Yacin Ameur, Haakan Hedenmalm, Nikolai Makarov
We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion) paper.
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Berezin transforms on modified Bergman spaces
本文研究了修正Bergman空间上Berezin变换的连续性，并基于Bergman-Poincaré度量建立了Lipschitz估计。
On the Berezin range of Toeplitz and weighted composition operators on weighted Bergman spaces
本文刻画了加权Bergman空间上调和符号Toeplitz算子的Berezin值域，引入一类新的加权复合算子并确定其Berezin值域与Berezin数，同时研究了复合算子Berezin值域的凸性及原点与闭包的关系。
Toeplitz Operators and Berezin-type Operators on Different Bergman Spaces
本文研究管状域上不同加权Bergman空间之间Toeplitz算子和Berezin型算子的有界性与紧性，建立其与Carleson测度的联系并给出刻画。
Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class …
We characterize Schatten class membership of positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space in terms of averaging functions and Berezin transforms in the range of $0<p<\infty$.
On the setting of the Siegel upper half-space we study the spaces of bounded and vanishing mean oscillations which are defined in terms of the Berezin transform, and we use them to characterize bounded and compact Hankel operators on Bergman space.
Harmonic Berezin transform on half-space with vertical weights
该研究针对上半空间带垂直权重的调和Bergman空间，借助Siegel域全纯情形的已知结论，推导得到加权调和Bergman核与垂直坐标函数调和Berezin变换的完整渐近展开式。
Compactness and the essential norm on Bergman spaces of the Siegel upper half-space
本文利用Berezin变换刻画紧性，得到西格尔上半空间伯格曼空间上Toeplitz代数中算子的本质范数估计。
Positive Berezin liminf does not imply essential positivity for radial Toeplitz operators on Bergman and Fock spaces
研究径向Toeplitz算子在Bergman和Fock空间上的本质正性是否可通过Berezin变换的渐进行为检测。构造了具有正Berezin liminf但本质谱包含负点的径向符号，推翻Perälä-Virtanen猜想。
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.
When the Berezin transform fails to detect compactness: Toeplitz operators with $L^1$ symbols on weighted Bergman spaces
该研究针对加权Bergman空间，构造了具有$L^1$符号的Toeplitz算子，其Berezin变换在边界消失但算子非紧，回答了Bauer与Isralowitz的问题，还在无权重圆盘上验证了相关性质并确定Zorboska假设的最优性。
In this article we characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the so-called Bekolle-Bonami condition in terms of the Berezin transform.
Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
该研究针对径向$\widehat{\mathcal{D}}$-加权Bergman空间，证明了Toeplitz算子紧性等价于其Berezin变换在边界处趋于零，同时构造反例说明该刻画不适用于Toeplitz算子乘积及对应代数。
Convexity of the Berezin range of composition operators induced by automorphisms of the Unit Ball in $\mathbb{C}^n$
本文研究单位圆盘和单位球自同构诱导的复合算子的 Berezin 值域凸性，给出了与酉映射复合的线性分式自同构情形下凸性的新刻画。
We define positive Toeplitz operators between harmonic Bergman-Besov spaces $b^p_α$ on the unit ball of $\mathbb{R}^n$ for the full ranges of parameters $0<p<\infty$, $α\in\mathbb{R}$. We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman-Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on $b^{2}_α$ to be a Schatten class operator $S_{p}$ in terms of averaging functions and Berezin transforms for $1\leq p<\infty$, $α\in\mathbb{R}$. Our results extend those kno…
We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin-Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.
We prove approximation results about sequences of Berezin transforms of finite sums of finite product of Toeplitz operators (and bounded linear maps, in general) in the spirit of Ramadanov and Skwarczynski theorems that are about convergence of Bergman kernels.
In this paper, we characterize Carleson measure and vanishing Carleson measure on Bergman spaces with admissible weights in terms of {\it t-Berezin transform} and {\it averaging function} as key tools. Moreover, power bounded and power compact weighted composition operators are characterized as application of Carleson measure and vanishing Carleson measure respectively on Bergman spaces with admissible weights.
We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying…
We characterise the boundedness of a Toeplitz operator on the Bergman space with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on the Bergman space with an L^1 symbol is completely determined by the boundary behaviour of itss Berezin transform. This result extends known results in the cases when the symbol is either a positive L^1 function, an L^\infty function or a general BMO^1 function.
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。
We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(ω)$ belong to the Schatten class $S_p$, $0<p<\infty$.
