On Bergman completeness of pseudoconvex Reinhardt domains

Explicit formulas of the Bergman kernel for some Reinhardt domains

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains $\{|z_3|^λ < |z_1|^{2p} + |z_2|^2, \quad |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and $\{|z_4|^λ < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.

具有有限维伯格曼空间的无界雷纳尔德域

Unbounded Reinhardt domains with finite-dimensional Bergman spaces in $\C^n$

本文研究了具有有限维伯格曼空间的无界雷纳尔德域，证明其伯格曼度量具有正曲率且自同构群为线性映射。

The Serre problem with Reinhardt fibers

The Serre problem for a class of hyperbolic pseudoconvex Reinhardt domains in $\Bbb C^2$ as fibers is solved.

光滑Reinhardt域上的丘成桐猜想

Yau's conjecture on smooth Reinhardt domains

该研究证明了复欧氏空间中带光滑边界与完备Bergman-Einstein度量的无界Reinhardt域双全纯等价于复单位球，明确这类域不存在对应的完备度量。

关于角形Reinhardt域

On horn-shaped Reinhardt domains

本文在n≥2的复空间ℂⁿ中引入一类新的无界Reinhardt域（角形域），研究其Bergman空间维数、Bergman度量全纯截面曲率及全纯等价问题。

Finite Type Conditions on Reinhardt Domains

In this paper we prove that, if $p$ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in $\C^n$, then the variety type at $p$ is identical to the regular type.

Variation of Bergman kernels of pseudoconvex domains

We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.

Hankel Operators on the Bergman spaces of Reinhardt Domains and Foliations of Analytic Disks

Let $Ω\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $Ω$ and obtain a compactness result for Hankel operators on the Bergman space of $Ω$.

具有一维Bergman空间的复欧几里得空间中的拟凸域

Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces

该研究针对任意n≥2，构造了复欧几里得空间中的拟凸域，其Bergman空间为一维，从而否定了Wiegerinck猜想。

Proper holomorphic mappings in the special class of Reinhardt domains

A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in $\C^2$ with the logarithmic image equal to a strip or a half-plane is given.

Parametrix for the localization of the Bergman metric on strictly pseudoconvex domains

We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of Graham-Kerzman is proved in the hereby paper, as well.

Reinhardt domains with a cusp at the origin

Let V be a bounded pseudoconvex Reinhardt domain in C^2 with many strictly pseudoconvex points and logarithmic image W. It was known that the maximal ideal in $H^{\infty}(V)$ consisting of all functions vanishing at (p,q) in V is generated by the coordinate functions z-p, w-q (meaning that one can solve the Gleason problem for $H^{\infty}(V)$) if W is bounded. We show that one can solve Gleason's problem for $H^{\infty}(V)$ as well if there are positive numbers $a$, $b$ and a positive rational number k/l such that V looks like {(z,w) in C^2 : a |w|^l <= |z|^k = b |w|^l} for small (z,w).

Bergman核及其相关不变量在弱伪凸域中的渐进行为

Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains

本文研究了弱伪凸域中Bergman核及其相关不变量在逼近h-可延展边界点序列时的渐进行为。

Compactness of composition operators on the Bergman space of bounded pseudoconvex domains in $\mathbb{C}^n$

We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.

Estimates of the $L^p$ norms of the Bergman projection on strongly pseudoconvex domains

We give estimates of the $L^p$ norm of the Bergman projection on a strongly pseudoconvex domain in $\mathbb{C}^n$. We show that this norm is comparable to $\frac{p^2}{p - 1}$ for $1 <p< \infty$.

Asymptotic expansion of the Bergman kernel for weakly pseudoconvex tube domains in C^2

In this paper we give an asymptotic expansion of the Bergman kernel for certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic formula asserts that the singularity of the Bergman kernel at weakly pseudoconvex points is essentially expressed by using two variables; moreover certain real blowing-up is necessary to understand its singularity. The form of the asymptotic expansion with respect to each variable is similar to that in the strictly pseudoconvex case due to C. Fefferman. We also give an analogous result in the case of the Szego kernel.

关于有界强拟凸域中若干新解析函数空间上的Bergman型投影

On Bergman type projection in some new analytic function spaces in bounded strongly pseudoconvex domains

本文将单位圆盘上Bergman型投影的有界性结果推广到具有光滑边界的有界强拟凸域中的两类解析函数空间。

Asymptotic boundary behavior of the Bergman curvatures of a pseudoconvex domain

We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the bisectional curvatures of the Bergman metric of a smooth bounded pseudoconvex domain near the boundary with constant Levi rank.

