L. D. Edholm, J. D. McNeal
A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.
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The purpose of this paper is to give an estimate of the $L^p$-norm of the Bergman projection on the Hartogs triangle.
We study weighted Bergman projections on the Hartogs triangle in $\mathbb{C}^2$. We show that projections corresponding to exponentially vanishing weights have degenerate $L^p$ mapping properties.
We prove a weighted Sobolev estimate of the Bergman projection on the Hartogs triangle, where the weight is some power of the distance to the singularity at the boundary. This method also applies to the $n$-dimensional generalization of the Hartogs triangle.
Zeros of the Bergman kernel of the Fock-Bargmann-Hartogs domain and the interlacing property
本文考虑Fock-Bargmann-Hartogs域$D_{n,m}$的Bergman核的零点.我们描述了如何存在的零的Bergman内核依赖于整数$m$和$n$的交错属性的帮助。
In this paper we investigate a two classes of domains in $\mathbb{C}^n$ generalizing the Hartogs triangle. We prove optimal estimates for the mapping properties of the Bergman projection on these domains.
The Bergman kernel of a certain Hartogs domain and the polylogarithm function
考虑一类与Fock-Bargmann空间相关的Hartogs域.我们给出了用多对数函数表示的域的Bergman核的一个显式公式。此外，在某些情况下，我们还解决了该区域的陆启铿问题。
A remark on the dimension of the Bergman space of some Hartogs domains
设D是Hartogs域，其形式为D={（z，w）\in CxC^N：|w| < e^{-u（z）}}，其中u是C上的下调和函数.证明了D上全纯平方可积〔的〕函数的Bergman空间是平凡的或无穷维的。
Relationships Between the Bergman Kernels of Hartogs Domains and Their Base
本文研究Hartogs域Bergman核与基域加权Bergman核的关系，提出级数表示和变换公式，并给出其他相关关系。
Polynômes de Hua, noyau de Bergman des domaines de Cartan-Hartogs et problème de Lu Qikeng
本文将Cartan-Hartogs域上的鲁其坑问题化为一个华氏多项式代数问题，并给出了Cartan-Hartogs域建立在维数不超过4的有界对称域上时的完全解.
We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted $L^p$ spaces when $p>\frac{4}{3}$, where the weight is a power of the distance to the singular boundary point. For $1<p\leq\frac{4}{3}$ we show that no such weighted estimates are possible.
The Bergman theory of domains $\{ |{z_{1} |^γ} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $γ$, including all positive integers. For such $γ$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_γ$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_γ(z,z)$.
In this paper we investigate a class of domains $Ω^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. we first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.
Density in weighted Bergman spaces and Bergman completeness of Hartogs domains
本文研究带多重次调和权的加权Bergman空间中边界邻域全纯函数的密度，并证明由平面域边界距离定义的Hartogs域Bergman完备当且仅当底域边界点均非孤立。
Let $ϕ$ be a globally defined real Kähler potential on a domain $Ω\subset \mathbb{C}^d$, and $g_{F}$ be a Kähler metric on the Hartogs domain $ M=\{(z,w)\in Ω\times\mathbb{C}^{d_0}: \|w\|^2<e^{-ϕ(z)}\}$ associated with the Kähler potential $Φ_{F}(z,w)=ϕ(z)+F(ϕ(z)+\ln\|w\|^2)$. Firstly, we obtain explicit formulas of the coefficients $\mathbf{a}_j\;(j=1,2)$ of the Bergman function expansion for the Hartogs domain $( M,g_F)$ in a momentum profile $φ$. Secondly, using explicit expressions of $\mathbf{a}_j\;(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $\mathbf{a}_j\…
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 $$\{(z_1,z_2,w)\in\mathbb C^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\}$$ and on $$\{(z_1,z_2,w)\in\mathbb C^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;{\rm and} \;|w|<1\}.$$ We use our formulas to determine th…
We generalize the Hartogs triangle to a class of bounded Hartogs domains, and we prove that the corresponding Bergman projections are bounded on $L^p$ if and only if $p$ is in the range $(\frac{2n}{n+1},\frac{2n}{n-1})$.
The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is defined by $$ D_{n, m}(μ):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w \Vert^2<e^{-μ\Vert z \Vert^2}\},$$ where $μ>0.$ The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n, m}(μ)$ with respect to the weight $(-ρ)^α$, where $ρ(z,w):=\|w\|^2-e^{-μ\|z\|^2}$ is a defining function for $D_{n, m}(μ)$ and $α>-1$. Then, for $p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P…
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
Two Kahler manifolds are called relatives if they admit a common Kahler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in [5, 4] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic.
We apply modern techniques of dyadic harmonic analysis to obtain sharp estimates for the Bergman projection in weighted Bergman spaces. Our main theorem focuses on the Bergman projection on the Hartogs triangle. The estimates of the operator norm are in terms of a Bekollé-Bonami type constant. As an application of the results obtained, we give, for example, an upper bound for the $L^p$ norm of the Bergman projection on the generalized Hartogs triangle $\mathbb H_{m/n}$ in $\mathbb C^2$.