$L^p$-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$p>m+nl$。
Localization operators on Bergman and Fock spaces
本文研究了Bergman和Fock空间上的定位算子，证明其在特定条件下的弱收敛性，并推导了相关应用，包括Toeplitz算子的范数估计、加窗Berezin变换和Szegö型定理。
Invertibility of Bergman Toeplitz operators
本文研究了 Bergman 空间上 Toeplitz 算子的可逆性问题，证明了符号的 Berezin 变换的可逆性是 Toeplitz 算子可逆的充要条件，并给出了特定函数形式下基于 Berezin 变换下确界的等价判据。
Compactness of Toeplitz Operators on the Bergman Space
本文用符号的局部平均值刻画 Bergman 空间上 Toeplitz 算子的紧性，证明紧性等价于 Bergman 圆盘平均值趋于零，并给出 Carleson 盒子平均的等价判据及反例。
Berezin density and planar orthogonal polynomials
本文提出利用非线性势能理论研究正交多项式渐进行为，通过Berezin密度与归一化正交多项式的关系，探讨多项式伯格曼核的显式展开公式。
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。
We study the off-diagonal decay of Bergman kernels $Π_{h^k}(z,w)$ and Berezin kernels $P_{h^k}(z,w)$ for ample invariant line bundles over compact toric projective \kahler manifolds of dimension $m$. When the metric is real analytic, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ where $D(z,w)$ is the diastasis. When the metric is only $C^{\infty}$ this asymptotic cannot hold for all $(z,w)$ since the diastasis is not even defined for all $(z,w)$ close to the diagonal. We prove that for general $C^{\infty}$ metrics, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$ as long as $w$ lies on the ${\mathbb R}_+…
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
Transformation formula for the Reduced Bergman kernel and its Application
本文证明了复平面有界域间真全纯对应及映射下约化Bergman核与加权约化Bergman核的变换公式，并给出了其应用。
Let $ϕ$ be an analytic self-map of the open unit disk $\mathbb{D}$ and $g$ analytic in $\mathbb{D}$. We characterize boundedness and compactness of generalized Volterra type integral operators $$GI_{(ϕ,g)}f(z)= \int_{0}^{z}f'(ϕ(ξ))\,g(ξ)\, dξ$$ and $$GV_{ (ϕ, g)}f(z)= \int_{0}^{z} f(ϕ(ξ))\,g(ξ)\, dξ, $$ acting between large Bergman spaces $A^p_ω$ and $A^q_ω$ for $0<p,q\le \infty$. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independe…
It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $Π_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $Π_N f(N^{-1} D_θ)$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over $P$ gives a complete asymptotic expansion for Dedekind-Riemann sums of smooth functions over lattice points in $…
The range of the Bergman space B_2(G) under the Cauchy transform K is described for a large class of domains. For a quasidisk G the relation K(B_2^*(G))=B_2^1(\mathbb C\setminus\bar{G}) is proved.
The so-called weighted solid Cauchy transform, from inside the unit disc into the complement of its closure, is considered and their basic properties such as boundedness is studied for appropriate probability measures. The action the disc polynomials is explicitly computed and used to describe the range of its restriction on the weighted true poly-Bergman spaces.
On the Commutativity of the Berezin Transform
本文研究加权Fock空间上Berezin变换的交换性，证明对所有有界函数及正参数，交换关系成立当且仅当参数$m=2$。
We consider a weighted Berezin transform: $$ B_α : L^{\infty} (\mathbb{B}^n) \to \ \mathcal{B},\quad α>-1,$$ defined, for $f \in L^{\infty} \left( \mathbb{B}^n \right)$ and $z \in \mathbb{B}^n$, by $$(B_αf) (z) = c_α\int_{\mathbb{B}^n} \frac{\left( 1-|z|^2 \right)^{n+1}}{|1 - \langle z, w \rangle|^{2n+2}} f(w) \left( 1-|w|^2 \right)^α\ d v(w),$$ where $c_α = \frac{Γ(α+n+1)}{Γ(α+1)π^n}$ , $v$ is the Lebesque measure and $\mathcal{B}$ is a Bloch-type space. We prove that $B_α$ is bounded iff $α>0$ and give the exact semi-norm of $ B_α$ for $0\leqα\leq 2n+3.$
Composition and Volterra type operators on large Bergman spaces with rapidly decreasing weights
本文刻画了快速递减权大 Bergman 空间之间广义 Volterra 型积分算子的有界性、紧性与 Schatten 类性质，利用 Littlewood-Paley 公式和嵌入定理推广了 Pau 与 Peláez 的积分算子研究。
This paper deals with some special integral transforms of Bargmann-Fock type in the setting of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. The construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. The ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic setting are obtained in both the Fock…
