Robert Berman, Bo Berndtsson, Johannes Sjoestrand
We give an elementary proof of the existence of an asymptotic expansion in powers of $k$ of the Bergman kernel associated to $L^k$, where $L$ is a positive line bundle. We also give an algorithm for computing the coefficients in the expansion.
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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.
Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.
Actually we will discuss some topics related to Bergman kernel on Cartan- Hartogs domain.
We investigate Lp regularity of weighted Bergman projections and zeros of weighted Bergman kernels for the weights that are radially symmetric and comparable to 1 on the unit disc.
We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal-Bargmann spaces of the complex plane are briefly discussed.
On the $p$-Bergman kernel with respect to a functional $ξ$
本文将p-Bergman核与ξ-Bergman核推广为关于泛函ξ的p-Bergman核，建立其性质，研究高阶Bergman核的L^p版本与ξ-Bergman核的关系，并推广了Blocki和Zwonek的先前结果。
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.
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.
We obtain an asymptotic expansion and some regularity results for the Bergman kernel on the intersection of two balls in C^2.
We solved the problem of the best rational approximation of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.
We study the connection between weighted Bergman kernel and Green's function on a domain W lying in C for which the Green's function exists.
In this paper we consider the zeros of the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$. We describe how the existence of zeros of the Bergman kernel depends on the integers $m$ and $n$ with the help of the interlacing property.
In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which deduces a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson-Mustcată and the effectiveness result of strong openness property of the modules at boundary points.
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。
Let $AL^{2}_ϕ(\mathbb{D})$ denote the closed subspace of $L^{2}(\mathbb{D},e^{-2ϕ}dλ)$ consisting of holomorphic functions in the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $ϕ: {\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the Bergman kernel for $AL^{2}_ϕ(\mathbb{D})$
We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros.
We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
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。
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.
We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.
In this work we provide an asymptotic expansion for the Szegő kernel associated to a suitably defined Hardy space on the the non-smooth worm domain $D'_β$. After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegő kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.
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.
We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.
We construct inner products by the Bernstein-Markov inequality on spaces of holomorphic sections of high powers of a line bundle. The corresponding weighted Bergman kernel functions converge to an extremal function. We obtain a uniform convergence speed.
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。
We obtain a quantitative estimate of Bergman distance when $Ω\subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $α_l(Ω)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of the Bergman kernel $K_Ω(\cdot, w)$ when $α_l(Ω)>0$.
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 $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ by using isoperimetric inequalities.
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds
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。
We consider a certain Hartogs domain which is related to the Fock-Bargmann space. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover we solve the Lu Qi-Keng problem of the domain in some cases.
In this paper, we study the Bergman kernel $B_φ(x,y)$ of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of $L^2$-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of $B_φ(x,y)$ and then give some estimates on and off the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.
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。
In this paper we investigate the Bergman kernel function for intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb{C}^{n+2} : |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \quad |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.
The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behaviour is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying planar measures on the complex plane. We obtain u…
A representation for the reproducing kernel of a weighted Bergman space
本文针对单位圆盘上由Blaschke因子幂乘积的模构成的权函数，给出了加权Bergman空间再生核的典范表示，并由此推导出有限零点集对应的压缩零除子公式。
A Canonical Positive Definite Kernel Associated with the $ξ$-Bergman Kernel
本文构造了与ξ-伯格曼核相关的典范埃尔米特正定核，证明其正定并恢复ξ-伯格曼核为其对角限制，还得到该核的显式公式等性质，揭示了微分泛函选择的影响。
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.
In this article, we consider Bergman kernels related to modules at boundary points on Stein manifolds, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted $L^2$ integrals on Stein manifolds, and reprove an effectiveness result of strong openness property of modules at boundary points on Stein manifolds.
Carleson-Type Measures and Kernel Estimates for Potential-Harmonic Weighted Bergman Spaces on the Unit Ball
本文研究了C^n单位球上的加权Bergman空间，建立了Carleson嵌入的刻画，并证明了再生核函数的逐点和范数估计。
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.
In this article, we derive off-diagonal estimates of the Bergman kernel associated to the tensor-powers of the cotangent bundle defined on a hyperbolic Riemann surface of finite volume, when the distance between the points is less than injectivity radius. We then use these estimates to derive estimates of the Bergman kernel along the diagonal.
Remote temperature sensing in 2D and the Bergman kernel
研究通过另一点高阶温度信息估计二维域内某点稳态温度，揭示与伯格曼核、鲁涅定理及近似零四次积分恒等式的关系。
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.
Let $\{Ω_t:-1<t<1\}$ be a family of bounded pseudoconvex domains and $φ_t\in PSH(Ω_t)$. Let $K_t(z,w)$ denote the Bergman kernel with weight $φ_t$ on $Ω_t$. We study the continuity and Hölder continuity of $K_t(z,w)$ in $t$. Several applications to singularity theory of psh functions are given, including a new proof of the openness theorem.
In this paper we determine explicitly the kernels $\mathbb K_{α,β}$ associated with new Bergman spaces $\mathcal A_{α,β}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $α\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{α,β}$. Some numerical results are given throughout the paper.
Boundary behaviour of the Bergman and Szegő kernels on generalized decoupled domains
本文在广义解耦伪凸域上证明了Bergman核、Szegő核及Bergman度量在边界附近的最优估计，该域允许复切方向非独立解耦且边界点可兼具有限与无限型方向。
Fiberwise Bergman kernels, vector bundles, and log-subharmonicity
本文研究全纯向量丛上奇异Hermitian度量边界点处模相关的Bergman核，证明其对数次调和性，并应用于加权$L^2$积分下界估计及强开性有效性结果的重新证明。
A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function known as the Schwarz function. Simplified proofs of several other well known facts about quadrature domains will fall out along the way. Finally, Bergman representative coordinates will be defined that make subtle alterations to a domain to convert it to a quadrature domain. In such coordinates, biholomorphic mappings become algebraic.
It is proved that for any domain in $\mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the "multidimensional Suita conjecture" theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel.
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.
We give a new proof of Dennis Hejhal's theorem on the nondegeneracy of the matrix that appears in the identity relating the Bergman and Szego kernels of a smoothly bounded finitely connected domain in the plane. Mergelyan's theorem is at the heart of the argument. We explore connections of Hejhal's theorem to properties of the zeroes of the Szego kernel and propose some ideas to better understand Hejhal's original theorem.
Asymptotic behaviour of the Bergman kernel and metric
本文研究光滑有界拟凸域在指数平坦无限型边界点处的 Bergman 不变量，通过局部化与缩放方法证明了 Bergman 核、度量及全纯截曲率的非切向渐近极限。
Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is infinitely differentiable but not necessarily real analytic. The bounds obtained are the best possible for this class of metrics.
We give a precise estimate of the Bergman kernel for the model domain defined by $Ω_F=\{(z,w)\in \mathbb{C}^{n+1}:{\rm Im}w-|F(z)|^2>0\},$ where $F=(f_1,...,f_m)$ is a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^m$, in terms of the complex singularity exponent of $F$.
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…
We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2ϕ})$ where $ϕ$ is a subharmonic function with $Δϕ$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $Δϕ$.
We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel. The characterization leads to a partial fractions-like decomposition for such rational functions and new ways to describe such Jordan curves. The multiply connected case is also explored.
We study the lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. We show that it implies a bound in terms of volume of the Azukawa indicatrix which can be treated as a multidimensional version of the Suita conjecture. We also prove that the corresponding upper bound holds for convex domains and discuss it in bigger detail on some convex complex ellipsoids.
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