[proofplan] The proof is a microlocal projection argument in a boundary collar. We use the standard microlocal Poisson [extension theorem](/theorems/59) for strictly pseudoconvex CR boundaries to construct $P$, then compute the boundary Gram operator $G=P^*P$ on the positive Hardy component. The normal integral of the factor $e^{-\lambda r}$ gives an elliptic symbol of order $-1$, so its microlocal inverse is the order-one tangential operator $\Lambda$. The Bergman projector is then the orthogonal projector onto the microlocal range of $P\Pi$, and the Hermite Fourier integral composition calculus converts $P\Pi\Lambda\Pi P^*$ into the stated Bergman oscillatory kernel with one higher symbol order than the Szegő kernel. [/proofplan]

[step:Choose the positive microlocal Poisson extension in the boundary collar] Set $U_\varepsilon:=\kappa(M\times[0,\varepsilon))$ after shrinking $\varepsilon>0$ so that $\kappa$ is defined on $M\times[0,\varepsilon)$ and so that $\overline{U_\varepsilon}$ is contained in the chosen collar. Since $n\ge 2$, the boundary CR dimension is $n-1\ge 1$, so the Boutet de Monvel microlocal boundary calculus for strictly pseudoconvex CR boundaries applies. By the positive microlocal Poisson extension hypothesis in the formalized statement, applied to the contact form $\theta$ and the positive cone $\Sigma_+$, there is a properly supported Hermite Fourier integral operator \begin{align*} P:\mathcal D'(M)\to\mathcal D'(U_\varepsilon) \end{align*} whose canonical relation is the relation $C_P$ from $\Sigma_+$ to the positive holomorphic collar relation supplied by the assumed Boutet de Monvel calculus, whose boundary trace operator \begin{align*} \gamma P:\mathcal D'(M)\to\mathcal D'(M) \end{align*} satisfies \begin{align*} \gamma P\equiv \Pi \end{align*} modulo a smoothing operator, and whose principal normal factor over $(x,\lambda\theta_x)\in\Sigma_+$ is $e^{-\lambda r}$ in the collar coordinate $r$. Here $\gamma:C^\infty(U_\varepsilon)\to C^\infty(M)$ denotes the trace map defined as follows: for each $u\in C^\infty(U_\varepsilon;\mathbb C)$, $\gamma u:M\to\mathbb C$ is the function $x\mapsto u(\kappa(x,0))$. The invoked Poisson extension input is one of the microlocal calculus hypotheses now stated in the theorem: $M$ is $C^\infty$, strictly pseudoconvex, oriented by $\theta$, and the collar coordinate satisfies $\rho(\kappa(x,r))=-r$. Since $\Pi$ is the orthogonal projector onto $H^2(M)$, the microlocal range of $\Pi$ over $\Sigma_+$ is the positive CR component. Thus $P\Pi$ maps boundary Hardy data to microlocal holomorphic functions in the collar, and $\gamma P\Pi\equiv\Pi$ modulo smoothing. [/step]

[step:Compute the boundary Gram operator and extract its elliptic inverse]Define the microlocal boundary Gram operator \begin{align*} G:\mathcal D'(M)&\to\mathcal D'(M) \end{align*} \begin{align*} u&\mapsto P^*Pu. \end{align*} Here \begin{align*} P^*:C_c^\infty(U_\varepsilon)\to\mathcal D'(M) \end{align*} is the Hilbert-space adjoint determined by $\mathcal L^{2n}$ on $U_\varepsilon$ and $\mathcal H^{2n-1}$ on $M$. In collar coordinates, the Euclidean measure has the form \begin{align*} d\mathcal L^{2n}(\kappa(x,r))=J(x,r)\,d\mathcal H^{2n-1}(x)\,d\mathcal L^1(r), \end{align*} where \begin{align*} J:M\times[0,\varepsilon)\to(0,\infty) \end{align*} is a $C^\infty$ density factor. On the positive cone $\Sigma_+$, the principal normal