---
tag: 6J1X63Z7QZ
kind: concept
name: Bergman projection
msc: [32A36, 32A25, 30H20]
specializes: [0B8KZRT28M]
sources:
  - url: https://en.wikipedia.org/wiki/Bergman_kernel
---

## Summary

The Bergman projection of a domain $\Omega \subseteq \C^n$ is the [[orthogonal projection|0B8KZRT28M]] of $L^2(\Omega)$ onto its closed subspace of holomorphic functions, the [[Bergman space|2D619WTPYC]].
It is an integral operator whose kernel is the [[Bergman kernel|8RWDHW0CXC]].
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 $P = I - \bar\partial^* N \bar\partial$.

## Explanations

### L1 human

Among all square-integrable functions on a domain, the holomorphic ones form a closed subspace.
The Bergman projection sends each function to the holomorphic function closest to it in the $L^2$ sense.
It does so by integrating the function against the Bergman kernel.
For example, on the unit disc it sends $\bar z$ to $0$ and $z^k$ to itself.