Bergman space

Summary

Let $\Omega$ be a domain in $\C$ and let $0 < p < \infty$. The Bergman space $A^p(\Omega)$ is the space of holomorphic functions $f$ on $\Omega$ such that $\abs{f}^p$ is integrable with respect to area. Point evaluations are continuous on it, so for $p \ge 1$ it is a closed subspace of $L^p(\Omega)$ and a Banach space. The space $A^2(\Omega)$ is a Hilbert space with a reproducing kernel, the Bergman kernel. Bergman spaces differ from Hardy spaces by measuring size over the whole domain instead of on circles near the boundary.