Abstract
A theorem of Wiegerinck says that the Bergman space over any domain in ℂ is either trivial or infinite dimensional. We generalize this theorem in the following form. Let E be a Hermitian, holomorphic vector bundle over ℙ1, the later equipped with a volume form and D an arbitrary domain in ℙ1. Then the space of holomorphic L2 sections of E over D is either equal to H0(M, E) or it has infinite dimension.
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Acknowledgement
We thank László Fehér for discussions on totally disconnected spaces. We also thank the referee for a very careful reading of the manuscript and suggestions to improve the presentation of the paper.
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Dedicated to László Lempert on the occasion of his 70th birthday
This research was supported by NKFI grant K112703 and the Rányi Institute of Mathematics.
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Szőke, R. On a Theorem of Wiegerinck. Anal Math 48, 581–587 (2022). https://doi.org/10.1007/s10476-022-0137-7
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DOI: https://doi.org/10.1007/s10476-022-0137-7