Bergman projection

Summary

The Bergman projection of a domain $\Omega \subseteq \C^n$ is the orthogonal projection of $L^2(\Omega)$ onto its closed subspace of holomorphic functions, the Bergman space. It is an integral operator whose kernel is the Bergman kernel. Its regularity, for example whether it maps smooth functions on the closure to smooth functions, governs the boundary behaviour of biholomorphic maps. It is closely tied to the $\bar\partial$-Neumann problem, through Kohn's formula $P = I - \bar\partial^* N \bar\partial$.