Let $n\ge 2$, and let $M$ be a compact smooth CR manifold of hypersurface type with real dimension $2n-1$. Assume that the CR structure on $M$ is strictly pseudoconvex and CR-embeddable. Let $d\mu$ be a smooth positive density on $M$, and define the CR Hardy space

\begin{align*} H_b^2(M,d\mu):=\{u\in L^2(M,d\mu):\bar\partial_b u=0\text{ in }\mathcal D'(M)\}. \end{align*}

Let

\begin{align*} \Pi:L^2(M,d\mu)\to H_b^2(M,d\mu) \end{align*}

be the [orthogonal projection](/theorems/437). Choose a positive contact form $\theta\in \Omega^1(M)$ annihilating the CR hyperplane distribution, and define the positive contact cone

\begin{align*} \Sigma^+:=\{(p,r\theta_p)\in T^*M:p\in M,\ r>0\}. \end{align*}

In the Boutet de Monvel-Guillemin Hermite calculus normalization in which the Szego projector has Hermite order $0$, the operator $\Pi$ is a Hermite Fourier integral operator of order $0$ with operator wavefront relation equal to the identity relation on $\Sigma^+$. Equivalently, if $K_\Pi\in \mathcal D'(M\times M)$ denotes the Schwartz kernel of $\Pi$, then $K_\Pi$ is a Hermite distribution associated with 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*}

Its Hermite principal symbol is an idempotent in the corresponding Hermite symbol algebra, and $\Pi$ is microlocally smoothing at every covector in $T^*M\setminus \Sigma^+$.