具有强拟凸边界的Stein空间上的Bergman-Einstein度量

Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary

该研究探讨带紧光滑强拟凸边界的Stein空间，证明当正则部分的Bergman度量为Kähler-Einstein时边界是球面的，丰富了复几何中度量与边界性质的关联。

某些伪凸域类上的Bergman-Fridman不变量

The Bergman-Fridman invariant on some classes of pseudoconvex domains

本文研究了由Bergman度量定义的Fridman不变函数变体在多种伪凸域（包括Levi余秩1域、强伪凸域、光滑有界凸域及多面体域）上的边界行为。

CSC Bergman 度量在伪凸域上是 Einstein 的

CSC Bergman metrics are Einstein on pseudoconvex domains

本文证明：在具有光滑强伪凸边界点的伪凸域上，若 Bergman 度量具有常数量曲率，则它必为 Kähler–Einstein 度量。

Compactness of Hankel Operators with Conjugate Holomorphic Symbols on Complete Reinhardt Domains in $\mathbb{C}^2$

In this paper we characterize compact Hankel operators with conjugate holomorphic symbols on the Bergman space of bounded convex Reinhardt domains in $\mathbb{C}^2$. We also characterize compactness of Hankel operators with conjugate holomorphic symbols on smooth bounded pseudoconvex complete Reinhardt domains in $\mathbb{C}^2$.

The degree of biholomorphisms of quasi-Reinhardt domains fixing the origin

We give a description of biholomorphisms of quasi-Reinhardt domains fixing the origin via Bergman representative coordinates, which are shown to be polynomial mappings with a degree bound given by the so-called "resonance order".

Reinhardt域的$L^p$-Bergman空间上的投影

Projections onto $L^p$-Bergman spaces of Reinhardt Domains

本文通过构造积分核，在Reinhardt域上建立了$L^p$-Bergman空间的绝对有界投影，并利用其识别了这些空间的对偶为加权Bergman空间。

由其内射映射确定的雷因哈特域

Reinhardt domains determined by their endomorphisms

本文研究二维伪凸雷因哈特域，证明其内射映射同构时在双全息或反双全息意义上等价，并展示所有能收缩到适当嵌入空心复线的斯特恩流形由其内射映射半群确定。

伪凸域上极值Bergman度量与cscK Bergman度量的刚性

Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains

本文证明有界伪凸域上Bergman度量的极值性、常数量曲率与Kähler--Einstein条件等价，并推广到无界情形，核心工具是ACH边界处的唯一延拓定理。

Reinhardt 域的加权再生核

The weighted reproducing kernels of the Reinhardt domain

该研究建立包含原点的 Reinhardt 域上加权 Bergman 空间及其 Bergman 核的一般表示公式，并据此求出三类域中特殊权重对应核函数的具体表达式。

强伪凸域Bergman空间上具有调和符号的Hilbert-Schmidt Hankel算子

Hilbert-Schmidt Hankel operators with harmonic symbols on the Bergman space of strongly pseudoconvex domains in $\mathbb{C}^n$

研究C^n中光滑有界强伪凸域上Bergman空间的Hilbert-Schmidt Hankel算子，证明对于直到闭包的C^3类调和符号，H_φ是Hilbert-Schmidt当且仅当φ在域上全纯。

Reinhardt域中单项基核的稳定性

Stability of the Monomial Basis Kernel of Reinhardt domains

研究Reinhardt域中p-单项基核(p-MBK)的稳定性，证明其在参数p下连续，并探讨索引集S_p(Ω)的结构特性。

伪凸域的Bergman度量

On the Bergman metric of a pseudoconvex domain with a strongly pseudoconvex polyhedral boundary point

研究伪凸域的Bergman度量性质，证明其非Einstein度量

Singularities of the Bergman kernel for certain weakly pseudoconvex domains

Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n ; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and $m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point, $k \in \Natl$ the degenerate rank of the Levi form at $z^0$. An explicit formula for $K^B(z)$ modulo analytic functions is given in terms of the polar coordinates $(t_1, \ldots, t_k, r)$ around $z^0$. This formula provides detailed information about the singularities of $K^B(z)$, which improves the result of A. Bonami and N. Lohoué \cite{bol}. A similar result is e…

一类有限型伪凸域上 Bergman 投影的局部正则性

Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type

本文证明满足 Bell-Ligocka 条件 R 且具有良好各向异性伸缩的伪凸域上，Bergman 投影具备局部 $L^p$-Sobolev 与 Hölder 正则性，并说明 $h$-可延拓域适用该结论。

Hilbert-Schmidt Hankel operators over semi-Reinhardt domains

Let $Ω$ be an arbitrary bounded semi-Reinhardt domain in $\mathbb{C}^{m+n}$. We show that for $m \geq 2$, if a Hankel operator with an anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $L_a^2(Ω)$, then it must equal zero. This fact has previously been proved for Reinhardt domains.

$L^p$-regularity of the Bergman projection on quotient domains

We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.

Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a Kähler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.