part of $G=P^*P$ is obtained by multiplying the two normal factors $e^{-\lambda r}$ and integrating in the normal variable $r$. Thus the principal contribution is \begin{align*} \int_0^\varepsilon e^{-2\lambda r}J(x,r)\,d\mathcal L^1(r). \end{align*} Since $J(x,r)=J(x,0)+rJ_1(x,r)$ for some $J_1\in C^\infty(M\times[0,\varepsilon);\mathbb R)$, Watson's lemma gives \begin{align*} \int_0^\varepsilon e^{-2\lambda r}J(x,r)\,d\mathcal L^1(r)=\frac{J(x,0)}{2\lambda}+S_{-2}(x,\lambda) \end{align*} as $\lambda\to+\infty$, where $S_{-2}$ is a classical symbol of order $-2$ in $\lambda$. Therefore $G$ is a classical tangential pseudodifferential operator of order $-1$ on the Hardy microlocal component, elliptic on $\Sigma_+$, with positive principal symbol \begin{align*} \sigma_{-1}(G)(x,\lambda\theta_x)=\frac{J(x,0)}{2\lambda}. \end{align*} Let $\Lambda\in\Psi^1(M)$ be a properly supported microlocal parametrix for $G$ on $\Sigma_+$, restricted to the Hardy component by $\Pi$. Thus \begin{align*} \Pi\Lambda\Pi G\Pi\equiv\Pi \end{align*} and \begin{align*} \Pi G\Pi\Lambda\Pi\equiv\Pi \end{align*} modulo smoothing operators microlocally near $\Sigma_+$. Its principal symbol has the form \begin{align*} \sigma_1(\Lambda)(x,\lambda\theta_x)=c(x)\lambda, \end{align*} where, for the normalization above, \begin{align*} c(x)=\frac{2}{J(x,0)}. \end{align*} Because $J(x,0)>0$, one has $c(x)>0$ for all $x\in M$. A different normalization of $P$ multiplies this positive function by the corresponding positive elliptic symbol factor, which is precisely the normalization dependence allowed in the statement.[/step]

[guided]The point of this step is to identify the one extra tangential derivative that separates the Bergman kernel from the Szegő kernel. We first define the operator being measured. The boundary Gram operator is \begin{align*} G:\mathcal D'(M)&\to\mathcal D'(M) \end{align*} \begin{align*} u&\mapsto P^*Pu. \end{align*} It is called a Gram operator because it records the $L^2(\Omega,\mathcal L^{2n})$ [inner product](/page/Inner%20Product) of two Poisson extensions. More explicitly, for smooth microlocal test data $u,v\in C^\infty(M;\mathbb C)$ supported in the chosen coordinate patch, the defining adjoint relation gives \begin{align*} (G u,v)_{L^2(M)}=(Pu,Pv)_{L^2(U_\varepsilon)}. \end{align*} We now compute its leading symbol over a covector $(x,\lambda\theta_x)\in\Sigma_+$. The Poisson operator was normalized so that the normal dependence of a positive-frequency boundary mode is $e^{-\lambda r}$. Therefore the product of one Poisson factor and one adjoint Poisson factor contributes $e^{-2\lambda r}$. The only remaining normal contribution in the $L^2$ inner product comes from the Euclidean volume density in collar coordinates. Since $\kappa$ is a diffeomorphism, there is a smooth positive function \begin{align*} J:M\times[0,\varepsilon)&\to(0,\infty) \end{align*} such that \begin{align*} d\mathcal L^{2n}(\kappa(x,r))=J(x,r)\,d\mathcal H^{2n-1}(x)\,d\mathcal L^1(r). \end{align*} Thus the principal normal integral is \begin{align*} \int_0^\varepsilon e^{-2\lambda r}J(x,r)\,d\mathcal L^1(r). \end{align*} To read off its asymptotic order, expand $J$ at $r=0$. There is a smooth function \begin{align*} J_1:M\times[0,\varepsilon)&\to\mathbb R \end{align*} such that \begin{align*} J(x,r)=J(x,0)+rJ_1(x,r). \end{align*} The leading term is \begin{align*} J(x,0)\int_0^\varepsilon