[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.
[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]
[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*}
[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]
[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]
