Jordi Marzo, Joaquim Ortega-Cerdá
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 $Δϕ$.
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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 $AL^{2}_ϕ(\mathbb{D})$
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 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.
We give criteria for boundedness of the associated Bergman-type projections on Lp-spaces on Cn with respect to generalized Gaussian weights exp(-|z|^2m). Complete characterization is obtained for 0<m<=1 and 2n/(2n-1)<m<2; partial results are given for other values of m.
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…
This is a partly expository article for the volume "Algebraic and Analytic Microlocal Analysis" on pointwise Weyl laws for spectral projections kernels in the Kaehler setting. We prove a 2-term pointwise Weyl law for projections onto sums of eigenspaces of spectral width $\hbar=k^{-1}$ of Toeplitz quantizations $\hat{H}_k$ of Hamiltonians on a Kaehler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is `strongly hyperbolic' the leading coefficient defines a uniformly continuous measure on $\R$ and a …
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…
We prove new pointwise bounds for weighted Bergman kernels in $\mathbb{C}^n$, whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on $(0,1)$-forms. Our results extend the ones obtained in $\mathbb{C}$ by Christ. Our main idea is to develop a version of Agmon theory (originally introduced to deal with Schrödinger operators) for weighted Kohn Laplacians on $(0,1)$-forms, inspired by the fact that these are unitarily equivalent to certain generalized Schrödinger operators.
We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $\text{Re} \int_Ωf(z) \overline{k(z)}\, ν(z) \, dA(z)$ over all functions $f$ of unit norm, where $ν$ is the weight function and $Ω$ is the domain of the functions in the space. We consider the case where $ν(z)$ is a decreasing radial function satisfying some additional assumptions, and where $Ω$ is either a disc centered at the origin or the entire complex plane. We show that …
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。
Carleson embeddings and pointwise multipliers between Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper half-plane
本文给出涉及凹或凸增长函数的 Carleson 测度的一般刻画，并据此建立上半平面 Hardy-Orlicz 空间与 Bergman-Orlicz 空间之间的连续嵌入与逐点乘子集合的刻画。
In this paper we solve a problem posed by H. Bommier-Hato, M. Engliš and E.H. Youssfi in [3] on the boundedness of the Bergman-type projections in generalized Fock spaces. It will be a consequence of two facts: a full description of the embeddings between generalized Fock-Sobolev spaces and a complete characterization of the boundedness of the above Bergman type projections between weighted $L^p$-spaces related to generalized Fock-Sobolev spaces.
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $kφ$ in a $\frac{1}{\sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
It is shown that the formula for the Möbius pseudodistance for the annulus yields better estimates than previously known for the constant in the Bergman space maximum principle. A maximum principle for the Fock space is proved.
The usual examples of Bergman spaces consist of the closure of an algebra of holomorphic functions on a domain. One can also take the real part of such functions, but essentially one is looking at the same object. In this paper the author shows that the properties of real analyticity and bounded point evaluation can be preserved under closure in a weighted $L^p$ space, if one takes the algebra generated by $z$ and $(\bar z)g(z)$, where $g$ is an entire function satisfying some condition on the distribution of zeros near $\infty$. The condition is not difficult to satisfy with simple examples. …
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…
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。
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.
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。
In this master thesis, we give a new proof on the pointwise asymptotic expansion for Bergman kernel of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfy local spectral gap condition. The main point is to introduce a suitable semi-classical symbol space and related symbolic calculus inspired from recent work of Hsiao and Savale. Particularly, we establish the existence of pointwise asymptotic expansion on the positive part for certain semi-positive line bundles.
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 introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0,R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.
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。
In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.
Firstly we establish a sharp pointwise estimate for the arbitrary derivative of the function $f\in F_α^{p},$ where $F_α^{p}$ denotes the Fock space for $1\leq p<\infty.$ Then, in a particular Hilbert case when $p=2$ we establish another specific pointwise sharp estimate. We also consider the differential operator between $F_α^{p}$ and $F_β^{p}$ for $β>α$ and its adjoint.
We study the action of some generalized integral operators of Bergman type on pointwise multipliers of holomorphic Triebel-Lizorkin spaces. We construct nontrivial examples of pointwise multipliers in Hardy-Sobolev spaces and give applications of all these results.
We study composition operators on the Fock spaces $\mathcal{F}^2_α(\mathbb{C}^n)$, problems considered include the essential norm, normality, spectra, cyclicity and membership in the Schatten classes. We give perfect answers for these basic properties, which present lots of different characterizations with the composition operators on the Hardy space or the weighted Bergman spaces.
Weighted contractivity for derivatives of functions in the Bergman space on the unit disk
本文推广了Ramos和Tilli关于单位圆盘解析函数的锐利不等式，将其扩展到Bergman空间中函数导数关于两种不同测度的情形，并讨论与Fock空间类似结果的联系。
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…
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.
Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
本文在多复变量加权Fock空间中，通过正则化曲率假设建立Bergman核估计，并构造\bar\partial解算子，进而用加权IDA空间刻画Hankel算子的有界性与紧性。
We discuss sampling constants for dominating sets in Bergman spaces. Our method is based on a Remez-type inequality by Andrievskii and Ruscheweyh. We also comment on extensions of the method to other spaces such as Fock and Paley-Wiener spaces.
This research is concerned with the nonhomogeneous linear complex differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_{1}f'+A_{0}f=A_{k} $$ in the complex plane. In the higher order case, the mutual relations between coefficients and solutions in weighted Fock spaces are discussed, respectively. In particular, sufficient conditions for the solutions of the second order case $$ f"+Af=0 $$ to be in some weighted Fock space are given by Bergman reproducing kernel and coefficient $A$.