e^{-2\lambda r}\,d\mathcal L^1(r). \end{align*} This integral equals \begin{align*} \frac{J(x,0)}{2\lambda}(1-e^{-2\lambda\varepsilon}). \end{align*} The exponentially small factor does not affect the classical symbol expansion, and the part coming from $rJ_1(x,r)$ has one lower order in $\lambda$. Consequently the principal symbol of $G$ over $\Sigma_+$ is \begin{align*} \sigma_{-1}(G)(x,\lambda\theta_x)=\frac{J(x,0)}{2\lambda}. \end{align*} This computation explains why $G$ has order $-1$: integrating in the inward normal direction contributes one factor of $\lambda^{-1}$. Since $J(x,0)>0$, this principal symbol is nonzero and positive on $\Sigma_+$. Hence $G$ is elliptic of order $-1$ on the Hardy microlocal component. The classical pseudodifferential parametrix theorem then gives an order-one operator \begin{align*} \Lambda\in\Psi^1(M) \end{align*} such that \begin{align*} \Pi\Lambda\Pi G\Pi\equiv\Pi \end{align*} and \begin{align*} \Pi G\Pi\Lambda\Pi\equiv\Pi \end{align*} modulo smoothing operators near $\Sigma_+$. The principal symbol of this inverse is the reciprocal of the principal symbol of $G$, so with the present normalization \begin{align*} \sigma_1(\Lambda)(x,\lambda\theta_x)=\frac{2}{J(x,0)}\lambda. \end{align*} Setting $c(x):=2/J(x,0)$ gives $c\in C^\infty(M;\mathbb R)$ and $c(x)>0$ for every $x\in M$.[/guided]

[step:Identify the Bergman projector as the microlocal orthogonal projection onto Poisson extensions] Let \begin{align*} E:\mathcal D'(M)&\to\mathcal D'(U_\varepsilon) \end{align*} \begin{align*} u&\mapsto P\Pi u. \end{align*} Then $E$ parametrizes the microlocal holomorphic solutions in the collar with positive CR boundary values. By the Bergman boundary reduction hypothesis in the formalized statement, microlocally in $U_\varepsilon$ and over $\Sigma_+$, the Bergman projector is the orthogonal projector onto the range of $E$. We verify the projection algebra once this boundary reduction is granted. The Gram operator of $E$ is \begin{align*} E^*E=\Pi P^*P\Pi=\Pi G\Pi \end{align*} modulo smoothing operators, because $\Pi^*=\Pi$ and $\Pi^2=\Pi$. Since $\Pi\Lambda\Pi$ is a microlocal parametrix for $\Pi G\Pi$ on $\Sigma_+$, the operator \begin{align*} Q:=E\Pi\Lambda\Pi E^* \end{align*} is the microlocal orthogonal projector onto $\operatorname{Range}(E)$. Substituting $E=P\Pi$ and using $\Pi^2=\Pi$ gives \begin{align*} Q\equiv P\Pi\Lambda\Pi P^* \end{align*} modulo smoothing operators. Choose nested collars \begin{align*} U_{\varepsilon_2}\subset U_{\varepsilon_1}\subset U_\varepsilon \end{align*} with $0<\varepsilon_2<\varepsilon_1<\varepsilon$. If \begin{align*} \chi_1,\chi_2\in C_c^\infty(\Omega\cap U_{\varepsilon_1};\mathbb R) \end{align*} are equal to $1$ on $\Omega\cap U_{\varepsilon_2}$, then proper support of $P$, $\Lambda$, and $P^*$ makes all compositions properly supported after localization. Therefore \begin{align*} \chi_1B\chi_2\equiv \chi_1P\Pi\Lambda\Pi P^*\chi_2 \end{align*} modulo an operator with $C^\infty$ Schwartz kernel on $(\Omega\cap U_\varepsilon)\times(\Omega\cap U_\varepsilon)$. [/step]

