[proofplan] The proof uses the Boutet de Monvel-Guillemin realisation theorem for the Szegő projector as a Hermite Fourier integral operator associated to the positive contact cone. First one identifies the positive characteristic cone of the tangential Cauchy-Riemann operator with $\Sigma^+$, using the positive contact form $\theta$. The local normal form for Hermite Fourier integral kernels then gives a complex phase and a classical amplitude with the stated diagonal normalization and symbol expansion. Finally, the cited realisation theorem identifies this local Hermite Fourier integral kernel with the genuine orthogonal Szegő projection modulo a smooth kernel. [/proofplan]

[step:Identify the positive characteristic direction determined by the contact form]Let \begin{align*} T^{1,0}M\subset \mathbb C TM \end{align*} denote the holomorphic CR bundle induced on $M=\partial\Omega$, and let \begin{align*} T^{0,1}M:=\overline{T^{1,0}M}\subset \mathbb C TM \end{align*} denote its antiholomorphic conjugate. Let \begin{align*} H(M):=\operatorname{Re}(T^{1,0}M\oplus T^{0,1}M)\subset TM \end{align*} denote the real contact distribution. Since $\Omega$ is strictly pseudoconvex, $H(M)$ is a contact distribution and a positive contact form $\theta\in\Omega^1(M)$ is characterized by \begin{align*} \ker\theta=H(M) \end{align*} with the positive Levi orientation. The principal symbol of the first-order tangential Cauchy-Riemann operator $\bar\partial_b$ vanishes precisely on the real covectors annihilating $H(M)$. Thus its nonzero characteristic set has two connected components, \begin{align*} \{(x,\lambda\theta_x):x\in M,\lambda>0\} \end{align*} and \begin{align*} \{(x,\lambda\theta_x):x\in M,\lambda<0\}. \end{align*} The positive component is exactly the cone $\Sigma^+$ appearing in the statement.[/step]

[guided]The geometric point is that the Szegő kernel is singular only in the characteristic directions of $\bar\partial_b$. The CR bundle $T^{0,1}M$ is induced by the embedding $M\subset\mathbb C^n$, and its real part determines the contact hyperplane distribution \begin{align*} H(M):=\operatorname{Re}(T^{1,0}M\oplus T^{0,1}M)\subset TM. \end{align*} A positive contact form is a one-form \begin{align*} \theta\in\Omega^1(M) \end{align*} such that \begin{align*} \ker\theta=H(M) \end{align*} and whose sign agrees with the strictly pseudoconvex orientation. Because $H(M)$ is a hyperplane distribution, every real covector annihilating $H_x(M)$ is a real multiple of $\theta_x$. Therefore the nonzero characteristic covectors of $\bar\partial_b$ over $x$ are precisely the two rays \begin{align*} \{\lambda\theta_x:\lambda>0\} \end{align*} and \begin{align*} \{\lambda\theta_x:\lambda<0\}. \end{align*} The sign convention in the theorem selects the first ray. Hence the positive characteristic cone is \begin{align*} \Sigma^+=\{(x,\lambda\theta_x)\in T^*M\setminus\{0\}:x\in M,\lambda>0\}. \end{align*} This is the canonical direction that the phase must encode on the diagonal.[/guided]

[step:Apply the Hermite Fourier integral normal form near the positive cone]Fix $p\in M$. Since $\Omega$ is bounded and has $C^\infty$ boundary, $M=\partial\Omega$ is compact and smooth. The density $d\mu$ is positive and $C^\infty$, and strict pseudoconvexity gives a compact strictly pseudoconvex CR manifold with positive contact cone $\Sigma^+$. Therefore the [Boutet de Monvel Guillemin Realisation][citetheorem:9226] applies to the orthogonal Szegő projector \begin{align*} S:L^2(M,d\mu)\to H^2(M,d\mu). \end{align*} It says that $S$ is a Hermite Fourier integral operator of order $0$ associated to $\Sigma^+$, with the kernel measured using $d\mu$ in the second variable. The local normal form for a Hermite Fourier integral kernel associated to $\Sigma^+$ gives an open neighbourhood $U\subset M$ of $p$ and a phase function \begin{align*} \psi:U\times U\to\mathbb C \end{align*} in $C^\infty(U\times U;\mathbb C)$ satisfying \begin{align*} \operatorname{Im}\psi(x,y)\ge 0 \end{align*} for $(x,y)\in U\times U$, \begin{align*} \psi(x,x)=0 \end{align*} for $x\in U$, \begin{align*} \psi(x,y)=-\overline{\psi(y,x)} \end{align*} for $(x,y)\in U\times U$, and \begin{align*} d_x\psi(x,x)=\theta_x \end{align*} for $x\in U$. The last equality is the phase normalization corresponding to the chosen positive contact form: replacing $\theta$ by a positive multiple would rescale the homogeneous parameter, so the chosen parametrisation of $\Sigma^+$ fixes $d_x\psi(x,x)$ to be $\theta_x$. Consequently the associated positive diagonal canonical direction is precisely \begin{align*} \{(x,d_x\psi(x,x)):x\in U\}=\{(x,\theta_x):x\in U\}, \end{align*} and its homogeneous extension in the oscillatory parameter $t>0$ is \begin{align*} \{(x,t\,d_x\psi(x,x)):x\in U,\ t>0\}=\Sigma^+\cap T^*U. \end{align*}[/step]