Sharp pointwise and uniform estimates for $\bar\partial$
本文用加权 $L^2$ 方法，在严格凸域和 Cartan 经典域上获得 $\bar\partial$ 方程规范解的尖锐逐点估计，并证明秩 2 情形下最大爆破阶超过单位球结果，同时给出更强条件下的一致估计。
In this article we introduce Variable exponent Fock spaces and study some of their basic properties such as the boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality.
We show that under mild conditions, a Gaussian analytic function $\boldsymbol F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock space has the property that a.s. no non-zero function in that space vanishes where $\boldsymbol F$ does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann-Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro (1979) on Bergman spaces and allowing us to strengthen a result of Zh…
IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces
本文以 IDA 函数的不增重排刻画加权 Bergman 空间中 Hankel 算子奇异值的渐近行为，并推广至共轭符号联合刻画及加权 Fock 空间中的 Berger-Coburn 现象。
In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.
The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the canonical solution to the non-homogeneous $k$-Cauchy-Fueter equation in a weighted $L^2$-space, by establishing the weighted $L^2$ estimate. The weighted $k$-Bergman space is the space of weighted $L^2$ integrable functions annihilated by the $k$-Cauchy-Fueter operator, as the counterpart of the Fock space of weighted $L^2$-holomorphic functions on $\mathbb{C}^n…
Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms
本文提出一种新的缩放方法，在局部谱间隙条件下研究复流形上线丛张量幂的谱核与Bergman核，建立了点态渐近的简单证明，并获得了函数情形下指数衰减的Bergman核首项，且该渐近在曲率退化时仍有效。
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。
We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.
The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schrödinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman kernels in ${\mathbb C}^n$ (chapters 2 and 3). These two themes are connected by the observation that the weighted Bergman kernel may be effectively studied by means of the analysis of the weighted Kohn Laplacian, which in turn is unitarily equivalent to a Schödinger operator with matrix-valued potential.
Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak inequalities hold on any hermitian manifold X, regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in the standard complex space C^n.
The sharp radius in Korenblum's maximum principle for the Fock space
本文确定了 Fock 空间 $F^2$ 中 Korenblum 最大值原理的最优半径为 1，并给出等号成立条件，首次精确求解该类型空间的最优半径，且方法推广至加权情形。
Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.
We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several variables and Bergman spaces in balls and polydiscs.
We study two-dimensional eigenvalue ensembles close to certain types of singular points in the bulk of the droplet. We prove existence of a microscopic density which quickly approaches the classical equilibrium density, as the distance from the singularity increases beyond the microscopic scale. As a consequence we obtain asymptotics for the Bergman function of certain Fock-Sobolev spaces of entire functions.
Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series
本研究针对狄利克雷级数加权Bergman空间的点赋值泛函范数估计问题，引入作用于该空间的Riemann-Liouville半群版本作为新工具，完善了该空间理论中的核心问题研究。
We obtain Littlewood-Paley formulas for Fock spaces $\mathcal{F}^q_{β,ω}$ induced by weights $ω\in A^{restricted}_\infty=\cup_{1\le p<\infty}A^{restricted}_{p}$, where $A^{restricted}_{p}$ is the class of weights such that the Bergman projection $P_α$, on the classical Fock space $\mathcal{F}^2_α$, is bounded on $$\mathcal{L}^p_{α,ω}:=\left\{f:\, \int_{\mathbb{C}}|f(z)|^pe^{-p\fracα{2}|z|^2}\,ω(z)dA(z)<\infty \right\}. $$ Using these equivalent norms for $\mathcal{F}^q_{β,ω}$ we characterize the Carleson measures for weighted Fock-Sobolev spaces $\mathcal{F}^{q,n}_{β,ω}$.
Sharp stability of convex functionals on weighted Bergman spaces with applications
本文研究了凸函数在加权伯格曼空间上的稳定性，并应用该结果到Fock空间、Cauchy小波和Hardy空间等场景。
Sampling constants in generalized Fock spaces
该研究证明了广义加倍Fock空间中支配集的Logvinenko-Sereda型定理，给出了采样常数对相对密度参数$γ$的精确多项式依赖，方法基于Remez型不等式和覆盖引理。
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。
In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In particular, we give explicit expressions of their reproducing kernels.
We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.
Given a smooth positive measure $μ$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate version of the Bochner--Kodaira--Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded g…
We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock-Sobolev spaces on ${\mathbb C}^n$ with respect to the weight $(1+|z|)^ρe^{-\fracα2|z|^{2\ell}}$, for $\ell\ge 1$, $α>0$ and $ρ\in{\mathbb R}$. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood-Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.
We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. However, unlike previous work, we construct an infinite number of densities, naturally parameterized by $L^1_{loc} ([0,\…
The purpose of this paper is to systematically study compactness and essential norm properties of operators on a very general class of weighted Fock spaces over $\C$. In particular, we obtain rather strong necessary and sufficient conditions for a wide class of operators (which includes operators in the Toeplitz algebra generated by bounded symbols) to be compact and we obtain related estimates on the essential norm of such operators. Finally, we discuss interesting open problems related to our results, and in particular discuss the possibility of extending our results to other generally weigh…
It is well known that functions in the analytic Besov space $B_1$ on the unit disk $\D$ admits an integral representation $$f(z)=\ind\frac{z-w}{1-z\bar w}\,dμ(w),$$ where $μ$ is a complex Borel measure with $|μ|(\D)<\infty$. We generalize this result to all Besov spaces $B_p$ with $0<p\le1$ and all Lipschitz spaces $Λ_t$ with $t>1$. We also obtain a version for Bergman and Fock spaces.
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