On the Dimension of the Bergman Space of Some Hartogs Domains with Higher Dimensional Bases
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On the Dimension of the Bergman Space of Some Hartogs Domains with Higher Dimensional Bases Blake J. Boudreaux1 Received: 13 May 2020 / Accepted: 1 November 2020 © Mathematica Josephina, Inc. 2020
Abstract Let D be a Hartogs domain of the form D = Dϕ (G) = {(z, w) ∈ G × C N : w < e−ϕ(z) }, where ϕ is a plurisubharmonic function on G and G ⊆ C M is a pseudoconvex domain. We expand on the results of Jucha (J Geom Anal 22(1):23–37, 2012) and prove the infinite-dimensionality or triviality of the space of square integrable holomorphic functions on Dϕ (G) for various choices of ϕ and G. Keywords Hartogs domains · Balanced domains · Problem of Wiegerinck · Ohsawa-Takegoshi extension theorem · Pluripotential theory Mathematics Subject Classification Primary 32A36 · Secondary 32A25
1 Introduction Let L 2h () denote the Bergman space of a domain ⊂ C N ; in other words, the space of holomorphic functions f : → C satisfying
| f (ζ )|2 d V (ζ ) < ∞,
where d V denotes the Euclidean volume form. In the paper of Wiegerinck [13], it was shown that for ⊆ C, the space L 2h () is either trivial or infinite-dimensional. More precisely, Carleson [3] showed that for a domain ⊆ C, L 2h () is nontrivial if and only if the complement of has positive logarithmic capacity. The theorem of Suita—conjectured in [12] and later proved in [2] by Błocki—gives a quantitive version of one implication of this result. In [13], Wiegerinck also exhibited a Reinhardt domain in C2 with finite-dimensional but non-
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Blake J. Boudreaux [email protected] Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
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B. J. Boudreaux
trivial Bergman space. However, this domain was not pseudoconvex. This naturally raises the question of whether or not there exist pseudoconvex domains with finitedimensional but nontrivial Bergman space. This question motivates this article, but in general remains open. We will be mostly interested in Hartogs domains of the form D = Dϕ = Dϕ (G) = (z, w) ∈ G × C N : w < e−ϕ(z) ⊆ C M × C N , where G is a domain in C M and ϕ : G → [−∞, ∞) is upper-semicontinuous. Pseudoconvexity of Dϕ (G) is equivalent to the plurisubharmonicity of ϕ in addition to the pseudoconvexity of G. The primary case of study in [7] is when G is a subset of the complex plane. The main result therein is that Dϕ (C) is either trivial or infinite-dimensional; in fact, a necessary and sufficient condition for the nontriviality of L 2h (Dϕ (C)) is given in terms of the Riesz measure ϕ. Auxillary results consist of sufficient conditions for the infinite-dimensionality of L 2h (Dϕ (G)) in terms of G and ϕ. The chief goal of this paper is to generalize some of the results in [7] to when G ⊆ C M with M > 1. In the final section we will be working with balanced domains in C M , i.e., domains D ⊆ C M with the property that if z ∈ D and λ ∈ C with |λ| < 1, then λz ∈ D. We will also work with Hartogs domains with k-dimensional balanced fibers; loosely speaking these are domains which are fibered over
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