[guided]The input used here is not the theorem being proved in its kernel-expansion form. The cited realisation theorem states that the already-defined orthogonal Szegő projection is a Hermite Fourier integral operator associated to the positive cone $\Sigma^+$. We verify its hypotheses in the present setting: $M=\partial\Omega$ is compact because $\Omega$ is bounded with smooth boundary, the induced CR structure is strictly pseudoconvex by hypothesis, and $d\mu$ is a positive smooth density, exactly the density used to define the [Hilbert space](/page/Hilbert%20Space) $L^2(M,d\mu)$ and the kernel in the second variable. The local normal form for Hermite Fourier integral kernels then supplies coordinates near each $p\in M$, an [open set](/page/Open%20Set) $U\subset M$, and a complex phase \begin{align*} \psi:U\times U\to\mathbb C \end{align*} with non-negative imaginary part. The diagonal identities \begin{align*} \psi(x,x)=0 \end{align*} and \begin{align*} \psi(x,y)=-\overline{\psi(y,x)} \end{align*} encode that the operator is self-adjoint at the level of the phase. The differential condition \begin{align*} d_x\psi(x,x)=\theta_x \end{align*} is the normalization of the positive homogeneous parametrisation: the covectors generated by the phase on the diagonal are \begin{align*} (x,t\,d_x\psi(x,x)) \end{align*} with $t>0$, and these must equal \begin{align*} (x,t\theta_x) \end{align*} with $t>0$. Thus the phase parametrises exactly $\Sigma^+\cap T^*U$, not the negative cone and not a differently normalized positive ray.[/guided]

[step:Solve the transport equations to build a microlocal projection kernel] In the same local Hermite Fourier integral normal form, the symbol attached to an order-$0$ Hermite Fourier integral operator on a real hypersurface of dimension $2n-1$ is represented by a classical scalar symbol \begin{align*} s:U\times U\times(0,\infty)\to\mathbb C \end{align*} of order $n-1$ in $t$, smooth in $(x,y)$, with an asymptotic expansion \begin{align*} s(x,y,t)\sim\sum_{j=0}^{\infty}t^{n-1-j}s_j(x,y), \end{align*} where \begin{align*} s_j:U\times U\to\mathbb C \end{align*} belongs to $C^\infty(U\times U;\mathbb C)$ for every integer $j\ge 0$. The principal Hermite symbol of $S$ is the idempotent symbol supplied by the [Boutet de Monvel Guillemin Realisation][citetheorem:9226]. Choosing the local representative of that symbol relative to the density $d\mu$ fixes the leading coefficient $s_0$ in this kernel convention, and the lower coefficients $s_j$ represent the lower homogeneous terms of the same Hermite symbol. Thus the operator $A$ with kernel \begin{align*} A(x,y):=\int_0^\infty e^{it\psi(x,y)}s(x,y,t)\,d\mathcal L^1(t) \end{align*} has the same full Hermite symbol as $S$ over $\Sigma^+\cap T^*U$. Here $\Psi^{-\infty}(U)$ denotes the class of smoothing operators on $U$, namely operators whose distribution kernels belong to $C^\infty(U\times U;\mathbb C)$ with respect to the density $d\mu$ in the second variable. [/step]

[step:Compare the local normal form with the true Szegő kernel] Let $C_c^\infty(U)$ denote the space of smooth compactly supported complex-valued functions on $U$, and let $\mathcal D'(U)$ denote the space of distributions on $U$. Define \begin{align*} S_U:C_c^\infty(U)\to\mathcal D'(U) \end{align*} by restricting the distribution kernel $S(x,y)$ to $U\times U$, with integration against $d\mu$ in the second variable. The operator $A$ is the local Hermite Fourier integral representative of $S_U$ obtained from the full symbol and phase supplied by the normal form. Since two Hermite Fourier integral representatives with the same phase parametrising $\Sigma^+\cap T^*U$ and the same full classical symbol differ by a smoothing operator, their difference satisfies \begin{align*} S_U-A\in\Psi^{-\infty}(U). \end{align*} Here the displayed inclusion means that the distribution kernel of $S_U-A$ belongs to \begin{align*} C^\infty(U\times U;\mathbb C) \end{align*} with respect to the density $d\mu$ in the second variable. [/step]

[step:Read off the local kernel asymptotic] Since $p\in M$ was arbitrary, the preceding construction applies near every point of the diagonal in $M\times M$. On each such neighbourhood $U\times U$, the distribution kernel of $S$ satisfies \begin{align*} S(x,y)\equiv \int_0^\infty e^{it\psi(x,y)}s(x,y,t)\,d\mathcal L^1(t) \end{align*} modulo a $C^\infty$ kernel. The phase $\psi$ has the stated positivity, diagonal vanishing, Hermitian antisymmetry, and diagonal differential properties, and the amplitude $s$ is a classical symbol of order $n-1$ with coefficients \begin{align*} s_j\in C^\infty(U\times U;\mathbb C). \end{align*} This is exactly the asserted Boutet de Monvel-Sjöstrand local asymptotic expansion for the Szegő kernel. [/step]