Let $\ds dA=\frac{dxdy}π$ denote the normalized Lebesgue area measure on the unit disk $\disk$ and $u$, a summable function on $\disk$. $$B(u)(z)=\int_\disk u(ζ)\frac{(1-|z|^2)^2}{|1-ζ\oln z|^4}dA(ζ)$$ is called the Berezin transform of $u$. Ahern \cite{a} described all the possible triples $\{u,f,g\}$ for which $$B(u)(z)=f(z)\oln g(z)$$ where both $f,g$ are holomorphic in $\disk$. This result was crucial in solving a version of the zero product problem for Toeplitz operators on the Bergman space. The natural next question was to describe all functions in the range of Berezin Transform which a…
The problem of describing the range of a Bergman space B_2(G) under the Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of \bar G. The description of K(B_2(G)) for an integrable Jordan domain is given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127]. In the present paper we give a description of K(B_2(G,ω)) analogous to the one given in [S. Merenkov, "On …
Asymptotic results on modified Bergman-Dirichlet spaces and examples of Segal-Bargmann transforms
本文引入修正的Bergman-Dirichlet空间，研究参数α趋于无穷大和-1时的渐近行为，得到修正的Bargmann-Dirichlet与Hardy-Dirichlet空间及其再生核，并给出Segal-Bargmann变换的例子。
Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with Békollé and Bonami weights
本文针对加权贝科勒 - 博纳米伯格曼空间间的托普利兹算子，通过建立核p-范数估计来刻画其有界和紧性，借助广义贝雷津变换等描述性质，引入权重条件确保加权伯格曼投影有界，将相关结果从希尔伯特空间推广到巴拿赫空间。
In this short note we improve some of recent results of Cuckovic and Sahutoglu concerning regularity of the Berezin transform for a class of smooth pseudoconvex domains.
The aim of the present paper is three folds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius $R$ centred at the origin. The explicit expression of its Bergman kernel is given and can be written in terms of special hypergeometric functions of two non-commuting (quaternionic) variables. Secondly, we introduce and study some basic properties of an associated integral transform, the quaternionic analogue of the so-called second Bargmann transform for the holomorphic Bergman space. Finally, we establish the asymptotic behavior a…
Optimal off-diagonal upper estimates for Bergman reproducing kernels
研究单位圆盘上伯格曼再生核的非对角线上界估计，基于权重矩的单边加倍条件证明其精确性及刻画性。通过新方法得到修正伯格曼核的\(L^p\)均值估计，建立相关函数与测度新联系，还将结果推广到高维等情况。
Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results.
We study Bergman kernels $K_Π$ and projections $P_Π$ in unbounded planar domains $Π$, which are periodic in one dimension. In the case $Π$ is simply connected we write the kernel $K_Π$ in terms of a Riemann mapping $φ$ related to the bounded periodic cell $\varpi$ of the domain $Π$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_Π$ to a pro…
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。
Carleson measures, tent embeddings, and Volterra-type integral operators on the unit ball
本文通过比较Carleson立方体和Bergman度量球条件，结合Berezin型刻画，证明了Besov空间和Bergman空间到对数帐篷空间的嵌入定理，并应用于刻画Volterra型积分算子的有界性、紧性和本质范数。
Composition operators, convexity of their Berezin range and related questions
本文研究再生核 Hilbert 空间中复合算子 Berezin 值域的凸性，刻画 Hardy 与 Bergman 空间情形，并证明正自伴算子在超二次函数下的 Berezin 集映射定理。
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。
A sufficient condition for $L^p$ regularity of the Berezin transform
本文证明了Berezin变换在$\nathbb{C}^n$的一大类域上具有$L^p$正则性，但在Hartogs三角形上缺乏$L^2$正则性。
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…
Toeplitz and Hankel operators on Bergman spaces with doubling weights
本文研究倍权Bergman空间上Toeplitz与Hankel算子的有界性和紧性，通过Berezin变换刻画有界性，并给出共解析符号情形下的充要条件。
The Reduced Bergman Kernel and its Properties
本文研究平面区域上$n$阶加权约化Bergman核的性质，探讨了Ramadanov型定理、局部化与边界行为，并给出了核在双全纯变换下的变换公式。
Fixed points of the Berezin transform on Fock-type spaces
本文研究了Fock型空间上Berezin变换的固定点，证明多项式固定点必须为调和多项式，除了可能的可数多个参数情况。
Spectral properties of the Cauchy transform on modified Bergman spaces
本文研究了修正 Bergman 空间上 Cauchy 变换相关算子的奇异值，通过超几何函数级数表示，并证明其加权序列在无穷远处存在有限极限。
We consider a bounded radial function f. B. Korenblum and K.E.Zhu give a case where we have equality between the limit near the boundary of the unit disc of the Berezin transform and the limit of the normalized Mellin coefficient when one of them is 0. We previously describe the case where one limit point has modulus infinite norm of f. We also use the mean values of f near 1. The aim of this article is to show that between these two extreme cases, we can have distinct limits. In the same time, we also study the sets of limit points of these quantities.
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