The Bergman kernel on the intersection of two balls in C^2

单位球上加权调和 Bloch 空间中调和 Bergman Besov 空间的闭包

Closures of Harmonic Bergman Besov Spaces in the Weighted Harmonic Bloch Spaces on the Unit Ball

本文通过可逆径向微分算子，利用自然水平集刻画了Rn单位球上加权调和Bloch空间中调和Bergman Besov空间的闭包，并首先证明了该空间包含于加权调和小Bloch空间。

Theory of Bergman Spaces in the Unit Ball of $C^n$

There has been a great deal of work done in recent years on weighted Bergman spaces $\apa$ on the unit ball $\bn$ of $\cn$, where $0<p<\infty$ and $α>-1$. We extend this study in a very natural way to the case where $α$ is {\em any} real number and $0<p\le\infty$. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^2$, and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergm…

实双曲球上的$\mathcal H$-调和Bergman-Besov空间

$\mathcal H$-Harmonic Bergman-Besov Spaces on the Real Hyperbolic Ball

通过多种微分算子刻画，将实双曲球上$\mathcal H$-调和函数的Bergman空间族从$\alpha>-1$延拓到所有$\alpha\in\mathbb R$，并推广了投影、对偶、原子分解和包含关系等性质。

The Bergman kernel for intersection of two complex ellipsoids

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 intersection of two complex ellipsoids $\{z \in \mathbb{C}^3 \colon |z_1|^p + |z_2|^q < 1, \quad |z_1|^p + |z_3|^r < 1\}$. We consider cases $p=6, q= r= 2$ and $p=q=r=2$. We also investigate the Lu Qi-Keng problem for $p=q=r=2$.

Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent

Let $ψ$ be a plurisubharmonic function on the closed unit ball and $K_{tψ}(z)$ the Bergman kernel on the unit ball with respect to the weight $tψ$. We show that the boundary behavior of $K_{tψ}(z)$ is determined by certain directional Lelong number of $ψ$ for all $t$ smaller than the John-Nirenberg exponent of $ψ$ associated to certain family of nonisotropic balls, which is always positive.

Semi-norms of the Bergman projection

It is known that the Bergman projection operator maps the space of essentially bounded functions in the unit ball in the d-dimensional complex vector space onto the Bloch space of the unit ball. This paper deals with the various semi-norms of the Bergman projection. We improve some recent results.

具有常曲率Bergman度量的领域及Bergman可忽略子集

Domains with Bergman metrics of constant curvature and Bergman-negligible subsets

研究了具有常负曲率Bergman度量的领域，证明其与单位球体减去测度为零的闭集biholomorphic，并讨论了相关函数扩展性质及应用。

Strongly exposed points in the ball of the Bergman space

We investigate which boundary points in the closed unit ball of the Bergman space of integrable holomorphic functions on the unit disc are strongly exposed. This requires study of the Bergman projection and its kernel, the annihilator of Bergman space. We show that all polynomials in the boundary of the unit ball are strongly exposed.

Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball

We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.

The polyharmonic Bergman space for the union of rotated unit balls

In the paper we consider the polyharmonic Bergman space for the union of the rotated unit Euclidean balls. Using so called zonal polyharmonics we derive the formulas for the kernel of this space. Moreover, we study the weighted polyharmonic Bergman space. By the same argument we get the Bergman kernel for this space.

二维Rudin球商的Bergman投影的L^p正则性问题

The $L^p$- regularity problem for the Bergman projection of two-dimensional Rudin ball quotients

本文研究了二维Rudin球商的Bergman投影的L^p正则性问题，得出了相关结论。

近球域在伯格曼球中的等周不等式

Isoperimetric inequality for nearly spherical domains in the Bergman ball

本文在伯格曼球中研究近球域的等周不等式，证明了相关定理并首次揭示了该领域的等周现象。

Composition operators on Bergman-Orlicz spaces on the ball

We give embedding theorems for weighted Bergman-Orlicz spaces on the ball and then apply our results to the study of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman-Orlicz spaces, different from $H^{\infinity}$, on which every composition operator is bounded.

$Φ$-Carleson measures and multipliers between Bergman-Orlicz spaces of the unit ball of $\mathbb {C}^n$

We define the notion of $Φ$-Carleson measures where $Φ$ is either a concave growth function or a convex growth function and provide an equivalent definition. We then characterize $Φ$-Carleson measures for Bergman-Orlicz spaces, and apply them to characterize multipliers between Bergman-Orlicz spaces.

