The Bergman Kernel on Some Hartogs Domains
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- 作者：Zhenghui Huo
- 关键词：Bergman kernel ; Reinhardt domain ; Hartogs domain ; Boundary behavior
- 刊名：The Journal of Geometric Analysis
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：27
- 期：1
- 页码：271-299
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds;
- 出版者：Springer US
- ISSN：1559-002X
- 卷排序：27
文摘
We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on $$\begin{aligned} \left\{ (z_1,z_2,w)\in \mathbb {C}^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\right\} \end{aligned}$$and on $$\begin{aligned} \left\{ (z_1,z_2,w)\in \mathbb {C}^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;\mathrm{and} \;|w|<1\right\} . \end{aligned}$$We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal.KeywordsBergman kernelReinhardt domainHartogs domainBoundary behaviorMathematics Subject Classification32A0532A0732A2532A3632A40References1.Beberok, T.: An explicit computation of the Bergman kernel function. Complex Var. 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Elliptic Equ. 58, 783–793 (2013)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Mathematica Josephina, Inc. 2016Authors and AffiliationsZhenghui Huo1Email author1.Department of MathematicsUniversity of IllinoisUrbanaUSA About this article CrossMark Publisher Name Springer US Print ISSN 1050-6926 Online ISSN 1559-002X About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; display: inline-block; 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