[step:Compose the Szegő parametrix with the Poisson factors] By the Boutet de Monvel-Sjöstrand Szegő kernel theorem [citetheorem:9225], the kernel of $\Pi$ is, microlocally near the diagonal of $M\times M$, an Hermite Fourier integral kernel of the form \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*} where $\psi$ is a positive complex phase parametrizing $\Sigma_+$ and $s\in S_{\mathrm{cl}}^{n-1}$. The operators $P$ and $P^*$ are Poisson Fourier integral operators with positive normal factors $e^{-tr}$ and $e^{-tr'}$ in the left and right collar variables. The operator $\Lambda$ is a classical tangential pseudodifferential operator of order $1$ with elliptic symbol $c(x)t$ over $\Sigma_+$. By the clean-composition clause in the assumed Boutet de Monvel calculus, the canonical relation $C_P$, the Szegő diagonal relation on $\Sigma_+$, the graph relation of $\Lambda$, and the adjoint relation $C_P^t$ compose cleanly, with the expected excess, to the Bergman canonical relation in the collar. Therefore the Hermite Fourier integral operator composition theorem, applied to \begin{align*} P\circ\Pi\circ\Lambda\circ\Pi\circ P^*, \end{align*} gives a positive complex collar phase \begin{align*} \Psi:(U_\varepsilon\cap\Omega)\times(U_\varepsilon\cap\Omega)\to\mathbb C \end{align*} and a classical amplitude $b$ such that the localized kernel of $P\Pi\Lambda\Pi P^*$ is \begin{align*} \int_0^\infty e^{it\Psi(z,w)}b(z,w,t)\,d\mathcal L^1(t) \end{align*} modulo a $C^\infty$ kernel. The phase $\Psi$ restricts to $\psi$ when $z=\kappa(x,0)$ and $w=\kappa(y,0)$, and its imaginary part contains the normal decay coming from the two Poisson factors. Define the collar normal-coordinate function $r_\kappa:U_\varepsilon\to[0,\varepsilon)$ by declaring, for each $x\in M$ and $r_0\in[0,\varepsilon)$, \begin{align*} r_\kappa(\kappa(x,r_0))=r_0. \end{align*} In the normalization above this means that the exponential includes the factor corresponding to $e^{-t r_\kappa(z)}e^{-t r_\kappa(w)}$. The symbol order is obtained by adding the orders in the clean composition. The two Poisson normal factors do not change the tangential classical order of the boundary amplitude; the Szegő amplitude has order $n-1$, and $\Lambda$ contributes order $1$. Hence \begin{align*} b\in S_{\mathrm{cl}}^n. \end{align*} By the projector identity from the previous step, the same representation holds for the localized Bergman kernel $K_\Omega$ modulo $C^\infty$ kernels. Equivalent positive complex phases and equivalent Poisson normalizations are related by the standard nondegenerate changes of phase variable and elliptic symbol conjugations in the Hermite calculus. These operations preserve the resulting operator modulo smoothing kernels and transform $b$ by the usual classical symbol rules. Therefore the invariant conclusion is the stated one: relative to the Szegő representation of order $n-1$, the Bergman boundary amplitude has order $n$. [/step]

[step:Conclude the localized kernel relation and the boundary order shift] Combining the microlocal projection identity \begin{align*} \chi_1B\chi_2\equiv \chi_1P\Pi\Lambda\Pi P^*\chi_2 \end{align*} with the Hermite composition calculation gives the asserted equality modulo smooth kernels and the stated oscillatory representation of $K_\Omega$ in the collar. The positivity and ellipticity of $\Lambda$ on $\Sigma_+$ follow from the positive principal symbol $J(x,0)/(2\lambda)$ of $P^*P$ and its microlocal inverse. Finally, since the only order change between the boundary Szegő representation and the collar Bergman representation is the order-one factor $\Lambda$, the Bergman boundary amplitude has classical symbol order $n$ instead of $n-1$. This proves the theorem. [/step]