Carleson Measures and Toeplitz operators for weighted Bergman spaces on the unit ball

Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.

Interpolation and Sampling Hypersurfaces for weighted Bergman spaces on the unit ball

We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional…

Harmonic Bergman Spaces on the Real Hyperbolic Ball: Atomic Decomposition, Interpolation and Inclusion Relations

For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.

Norm of the Bergman projection

This paper deals with the the norm of the weighted Bergman projection operator $P_α:L^\infty\to \mathcal{B}$ where $α>-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space $\mathbf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

In this paper, we study the Bergman metric of a finite ball quotient $\mathbb{B}^n/Γ$, where $Γ\subseteq \mathrm{Aut}(\mathbb{B}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler--Einstein if and only if $Γ$ is trivial, i.e., when the ball quotient $\mathbb{B}^n/Γ$ is the unit ball $\mathbb{B}^n$ itself. As a consequence, we establish a characterization of the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman-Einstein metric.

Algebraicity of the Bergman Kernel

Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a counterexample in dimension three constructed in this paper. As a corollary of our main theorem, we prove, e.g., that a smoothly bounded strictly pseudoconvex domain G in $\mathbb{C}^2$ has rational Bergman kernel if and only if there is a rational biholomorphism from G to the 2-dimensional unit ball.

Maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

Further properties of the Bergman spaces of slice regular functions

In this paper we continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paper we mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitable weights are taken into account. Fina…

Difference of composition operators between weighted Bergman spaces on the unit ball

We obtain some estimates for norm and essential norm of the difference of two composition operators between weighted Bergman spaces $A^p_α$ and $A^q_β$ on the unit ball. In particular, we completely characterize the boundedness and compactness of $C_φ-C_ψ: A^p_α\to A^q_β$ for full range $0<p,q<\infty$, $-1<α,β<\infty$.

$\\(\mathbb C^n\\)$ 单位球上的修正 Bergman 空间及其应用

Modified Bergman spaces on the unit ball of $\mathbb C^n$ and applications

本文在复单位球上引入推广经典 Bergman 空间的新全纯函数空间，并研究相关算子性质及 Bergman 型投影和 Berezin 变换等积分表示。

A Direct Connection Between the Bergman and Szegő Kernels

We use Stokes's theorem to establish an explicit and concrete connection between the Bergman and Szegő projections on the disc, the ball, and on strongly pseudoconvex domains.

Tent spaces and Littlewood-Paley $g$-functions associated with Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, a family of holomorphic spaces of tent type in the unit ball of $\mathbb{C}^n$ is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type $g$-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient, and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Hardy…

单位球上加权Bergman空间上的加权复合算子差

Difference of weighted composition operators on weighted Bergman spaces over the unit Ball

本文在单位球上刻画了加权Bergman空间到Lebesgue空间的加权复合算子差的有界性与紧致性，推广了单位圆盘结果，并给出了q-Carleson测度的新刻画。

Bergman projections induced by doubling weights on the unit ball of Cn

The boundedness of $P_ω:L^\infty(\mathbb{B})\to \mathcal{B}(\mathbb{B})$ and $P_ω(P_ω^+):L^p(\mathbb{B},\upsilon dV)\to L^p(\mathbb{B},\upsilon dV)$ on the unit ball of $\mathbb{C}^n$ with $p>1$ and $ω,\upsilon\in \mathcal{D}$ are investigated in this paper.

Weighted Bergman kernels and virtual Bergman kernels

We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.

Toeplitz operators on Bergman spaces induced by doubling weights on the unit ball of $\mathbb{C}^n$

The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.