On the dimension of the Bergman space for some unbounded domains

A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D'Angelo.

On Bergman completeness of non-hyperconvex domains

We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman completeness and the convergence of the Bergman kernel to infinity while tending to the boundary are all bounded pseudoconvex balanced domains, all bounded Hartogs domains with balanced fibers over regular domains and some bounded Laurent-Hartogs domains.

强伪凸域上近最小光滑性下 Bergman 与 Szegő 投影的加权 $L^p$ 估计

Weighted $L^p$ Estimates for the Bergman and Szegő Projections on Strongly Pseudoconvex Domains with Near Minimal Smoothness

本文证明近最小光滑强伪凸域上 Bergman 与 Cauchy-Szegő 投影的加权 $L^p$ 正则性，将 $B_p/A_p$ 条件推广至基于齐型空间拟度量的球上。

Nilpotent Toeplitz Operators on Reinhardt Domains

We construct explicit examples of non-trivial nilpotent Toeplitz operators on Bergman spaces of certain Reinhardt domains in $\mathbb{C}^2$.

关于csc Bergman度量的注记

A note on csc Bergman metric

本文证明：若C^n（n≥3）中伪凸域的Bergman度量具有常数数量曲率，则该域的每个强伪凸边界点都是球面的。

The Log term in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in $\mathbb C^2$

In this paper, we consider bounded strictly pseudoconvex domains $D\subset \mathbb C^2$ with smooth boundary $M=M^3:=\partial D$. If we consider the asymptotic expansion of the Bergman kernel on the diagonal $$ K_B\sim \frac{ϕ_B}{ρ^{n+1}}+ψ_B\logρ, $$ where $ρ>0$ is a Fefferman defining equation for $D$, then it is well known that the trace of the log term $bψ_B:=(ψ_B)|_M$ on $M$ does not determine the CR geometry of $M$ locally; e.g., the vanishing of $bψ_B$ on an open subset of $M$ does not imply that $M$ is locally spherical there. Nevertheless, the main result in this paper is that if $D\s…

Estimates of invariant metrics on pseudoconvex domains near boundaries with constant Levi ranks

Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given.

Capacities, Green function and Bergman functions

Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies $K_Ω(z)\gtrsim δ_Ω(z)^{-2}$ for any bounded pseudoconvex domain with $C^0-$boundary. An application to holomorphic motions is given.

有界伪凸域与管型域上的Bergman型投影的最新进展

Recent advances in Bergman type projections in bounded pseudoconvex and tubular domains

本文为综述，汇总多复变中有界伪凸域、对称锥上管型域等场景下不同维数函数空间间Bergman型投影的最新成果，推广经典结论，提出新研究问题并展望其应用场景。

单项式多面体的Bergman核

Bergman kernels of Monomial Polyhedra

本文针对单项式多面体这类有界伪凸Reinhardt域，利用Bell变换公式显式计算其Bergman核，并证明所得核为有理函数。

Note on the paper of Fu and Wong on strictly pseudoconvex domains with Kähler--Einstein Bergman metrics

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.

On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains

We compute the leading and sub-leading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.

拟凸模型域上 Bergman 核的渐近行为

The asymptotic behavior of the Bergman kernel on pseudoconvex model domains

本文利用域定义函数的 Newton 多面体几何信息，刻画若干拟凸模型域边界处 Bergman 核的渐近行为，并将结果推广至有限型及部分无限型情形。

具有极小光滑性的强拟凸域上Bergman投影的交换子

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

本文研究了$C^2$类极小光滑性强拟凸域上Bergman投影交换子的有界性与紧性，利用平均振荡条件和二进Berezin变换等方法给出了等价刻画。

A note on smoothing properties of the Bergman projection

Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on a class of pseudoconvex domains. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.

Boundary Behavior of the Bergman Metric

More precise estimates for the Bergman metric on strongly pseudoconvex domains are given, based on the use of the squeezing function.

二维拟凸域上不变度量的Kähler性

K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

本文针对二维有界有限型拟凸域，利用三阶拟有限几何的Kähler-Kobayashi度量与Kähler-Carathéodory度量证明单值化定理，并给出Reinhardt域为单位球的Bergman度量标量倍数判据及全纯截面曲率与Lu常数的刚性定理。

Bergman-Toeplitz operators on weakly pseudoconvex domains

We prove that for certain classes of pseudoconvex domains of finite type, the Bergman-Toeplitz operator $T_ψ$ with symbol $ψ=K^{-α}$ maps from $L^{p}$ to $L^{q}$ continuously with $1< p\le q<\infty$ if and only if $α\ge\frac{1}{p}-\frac{1}{q}$, where $K$ is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains by Čučković and McNeal, and Abeta, Raissy and Saracco.

Growth of Sibony metric and Bergman kernel for domains with low regularity

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $\mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $\mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the …