On a theorem of Wiegerinck

Abstract

A theorem of Wiegerinck says that the Bergman space over any domain in $\mathbb{C}$ is either trivial or infinite dimensional. We generalize this theorem in the following form. Let E be a hermitian, holomorphic vector bundle over ${\mathbb{P}}^1$, the later equipped with a volume form and $D$ an arbitrary domain in ${\mathbb{P}}^1$. Then the space of holomorphic L2 sections of $E$ over $D$ is either equal to $H^0(M,E)$ or it has infinite dimension.

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Cited by in corpus (1)

• On the dimension of Bergman spaces on ${\mathbb{P}}^1$