Interpolating sequences for weighted Bergman spaces of the ball

Let $B_α^{p}$ be the space of $f$ holomorphic in the unit ball of $\Bbb C^n$ such that $(1-|z|^2)^αf(z) \in L^p$, where $0<p\leq\infty$, $α\geq -1/p$ (weighted Bergman space). In this paper we study the interpolating sequences for various $B_α^{p}$. The limiting cases $α=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and $A^{-α}$, the holomorphic functions with polynomial growth of order $α$, which have generated particular interest. In §1 we first collect some definitions and well-known facts about weighted Bergman spaces and then introduce the natural interpolation problem, alo…

New characterizations of Bergman spaces

We obtain several new characterizations for the standard weighted Bergman spaces $A^p_α$ on the unit ball of $\cn$ in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Quantum matrix ball: the Bergman kernel

In our preprint q-alg/9703005 q-analogues of bounded symmetric domains were defined to be homogeneous spaces of the associated quantum groups. The investigation of a simplest among those domains, the quantum matrix ball, was started in math.QA/9803110. This work presents a construction of q-analogues for Hardy-Bergman spaces of 'functions in those balls', together with an explicit form of the Bergman kernel. Besides that, two auxiliary results are also established: a boundedness of matrix balls is proved, and de Rham complexes of differential forms with finite coefficients in those balls are c…

Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, we show that every (weighted) Bergman space $\mathcal{A}^p_α (\mathbb{B}_n)$ in the complex ball admits an atomic decomposition of real-variable type for any $0 < p \le 1$ and $α> -1.$ More precisely, for each $f \in \mathcal{A}^p_α (\mathbb{B}_n)$ there exist a sequence of real-variable $(p, \8)_α$-atoms $a_k$ and a scalar sequence $\{λ_k \}$ with $\sum_k | λ_k |^p < \8$ such that $f = \sum_k λ_k P_α (a_k),$ where $P_α$ is the Bergman projection from $L^2_α (\mathbb{B}_n)$ onto $\mathcal{A}^2_α (\mathbb{B}_n).$ The proof is constructive, and our construction is based on some sh…

Spectra of some invertible weighted composition operators on Hardy and weighted Bergman spaces in the unit ball

In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_α^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_α^2(\mathbb{D})$.

On the boundary behavior of the holomorphic sectional curvature of the Bergman metric

We obtain a conceptually new differential geometric proof of P.F. Klembeck's result that the holomorphic sectional curvature of a strictly pseudoconvex domain approaches (in the boundary limit) the constant sectional curvature of the Bergman metric of the unit ball.

Weighted BMO and Hankel operators between Bergman spaces

We introduce a family of weighted BMO and VMO spaces for the unit ball and use them to characterize bounded and compact Hankel operators between different Bergman spaces. In particular, we resolve two problems left open by S. Janson in 1988 and R. Wallsten in 1990.

由球诱导的 Bergman 度量

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。

The Bergman kernel: explicit formulas, deflation, Lu Qi-Keng problem and Jacobi Polynomials

In this paper we investigate the Bergman kernel function for intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb{C}^{n+2} : |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \quad |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$

双曲球上的 $\mathcal H$-调和 Bergman 投影

$\mathcal H$-Harmonic Bergman Projection on the Hyperbolic Ball

本文刻画了双曲球上 Bergman 投影在 Lebesgue 空间与 $\mathcal H$-调和加权 Bergman 空间之间有界的精确条件，验证了 Stoll 猜想，并给出再生核估计及到 Bloch 空间的投影结果。

实双曲球上的调和 Bloch 空间

Harmonic Bloch Space on the Real Hyperbolic Ball

本文研究实双曲球上调和函数的 Bloch 空间，证明 Bergman 投影的满射性，确定双曲调和 Bergman 空间的对偶与预对偶，并给出 Bloch 函数的原子分解。

Quantum matrix ball: the weighted Bergman kernels

We proceed with studying the q-analogues of Cartan domains introduced in q-alg/9703005 and turn to the case of a ball in the space of complex matrices. An explicit expression for a positive $U_q\frak{su}_{nm}$-invariant integral (see math.QA/9803110) is used to define q-analogues for weighted Bergman spaces. The orthogonal projections onto these spaces are integral operators; this work presents an explicit formula for their kernels.

M-harmonic reproducing kernels on the ball

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szegö and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on the unit ball of the complex $n$-space. This yields, among others, an explicit formula for the $M$-harmonic Szegö kernel in terms of multivariable as well as single-variable hypergeometric functions, and also shows that most likely there is no explicit (``closed'') formula for the corresponding weighted Bergman kernels.

