The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where $$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ \textrm{in the sense of distributions}\}.$$
If $M = \partial\Omega$ is a domain boundary, then elements of $H^2$ are a.e. boundary values of holomorphic functions on $\Omega$---in particular, the boundary values of their Poisson integrals. So if we define for each $z \in \Omega$ the continuous functional $$\Psi_{z}\colon f \to Pf(z), \qquad f \in H^2(\partial\Omega)$$ with $k_{z}(\zeta)$ its $H^2(\partial\Omega)$ Hilbert space representative; and a kernel $S$ by $$S(z,\zeta) = \overline{k_{z}(\zeta)},\qquad z\in\Omega, \zeta \in \partial\Omega,$$ then it follows $S$ is a reproducing kernel for $H^2$ and the Szego projection on $L^2(\partial\Omega)$ is exactly $$f \mapsto \int_{\partial\Omega} f(\zeta)S(z,\zeta)\,d\sigma(\zeta).$$ Fine. But what if codim($M$) > 1? Is there any reason to think that the Szego projection is still an integral operator? I have tried to find a reference for this, but to no avail...
