo a smoothing operator. [/proofplan] [step:Use embeddability and strict pseudoconvexity to obtain the Hermite Szego parametrix] Let $H(M)\subset TM$ denote the real CR hyperplane distribution, and let $\theta\in\Omega^1(M)$ be the chosen positive contact form with $\ker\theta=H(M)$. Since $M$ is compact, smooth, strictly pseudoconvex, and CR-embeddable, the Boutet de Monvel-Sjostrand Szego kernel parametrix, in its Boutet de Monvel-Guillemin Hermite formulation, applies to the CR Hardy projection associated with any smooth positive density $d\mu$. Concretely, this theorem supplies an operator \begin{align*} S:L^2(M,d\mu)\to L^2(M,d\mu) \end{align*} whose Schwartz kernel $K_S\in\mathcal D'(M\times M)$ is locally represented near the diagonal by a positive complex phase oscillatory integral parametrizing the positive contact cone. In the normalization fixed in the statement, $S$ belongs to the Hermite class of order $0$ associated with \begin{align*} \Delta_{\Sigma^+}:=\{(p,r\theta_p;p,-r\theta_p)\in T^*(M\times M):p\in M,\ r>0\}. \end{align*} The same theorem gives the microlocal identities \begin{align*} \bar\partial_b S\equiv 0,\qquad S\bar\partial_b^*\equiv 0,\qquad S^2\equiv S,\qquad S^*\equiv S \end{align*} modulo smoothing operators, where $\bar\partial_b^*$ denotes the Hilbert-space adjoint determined by $d\mu$ and the chosen Hermitian metric on tangential forms. Here $\equiv$ means equality modulo an operator with smooth Schwartz kernel on $M\times M$. [guided] The first point is that we are not constructing the Szego kernel from elementary functional analysis. The relevant input is the Boutet de Monvel-Sjostrand parametrix, refined by Boutet de Monvel and Guillemin into the Hermite Fourier integral operator calculus. Its hypotheses are exactly the geometric hypotheses in the theorem: $M$ is smooth, compact, strictly pseudoconvex, and CR-embeddable, and a positive density $d\mu$ has been fixed so that the $L^2$ orthogonal projection is a Hilbert-space object. Let $H(M)\subset TM$ be the CR hyperplane distribution. The chosen one-form \begin{align*} \theta\in\Omega^1(M) \end{align*} is positive and satisfies $\ker\theta=H(M)$. This defines the positive characteristic cone \begin{align*} \Sigma^+:=\{(p,r\theta_p)\in T^*M:p\in M,\ r>0\}. \end{align*} The Hermite calculus records kernels on $M\times M$, so the corresponding conic subset of $T^*(M\times M)$ is the positive contact diagonal \begin{align*} \Delta_{\Sigma^+}:=\{(p,r\theta_p;p,-r\theta_p)\in T^*(M\times M):p\in M,\ r>0\}. \end{align*} The minus sign in the second covector is the standard kernel convention: an operator relation from the right variable to the left variable is represented in the cotangent bundle of the product by reversing the sign on the right covector. The Boutet de Monvel-Guillemin realization theorem then gives an operator \begin{align*} S:L^2(M,d\mu)\to L^2(M,d\mu) \end{align*} whose Schwartz kernel $K_S\in\mathcal D'(M\times M)$ is a Hermite distribution associated with $\Delta_{\Sigma^+}$. In the convention fixed in the statement, this means $S$ has Hermite order $0$. The theorem also states that $S$ is a microlocal Szego projector: \begin{align*} \bar\partial_b S\equiv 0,\qquad S\bar\partial_b^*\equiv 0,\qquad S^2\equiv S,\qquad S^*\equiv S \end{align*} modulo smoothing operators. These four relations say respectively that the range is microlocally CR, the operator is microlocally orthogonal to the $\bar\partial_b$-range, it is a projection microlocally, and it is self-adjoint microlocally. [/guided] [/step] [step:Identify the parametrix with the Hilbert-space Szego projection modulo smoothing] Because $M$ is compact