单位球上带指数权重的 Bergman 核的逐点估计

Pointwise estimates of the Bergman kernel with an exponential weight on the unit ball

本文在单位球上研究带指数权重 $e^{-1/(1-|z|^2)}$ 的加权 Bergman 空间，证明了 Bergman 核的逐点估计，用黎曼距离球体积和指数衰减项控制核的大小，结果受 Christ 工作启发。

Weighted norm inequalities in the variableLlebesgue spaces for the Bergman projector on the unit ball of $\mathbb{c}^n$

In this work, we extend the theory of Békollè-Bonami $B_p$ weights. Here we replace the constant $p$ by a non-negative measurable function $p(\cdot),$ which is log-Hölder continuous function with lower bound $1$. We show that the Bergman projector on the unit ball of $\mathbb C^n$ is continuous on the weighted variable Lebesgue spaces $L^{p(\cdot)}(w)$ if and only if $w$ belongs to the generalised Békollè-Bonami class $B_{p(\cdot)}$. To achieve this, we define a maximal function and show that it is bounded on $L^{p(\cdot)}(w)$ if $w\in B_{p(\cdot)}$. We next state and prove a weighted extrapol…

单位球 Bergman 空间上一类具有奇异符号的 Toeplitz 算子的 Szegö 极限定理

A Szegö limit theorem for a class of Toeplitz operators on the Bergman space of the unit ball with singular symbols

本文针对单位球加权 Bergman 空间上符号支撑于迷向或余迷向子流形且关于 Lebesgue 测度奇异的一类 Toeplitz 算子，建立了 Szegö 极限定理，刻画了参数趋于无穷时的渐近行为。

Weighted Bergman spaces induced by doubling weights in the unit ball of $\mathbb{C}^n$

This paper is devoted to the study of the weighted Bergman space $A_ω^p $ in the unit ball $\mathbb{B}$ of $\mathbb{C}^n$ with doubling weight $ω$ satisfying $$\int_r^1ω(t)dt <C \int_{\frac{1+r}{2}}^1ω(t)dt ,\,\, 0\leq r<1.$$ The $q-$Carleson measures for $A_ω^p$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_ω^p$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_g:A_ω^p\to A_ω^q$ are also investigated in this paper with $0<p\l…

The Bergman Kernel on Forms: General Theory

The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(Ω)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions.

High-power asymptotics of some weighted harmonic Bergman kernels

For~weights $ρ$ which are either radial on the unit ball or depend only on the vertical coordinate on the upper half-space, we describe the asymptotic behaviour of the corresponding weighted harmonic Bergman kernels with respect to $ρ^α$ as $α\to+\infty$. This can be compared to the analogous situation for the holomorphic case, which is of importance in the Berezin quantization as well as in complex geometry.

通过伯格曼球上的等周稳定性研究全纯函数的收缩性质

Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball

本文研究全纯函数在近常数情况下的局部收缩性质，通过伯格曼球上的等周稳定性结果，将加权伯格曼空间的缺陷不等式与全纯函数的收缩性质联系起来。

具有有限型边界的Stein空间上Bergman核的局部代数性与局部化

Local algebraicity and localization of the Bergman kernel on Stein spaces with finite type boundaries

本文研究二维Stein空间中Bergman核的局部代数性，借助伪凸有限型区域上的新局部化结果建立边界类型与局部代数次数的估计，并刻画二维球商。

Little Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball

In this paper, we study the boundedness and the compactness of the little Hankel operators $h_b$ with operator-valued symbols $b$ between different weighted vector-valued Bergman spaces on the open unit ball $\mathbb{B}_n$ in $\mathbb{C}^n.$ More precisely, given two complex Banach spaces $X,Y,$ and $0 < p,q \leq 1,$ we characterize those operator-valued symbols $b: \mathbb{B}_{n}\rightarrow \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_{b}: A^p_α(\mathbb{B}_{n},X) \longrightarrow A^q_α(\mathbb{B}_{n},Y),$ is a bounded operator. Also, given two reflexive complex Banach s…

Bergman-type Singular Operators and the Characterization of Carleson Measures for Besov--Sobolev Spaces on the Complex Ball

The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle "Bergman--type" singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov--Sobolev space of analytic functions $B^σ_2$ on the complex ball of $\mathbb{C}^d$. In particular, we demonstrate that for any $σ> 0$, the Carleson measures for the space are ch…

变指数Bergman空间上的Toeplitz算子与加权复合算子

Toeplitz operators and weighted composition operators on variable exponent Bergman spaces

本文将Choe等人关于加权复合算子之差有界性与紧性的Carleson测度刻画推广到单位球上的变指数Bergman空间，并给出更简单的新证明，同时刻画了Toeplitz算子的有界性与紧性。