Let $n\ge 2$, let $\Omega\subset\mathbb C^n$ be a bounded strictly pseudoconvex domain with $C^\infty$ boundary $M:=\partial\Omega$, and let $\rho\in C^\infty(\overline\Omega;\mathbb R)$ be a defining function satisfying

\begin{align*} \Omega=\{z\in\mathbb C^n:\rho(z)<0\} \end{align*}

and $d\rho_x\ne 0$ for every $x\in M$. Equip $\Omega$ with Euclidean [Lebesgue measure](/page/Lebesgue%20Measure) $\mathcal L^{2n}$ and equip $M$ with the induced hypersurface measure $\mathcal H^{2n-1}$. Let $H^2(M)\subset L^2(M,\mathcal H^{2n-1})$ be the Hardy space of boundary values of holomorphic functions on $\Omega$, let $A^2(\Omega)\subset L^2(\Omega,\mathcal L^{2n})$ be the Bergman space, and let $\Pi:L^2(M,\mathcal H^{2n-1})\to H^2(M)$ and $B:L^2(\Omega,\mathcal L^{2n})\to A^2(\Omega)$ be the orthogonal projectors. Define

\begin{align*} \theta:=\frac{i}{2}(\bar\partial\rho-\partial\rho)|_M \end{align*}

with the strictly pseudoconvex orientation, and define

\begin{align*} \Sigma_+:=\{(x,\lambda\theta_x)\in T^*M\setminus\{0\}:x\in M,\lambda>0\}. \end{align*}

Let $\kappa:M\times[0,\varepsilon_0)\to\overline\Omega$ be a collar diffeomorphism satisfying $\rho(\kappa(x,r))=-r$.

Assume the standard Boutet de Monvel microlocal boundary calculus for strictly pseudoconvex domains in CR dimension $n-1\ge 1$ in the following precise form. The positive microlocal holomorphic Poisson extension exists in such a collar as a properly supported Hermite Fourier integral operator with canonical relation $C_P$ from $\Sigma_+$ to the positive holomorphic collar relation, its boundary trace is the Szegő projector modulo smoothing operators, and its positive normal principal factor over $(x,\lambda\theta_x)\in\Sigma_+$ is $e^{-\lambda r}$. The adjoint relation $C_P^t$, the Szegő diagonal relation on $\Sigma_+$, and the graph relation of any tangential pseudodifferential operator elliptic on $\Sigma_+$ compose cleanly, with the expected excess, to the Bergman canonical relation in the collar. The boundary Gram operator $P^*P$ is a classical tangential pseudodifferential operator on the Hardy microlocal component, its principal symbol is obtained by the normal density integral of the two Poisson factors, and elliptic parametrices in this microlocal calculus exist. Finally, the Bergman projector is microlocally the [orthogonal projection](/theorems/437) onto the corresponding Poisson extension range with inverse boundary Gram operator.

Then, after possibly replacing $\varepsilon_0$ by a smaller positive number $\varepsilon$ and setting $U_\varepsilon:=\kappa(M\times[0,\varepsilon))$, there exist a properly supported microlocal holomorphic Poisson operator $P:\mathcal D'(M)\to\mathcal D'(U_\varepsilon)$ and a properly supported classical tangential pseudodifferential operator $\Lambda\in\Psi^1(M)$ with the following properties. The boundary trace of $P$ equals $\Pi$ modulo a smoothing operator on $M$. In collar coordinates, the positive normal principal factor of $P$ over $(x,\lambda\theta_x)\in\Sigma_+$ is $e^{-\lambda r}$. The operator $\Lambda$ is elliptic on $\Sigma_+$ and has principal symbol

\begin{align*} \sigma_1(\Lambda)(x,\lambda\theta_x)=c(x)\lambda \end{align*}

for some $c\in C^\infty(M;\mathbb R)$ satisfying $c(x)>0$ for every $x\in M$.

Moreover, if $\chi_1,\chi_2\in C_c^\infty(\Omega\cap U_\varepsilon;\mathbb R)$ are supported in a sufficiently small boundary collar and are equal to $1$ on a smaller boundary collar, then

\begin{align*} \chi_1 B\chi_2\equiv \chi_1P\Pi\Lambda\Pi P^*\chi_2 \end{align*}

modulo an operator whose Schwartz kernel is $C^\infty$ on $(\Omega\cap U_\varepsilon)\times(\Omega\cap U_\varepsilon)$. Here $P^*:C_c^\infty(U_\varepsilon)\to\mathcal D'(M)$ denotes the Hilbert-space adjoint with respect to $\mathcal L^{2n}$ on $\Omega$ and $\mathcal H^{2n-1}$ on $M$, with the collar localization understood in both variables.

Consequently, suppose that the Szegő kernel is written microlocally near the diagonal of $M\times M$ in Boutet de Monvel-Sjöstrand 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 $s\in S_{\mathrm{cl}}^{n-1}$. Then the Bergman kernel has, in the collar and modulo $C^\infty$ kernels, an oscillatory representation

\begin{align*} K_\Omega(z,w)\equiv \int_0^\infty e^{it\Psi(z,w)}b(z,w,t)\,d\mathcal L^1(t), \end{align*}

where $\Psi$ is a positive complex collar phase compatible with the canonical relation of the Poisson construction, restricts to $\psi$ on $M\times M$, and contains the normal exponential decay determined by $\rho$ and by the chosen normalization of $P$. The amplitude satisfies $b\in S_{\mathrm{cl}}^n$. Equivalent choices of phase, collar, or Poisson normalization change $b$ only by the standard classical symbol transformations and by smoothing remainders. In particular, relative to the corresponding Szegő representation, the Bergman boundary amplitude has classical symbol order $n$, while the Szegő boundary amplitude has classical symbol order $n-1$.