Siqi Fu, Alexander V. Isaev, Steven G. Krantz
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.
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We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.
We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.
In this paper, we define what is called a quasi-Reinhardt domain and study biholomorphisms between such domains. We show that all biholomorphisms between two bounded quasi-Reinhardt domains fixing the origin are polynomial mappings, and we give a uniform upper bound for the degree of such polynomial mappings. In particular, we generalize the classical Cartan's linearity theorem for circular domains to quasi-Reinhardt domains.
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。
We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.
We describe all possibilities of existence of non-elementary proper holomorphic maps between non-hyperbolic Reinhardt domains in $\mathbb C^2$ and the corresponding pairs of domains.
We characterize pairs of bounded Reinhardt domains in $\CC^2$ between which there exists a proper holomorphic map and find all proper maps that are not elementary algebraic.
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.
A characterization of non-hyperbolic pseudoconvex Reinhardt domains in $\mathbb C^2$ for which the answer to the Serre problem is positive is given.
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".
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.
We show that every bounded hyperconvex Reinhardt domain can be approximated by special polynomial polyhedra defined by homogeneous polynomial mappings. This is achieved by means of approximation of the pluricomplex Green function of the domain with pole at the origin.
In the paper we find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.
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 $\mathbb{C}^2$.
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。
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.
We give a complete description of bounded Reinhardt domains of finite boundary smoothness that have non-compact automorphism group. As part of this program, we show that the classification of domains with non-compact automorphism group and having only finite boundary smoothness is considerably more complicated than the classification of such domains that have infinitely smooth boundary.
On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert-Schmidt. We also present examples of unbounded non-pseudoconvex domains that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.
We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $L^p$ spaces for all $p\not=2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.
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。
We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image.
Let E denote a bundle with fiber D and with basis B. Both D and B are assumed to be Stein. For D a Reinhardt bounded domain of dimension d=2 or 3, we give a necessary and sufficient condition on D for the existence of a non-Stein such E (Theorem 1); for d=2, we give necessary and sufficient criteria for E to be Stein (Theorem 2). For D a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for E to be Stein (Theorem 3).
The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $\mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these res…
We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $Ω$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman kernel on $Ω$ satisfies appropriate estimates, then the Bergman projection on the successor is $L^p$ bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on $L^p$ for $1<p<\infty$. The successor domains need not have smooth boundary nor be strictly pseudoconvex.
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 $\{|z_4|^λ < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.
We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log d_D)+C$. The lower estimate by $-1/2\log d_D+C$ is obtained under additional assumptions of $\mathcal C^1$-smoothness of a domain and a non-tangential convergence.
The Serre problem for a class of hyperbolic pseudoconvex Reinhardt domains in $\Bbb C^2$ as fibers is solved.
We give an explicit description of hyperbolic Reinhardt domains D in C^2 such that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii) D intersects at least one of the coordinate complex lines $\{z_1=0\}$, $\{z_2=0\}$, and (iii) D has noncompact automorphism group. We also give an example that explains why such a setting is natural for the case of hyperbolic domains and an example that indicates that the situation in C^n for n greater than or equal to 3 is essentially more complicated than that in C^2.
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。
Zero products of Toeplitz operators on Reinhardt domains
研究有界Reinhardt域Bergman空间上Toeplitz算子的零乘积问题，针对有界拟齐次函数有限和符号，证明乘积为零时必有一个算子为零。
We construct explicit examples of non-trivial nilpotent Toeplitz operators on Bergman spaces of certain Reinhardt domains in $\mathbb{C}^2$.
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).
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。
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。
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 $Ω$.
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.
We show the following symmetry property of a bounded Reinhardt domain $Ω$ in $\mathbb{C}^{n+1}$: let $M=\partialΩ$ be the smooth boundary of $Ω$ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In Appendix we state the result from an hamiltonian point of view.
Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume that the symbols f and g are continuous on the closure of D and harmonic on the disks in the boundary of D. We show that if the product of Hankel operators H^*_f H_g is compact on the Bergman space of D, then on any disk in the boundary of D, either f or g is holomorphic.
Compactness of Hankel and Toeplitz operators on convex Reinhardt domains in $\mathbb{C}^2$
本文针对 $\mathbb{C}^2$ 中凸 Reinhardt 域的 Bergman 空间，刻画了闭包连续符号下 Toeplitz 算子及 Hankel 算子 Hermitian 平方的紧性判据。
On Stable Univalence and Coefficient Estimates for a Class of Pluriharmonic Mappings in Convex Reinhardt Domains
本文研究了凸Reinhardt域中双曲映射的稳定性与系数估计，通过多维Noshiro-Warschawski定理的推广，建立了双曲映射与全纯函数的一一对应关系，并给出了尖锐的系数估计。
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。
Extensions of Daubechies' theorem: Reinhardt domains, Hagedorn wavepackets and mixed-state localization operators
本文扩展了Daubechies定理，研究Reinhardt领域、Hagedorn波包和混合态定位算子在量子调和分析中的应用。
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.
We give an example of a bounded, pseudoconvex, circular domain in ${\mathbb C}^3$ with smooth, real-analytic boundary and non-compact automorphism group, which is not biholomorphically equivalent to any Reinhardt domain.
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。
For a Reinhardt domain $Ω$ with the smooth boundary in $\mathbb{C}^{m+1}$ and a positive smooth measure $μ$ on the boundary of $Ω$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $γ_{N}$ which is induced by $L^{2}(\partialΩ,dμ)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\in\partialΩ$. First of all we apply stationary phase method to the Boutet de Monvel-Sjöstrand theorem to get the asymptotic for the partial szegö kernel, $S_{N}(z,z)$, and then we compute the s…
We present an effective formula for the Sibony function for all Reinhardt domains.
It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin.
We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.
It is shown that the Laurent series of a holomorphic function smooth up to the boundary on a Reinhardt domain in $\mathbb{C}^n$ converges unconditionally to the function in the Fréchet topology of the space of functions smooth up to the boundary.
We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends nontrivially on the parameters m, M and D. Our main result is that E_m(D,M) is Stein if and only if m log(r(M)) <= 2 π^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- …
We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in $\mathbb{C}^{2}$ are $c$-regular.
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.
We present a result on existence of some kind of peak functions for $\C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.
On integral representation of radial operators
本文利用积分表示刻画 Reinhardt 域上加权 Bergman 空间及单位圆盘 Dirichlet 和 Hardy 空间中的径向算子，研究其正规性、紧性、谱与数值域，并构造任意 Reinhardt 域上解析函数的 von Neumann 代数例子。
Local Banach Space Theoretic Approach to Bohr's Theorem for Vector Valued Holomorphic and Pluriharmonic Functions
本文通过局部巴拿赫空间理论研究向量值全纯和多值调和函数的波尔定理，证明了波尔半径的正性及渐进行为，并建立了相关系数型施瓦茨-皮克引理。
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。
Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
本文针对Hayashida-Kamimoto构造的具有常正全纯截面曲率2的Bergman度量的Reinhardt区域，证明三明治原理并构造大量不等价光滑非Reinhardt区域，还分类双全纯映射并确定角扰动族的自同构群。
In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and metrics on bounded domains in terms of squeezing functions. To study the boundary behavior of squeezing functions, we introduce the notions of (intrinsic) ball pinching radius, and give boundary estimate of squeezing functions in terms of these datum. Finally, we use these results to study geometric and analytic properties of some interesting domains, includin…
Let $Ω$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $ϕ\in C(\barΩ)$. We show that the Hankel operator $H_ϕ$ is compact if and only if $ϕ$ is holomorphic along every non-trivial analytic disc in the boundary of $Ω$.
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