$L^p$ boundedness of the Bergman Projection on generalizations of the Hartogs triangle in $\mathbb{C}^{n+1}$
本文研究一类推广的Hartogs三角形域$\Omega^{n+1}_\gamma$，得到了Bergman投影在该域上有界的精确$p$范围，将Edholm和McNeal在$n=1$时的结果推广到任意维数$n$。
Regularity of Canonical Operators and Nebenhülle: Hartogs Domains
我们将Bergman投影算子和标准解算子的正则性与完备Hartogs域的Nebenhülle联系起来。
On Bergman completeness of non-hyperconvex domains
本文研究了一类有界伪凸域中Bergman核的边界性态和Bergman完备性问题，其中伪凸域中也包含非超凸域。在我们证明Bergman完备性和Bergman核趋于边界时收敛到无穷大的类中，有界伪凸平衡域，正则域上具有平衡纤维的有界Hartogs域和某些有界Laurent-Hartogs域都是如此.
We prove the $L^p$ regularity of the weighted Bergman projection on the Hartogs triangle, where the weights are powers of the distance to the singularity at the boundary. The restricted range of $p$ is proved to be sharp. By using a two-weight inequality on the upper half plane with Muckenhoupt weights, we can consider a slightly wider class of weights.
An application of a Diederich-Ohsawa theorem in characterizing some Hartogs domains
应用Diederich和Ohsawa关于加权Bergman核的一个定理，我们用全纯自同构刻画了某些Hartogs域。
In this paper, we obtain some $L^{p}$ mapping properties of the Bergman-Toeplitz operator \[ f\longrightarrow T_{K^{-α}}\left(f\right):=\intop_ΩK_Ω\left(\cdot,w\right)K^{-α}\left(w,w\right)f\left(w\right)dV(w) \] on fat Hartogs triangles $Ω_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\left|z_{1}\right|^{k}<\left|z_{2}\right|<1\right\} $, where $α\in\mathbb{R}$ and $k\in \mathbb Z^+$.
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
In this paper, we investigate the weak-type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc we provide a proof that the weak-type behavior is of "$L\log L$" type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle we show that the operator is of weak-type $(4,4)$; settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak-type $(\frac{4}{3}, \frac{4}{3})$ …
The equivalent classical metrics on the Cartan-Hartogs Domains
本文研究了华氏域的特殊类型Cartan-Hartogs域上的完全不变度量。首先，我们在这些域上引入了一类新的完全不变度量，并证明了这些度量与Bergman度量是等价的。其次，在这些新度量下的里奇曲率由负常数从上到下有界。第三，我们估计了新度量的全纯截面曲率，证明了全纯截面曲率上下有界。最后，利用这些度量和Yau的Schwarz引理，证明了Bergman度量与Einstein-Kähler度量是等价的.这说明在Cartan-Hartogs域上Yau猜想成立。
We introduce a wider class of bounded Hartogs domains, which contains some generalizations of the classical Hartogs triangle. A sharp criteria for the $L^p-L^q$ boundedness of the Toeplitz operator with symbol $K^{-t}$ is obtained on these domains, where $K$ is the Bergman kernel on diagonal and $t\geq 0$. It generalizes the results by Chen and Beberok in the case $1<p<\infty$.
We compute explicitly the Bergman kernels of all two dimensional monomial polyhedra, a class of domains including the Hartogs triangle and some of its generalizations. The kernel is computed from the representation of such domains as quotient of the product of a disc and punctured disc.
Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains
本文证明了在覆盖有界同构域的Hartogs域上Bergman度量的刚性定理，展示了Bergman度量为Kähler-Einstein度量、同构、与单位球体双全纯等条件的等价性，回答了Yau的问题。
On the Bergman metric of Cartan-Hartogs domains
本文在 Cartan-Hartogs 域上引入 Bergman 对偶，证明其双全纯等价于单位球、Bergman 度量是 Kahler-Ricci 孤子及对偶有限射影诱导三者等价，并与其他典范度量比较，提出开问题。
A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $Σ_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional gen…
If a bounded domain can be covered by the polydisk through a rational proper holomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in a certain range depending on the ramified rational covering. This result can be applied to the symmetrized polydisk and to the Hartogs triangle with exponent $γ$.
Sobolev regularity of the $\bar{\partial}$-equation on the Hartogs triangle
利用$L^2$方法研究了$\mathbb{C}^2$中定义域$\{|{z_1}|<|{z_2}|<1\}$上的$\bar{\partial}$ -问题的正则性.在加权的$L^2$ -索伯列夫空间中，当权在点$(0,0)$奇异时，得到了正则解的估计.在远离奇异点$(0,0)$的加权索伯列夫空间中，带权的$\dbar$的正则解是正合正则的.特别地，Hartogs三角形的Bergman投影的奇异性包含在奇点处，并且它不传播。
$L^p$-Boundedness of Forelli-Rudin Type Operators on Rational Hartogs Triangles
研究有理Hartogs三角形上Forelli-Rudin型算子的$L^p$有界性，给出了充要条件，并恢复了Bergman投影和Berezin变换的精确$L^p$范围。
Expected duality and approximation properties are shown to fail on Bergman spaces of domains in $\mathbb{C}^n$, via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Reinhardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc.
