On the dependence of the reproducing kernel on the weight of integration

Pasternak-Winiarski, Zbigniew

doi:10.1016/0022-1236(90)90030-o

Cited by 45 publications

(34 citation statements)

References 2 publications

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“…which is well-defined under the admissibility condition in the sense of [14] (the map is a priori continuous on the subspace dom(D) ∩ A 2 (M, h, e −ψ )).…”

Section: The ∂-Operators On Weighted Bergman Spacesmentioning

confidence: 99%

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

Haslinger

Son

2020

Journal of Mathematical Analysis and Applications

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In this paper, we generalize several results about the ∂-complex on the Segal-Bargmann space of C n to weighted Bergman spaces on Hermitian manifolds. We also study in detail the ∂-complex on the unit ball with the complex hyperbolic metric and a non-Kähler metric. The former case turns out to have duality properties similar to the Segal-Bargmann space while the later exhibits a different behavior. We apply these results to solve the ∂-equation on the Bergman spaces in the unit ball of C n with the "exponential" and "standard" weights.

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“…which is well-defined under the admissibility condition in the sense of [14] (the map is a priori continuous on the subspace dom(D) ∩ A 2 (M, h, e −ψ )).…”

Section: The ∂-Operators On Weighted Bergman Spacesmentioning

confidence: 99%

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

Haslinger

Son

2020

Journal of Mathematical Analysis and Applications

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“…This condition is sometimes called admissibility of the measure µ (see, e.g., [PW90] and [Zey13]). In this paper we restrict our attention to smooth positive measures, that is, measures having smooth positive density with respect to Lebesgue measure in local coordinates.…”

Section: Preliminaries On Bergman Kernels and The Complex Laplacian Onmentioning

confidence: 99%

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Berger

Dall’Ara

Son

2019

Complex Variables and Elliptic Equations

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Given a smooth positive measure µ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on (0, 1)-forms. Thanks to an appropriate version of the Bochner-Kodaira-Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold.Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam.We also show in an appendix that the "twisted basic identities" of, e.g., [MV15] are standard basic identities with respect to conformally Kähler metrics.

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“…Theorem 2.2 contains some necessary and sufficient conditions for a weight to be an a-weight. Point (iv) of this theorem shows that Definition 2.1 is equivalent to the definition of the a-weight given in [5]. Other by the scalar product [3] and [6]).…”

Section: Introductionmentioning

confidence: 99%

“…In (Winiarski [5]) the concept of the a-weight was introduced. The main purpose of the presented study is to give a more detailed characterization of a-weights.…”

Section: Introductionmentioning

confidence: 99%

On weights which admit the reproducing kernel of Bergman type

Pasternak-Winiarski

1992

International Journal of Mathematics and Mathematical Sciences

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ABSTRACT. In this paper we consider (1) the weights of integration for which the reproducing kernel of the Bergman type can be defined, i.e., the admissible weights, and (2) the kernels defined by such weights. It is verified that the weighted Bergman kernel has the analogous properties as the classical one. We prove several sufficient conditions and necessary and sufficient conditions for a weight to be an admissible weight. We give also an example of a weight which is not of this class. As a positive example we consider the weight #(z) (Imz) 2 defined on the unit disk in C.

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On the dependence of the reproducing kernel on the weight of integration

Cited by 45 publications

References 2 publications

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

On weights which admit the reproducing kernel of Bergman type

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