and strictly pseudoconvex with an embeddable CR structure, the closed range theorem for the tangential Cauchy-Riemann complex in degree $0$ implies that \begin{align*} H_b^2(M,d\mu)=\ker\bar\partial_b\cap L^2(M,d\mu) \end{align*} is a closed subspace of $L^2(M,d\mu)$. Hence the orthogonal projection \begin{align*} \Pi:L^2(M,d\mu)\to H_b^2(M,d\mu) \end{align*} is well-defined. The microlocal Szego parametrix $S$ constructed above has range in $\ker\bar\partial_b$ modulo smooth errors and is orthogonal to the range of $\bar\partial_b^*$ modulo smooth errors. By the [Hodge decomposition](/theorems/2745) for the closed range $\bar\partial_b$ complex on compact strictly pseudoconvex embeddable CR manifolds, these two microlocal conditions characterize the orthogonal projection onto $\ker\bar\partial_b$ modulo smoothing operators. Therefore \begin{align*} \Pi-S:R^s(M)\to C^\infty(M) \end{align*} is smoothing for every Sobolev order $s\in\mathbb R$, where $R^s(M)$ denotes the [Sobolev space](/page/Sobolev%20Space) $H^s(M)$ built from any fixed smooth positive density and elliptic reference operator on $M$. Equivalently, the Schwartz kernel $K_{\Pi-S}$ belongs to $C^\infty(M\times M)$. [/step] [step:Read the operator wavefront relation from the positive contact diagonal] Since $K_{\Pi-S}\in C^\infty(M\times M)$, the kernels $K_\Pi$ and $K_S$ have the same microlocal singularities. The kernel $K_S$ is a Hermite distribution associated with $\Delta_{\Sigma^+}$, so \begin{align*} \operatorname{WF}'(S)=\{((p,r\theta_p),(p,r\theta_p)):p\in M,\ r>0\}. \end{align*} Here $\operatorname{WF}'(S)$ denotes the operator wavefront relation obtained from the wavefront set of the Schwartz kernel by reversing the sign of the covector in the second factor. Since smoothing operators have empty operator wavefront relation, we obtain \begin{align*} \operatorname{WF}'(\Pi)=\operatorname{WF}'(S)=\{((p,r\theta_p),(p,r\theta_p)):p\in M,\ r>0\}. \end{align*} Thus the operator wavefront relation of $\Pi$ is the identity relation on $\Sigma^+$. [/step] [step:Use the Hermite composition calculus to identify the principal symbol as an idempotent] Let \begin{align*} \sigma_{\mathrm H}(\Pi) \end{align*} denote the Hermite principal symbol of $\Pi$ in the Boutet de Monvel-Guillemin symbol algebra over $\Sigma^+$. Since $\Pi-S$ is smoothing, $\Pi$ and $S$ have the same Hermite principal symbol: \begin{align*} \sigma_{\mathrm H}(\Pi)=\sigma_{\mathrm H}(S). \end{align*} The Hermite composition theorem states that principal symbols multiply under composition in the Hermite symbol algebra. Applying this theorem to the Hilbert-space identity \begin{align*} \Pi^2=\Pi \end{align*} gives \begin{align*} \sigma_{\mathrm H}(\Pi)^2=\sigma_{\mathrm H}(\Pi). \end{align*} Hence the Hermite principal symbol of $\Pi$ is an idempotent. [/step] [step:Conclude smoothing away from the positive contact cone] Let $(p,\xi)\in T^*M\setminus\Sigma^+$. Since the operator wavefront relation of $\Pi$ is the identity relation on $\Sigma^+$, there is no covector in $\operatorname{WF}'(\Pi)$ whose left or right component is $(p,\xi)$. Therefore, by the definition of microlocal smoothing for an operator, $\Pi$ is microlocally smoothing at $(p,\xi)$. Combining this smoothing-away statement with the Hermite description of $K_\Pi$ along $\Delta_{\Sigma^+}$ proves that $\Pi$ is a Hermite Fourier integral operator of order $0$ associated with the positive contact diagonal, has operator wavefront relation equal to the identity on $\Sigma^+$, and has idempotent Hermite principal symbol. This is precisely the claimed Boutet de Monvel-Guillemin realization of the Szego projector. [/step]