On the Dimension of the Bergman Space for Some Unbounded Domains

Gallagher, A.-K.; Harz, Tobias; Herbort, Gregor

doi:10.1007/s12220-016-9725-8

Cited by 19 publications

(28 citation statements)

References 24 publications

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“…In the case when the boundary ∂A of the domain A is not foliated by translations of the graph of an entire function, we know from our Main Theorem that its core c(A) is empty. It follows then from Theorem 1 in [11] that in this case dim A 2 (A) = ∞, where A 2 (A) is the Bergman space, i.e. the space of holomorphic functions on A which belong to L 2 (A).…”

Section: Remarkmentioning

confidence: 98%

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On the existence of Kobayashi and Bergman metrics for Model domains

Shcherbina

2020

Math. Ann.

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We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( F ) of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

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Section: Remarkmentioning

confidence: 98%

“…By the choice of the points Q 0 , Q m and points z 0 , z m , and in view of the Condition (1), we can also see that z 0 =z = z m . Then, we can finally obtain from (11) and (12)…”

Section: Lemma 11mentioning

confidence: 99%

On the existence of Kobayashi and Bergman metrics for Model domains

Shcherbina

2020

Math. Ann.

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“…Remark 10. Recall that the main result in [6] assures the infinite dimensionality of L 2 h (D) in the case the core of the domain is different from D. Note that domains from the class D S do not satisfy the equality c ′ (D) = D. Therefore, the infinite dimensionality of L 2 h (D) for these domains cannot be concluded from [6]. We shall need the following properties of ν that will follow directly from that of the standard Lelong number:…”

Section: Bergman Spaces In Two-dimensional Balanced Domainsmentioning

confidence: 99%

$$L_h^2$$ L h 2 -Functions in Unbounded Balanced Domains

Pflug

Zwonek

2017

J Geom Anal

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We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of L 2 h -domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in C 2 . This allows easily to decide which pseudoconvex balanced domain in C 2 has a positive Bergman kernel and which admits the Bergman metric.

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“…In particular, they proved that a Stein manifold X possesses a Bergman metric, provided that X carries a bounded continuous strictly plurisubharmonic function. Recently, a new characterization for the existence of the Bergman metric on unbounded domains has been given by Gallagher et al [13]. It was proved there that a pseudoconvex domain with empty core (see Sect.…”

mentioning

confidence: 99%

“…The construction of is motivated by [16], where a Kobayashi hyperbolic model domain with a nonempty core has been constructed. Using a characterization of the Bergman space in terms of the core (see [13,Remark 7(b)]), we show the existence of the Bergman metric on . For the completeness of this metric, we apply a criterion given by Chen [7, Theorem 1.1], which uses the asymptotic behavior at infinity of the volumes of sublevel sets of the pluricomplex Green function.…”

mentioning

confidence: 99%