Identifying holomorphic functions by moment conditions is a classical problem. This paper explores ideas introduced by Tumanov in a well known paper on the strip problem, and building on the author's previous work, to show that weighted Bergman projections can be computed via asymptotic formulas involving Szegő projections a fixed 1-dimensional family of curves. Various new operators are introduced; in addition, similar formulas are derived via dimension reduction, using the author's CR Hartogs theorem.
We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.
$L^p$-boundedness of Berezin transforms on generalized Hartogs triangles
研究有理广义Hartogs三角形上Berezin变换的$L^p$有界性，证明了其有界当且仅当$p>m+nl$。
The Kaehler-Einstein metric for some Hartogs domains over bounded symmetric domains
本文研究了Hartogs域$\widetilde Ω$的完备Kähler-Einstein度量，它是由不可约有界对称域$Ω$的膨胀得到的，并利用$Ω$的泛范数的幂$N^μ$. Kähler-Einstein度规的生成函数满足一个带边界条件的复蒙日-安培方程。域$\widetilde Ω$一般不是齐次的，但它有一个自同构子群，其轨道由$X\in\lbrack0,1[$参数化。这允许将蒙日-安培方程简化为具有极限条件的常微分方程。该方程可以显式求解为$μ$的特殊值$μ_{0}$，称为临界指数。我们工作的两个特殊的对称域的细节。临界指数似乎也是相关的其他不变度量的属性，如伯格曼度量;一个猜想，这是证明例外域。
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a Kähler metric $αg(μ;ν)$ $(α>0)$ on $D_{n,m}(μ)$, where $g(μ;ν)$ is the Kähler metric associated with the Kähler potential $Φ(z,w):=μν{\Vert z\Vert}^{2}-\ln(e^{-μ{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($ν>-1$) on $D_{n,m}(μ)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of th…
Restricted type estimates on the Bergman projection of some singular domains
本文对单项式多面体上的Bergman投影建立加权限制型估计，由此重得$L^p$有界性，并发现某些情形下弱型端点失效。
On the dimension of the $p$-Bergman spaces
本文研究了伪凸域中 $p$-Bergman 空间维度问题，通过构造 $L^p$ 版本的延伸定理，建立了确保空间无限维的几何和势理论准则，并给出了特定域的充分条件。
Real Variable Things in Bergman Theory
本文通过 Hardy 型不等式和 Sobolev-Morrey 不等式，建立了 Bergman 核的延拓、可积性、绝对连续性及最小值下界，并证明了平面域上 $κ(Ω)$ 与第一 Dirichlet 特征值成正比。
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …
On canonical metrics on Cartan-Hartogs domains
Cartan-Hartogs域定义为不可约有界对称域上的一类Hartogs型域。本文件有两个目的。首先，对于具有标准度量$g(μ)$的Cartan-Hartogs域$Ω^{B^{d_0}}(μ)$，我们得到了权为$\exp\{-αφ\}$的$(Ω^{B^{d_0}}(μ), g(μ))$上的平方可积全纯函数的加权Hilbert空间$\mathcal{H}_α$的Bergman核的显式表达式（其中$φ$是$α>0$的全球定义的Kähler势），此外，给出了$(Ω^{B^{d_0}}(μ), g(μ)).$的Rawnsley $\varepsilon$函数展开式的显式表达式。其次，利用Rawnsley $\varepsilon$函数展开式的显式表达式，我们证明了Rawnsley的$\varepsilon$ -函数展开的系数$a_2$对于Cartan-Hartogs域$(Ω^{B^{d_0}}(μ), g(μ))$在$Ω^{B^{d_0}}(μ)$上是常数当且仅当$(Ω^{B^{d_0}}(μ), g(μ))$与复双曲空间双全纯等距.从而对M.泽达
Bergman metrics induced by the ball
本文研究有界域 Bergman 度量经全纯等距浸入由单位球诱导的条件，证明二维严格拟凸域的刚性，并排除 Hartogs 域与蛋形域中的非球例子。
Rotational symmetries of domains and orthogonality relations
本文基于 Bergman 空间中单项式的 $L^2$ 正交关系，给出域具有 Reinhardt、圆形或 Hartogs 对称性及在环面线性群作用下不变的充分条件。
We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These functions are defined as limits, on the support of $T$, of sequences of plurisubharmonic functions decreasing on this support. In particular, we show that the poles of such functions are pluripolar sets. We also show that the maximum principle and the Hartogs's theorem remain valid in a weak sense. We study these functions by means of a class of measures, so-ca…
A transference principle for involution-invariant functional Hilbert spaces
本文提出对合不变再生核 Hilbert 空间的转移原理，通过 $\sigma$-不变映射构造新空间并证明其酉等价性，给出再生核公式，进而导出对称双圆盘等域上压缩元组的 von Neumann 不等式。
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