Hasi Wulan, Kehe Zhu
We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.
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n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer. Math. Soc., 129: 2039-2044, 2000).
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。
Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in $\CC^n$, $n > 1$, with Lipschitz boundary, but it holds for domains in $\CC^1$ with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings.
For $f$ analytic on the unit disc let $r_t(f)(z)=f(e^{it}z)$ and $f_r(z)=f(rz)$, rotations and dilations respectively. We show that for $f$ in the Bergman space $A^p$ and $0<α\leq 1$ the following are equivalent. \begin{itemize} \item[(i)] $\n{r_t(f)-f}_{A^p}=\og(|t|^α), \quad t\to 0$, \item[(ii)] $\n{(f')_r}_{A^p} =\og\left (1-r)^{α-1}\right ), \quad r\to 1^{-}$, \item[(iii)] $\n{f_r-f}_{A^p}=\og((1-r)^α),\quad r\to 1^{-}$. \end{itemize} The Hardy space analogues of these conditions are known to be equivalent by results of Hardy and Littlewood and of E. Storozhenko, and in that setting they d…
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…
In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p (Ω,$δ$ $γ$), $δ$ being the distance to the boundary, with gain on the index p and the exponent $γ$. These estimates allow us to extend the L p (Ω,$δ$ $γ$) and lipschitz regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.
We obtain estimates of commutators of singular integral operators in Lipschitz spaces and apply the results to boundary regularity of elliptic equations in the plane. We obtain an explicit asymptotic formula for the Bergman projection.
Regularity of the $p-$Bergman kernel
本文证明了有界域上p-Bergman核的局部C^{1,1}正则性，并给出了全局非正则性例子及Levi形式上界估计和对数Lipschitz连续性结果。
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.
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}\left(Ω,dμ_{0}\right)$ where $Ω$ is a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{n}$ and $μ_{0}=\left(-ρ_{0}\right)^{r}dλ$, $λ$ being the Lebesgue measure, $r\in\mathbb{Q}_{+}$ and $ρ_{0}$ a special defining function of $Ω$, are still valid for the Bergman projection of $L^{2}\left(Ω,dμ\right)$ where $μ=\left(-ρ\right)^{r}dλ$, $ρ$ being any defining function of $Ω$. In fact a stronger direct…
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 continue our investigation of function spaces on certain classes of complex-valued functions. In particular, we give characterizations on Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we present applications of our results to certain nonlinear PDEs.
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。
For appropriate domains $Ω_{1}, Ω_{2}$ we consider mappings $Φ_{\mathbf A}:Ω_{1}\toΩ_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(Ω_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $Φ_{\mathbf A}$. We then show that each subspace is isomorphic to a weighted Bergman space on $Ω_{2}$. This leads to a formula for the Bergman kernel on $Ω_{1}$ as a sum of weighted Bergman kernels on $Ω_{2}$
We show that any function in a Bergman space with exponential type weights admits a representation in terms of an infinite series of kernel functions.
We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1 < p < $\infty$. We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space A p (D) and in its generalized version A p f (D), that consists in approximating a function in subsets of D by the restriction of a function belonging to A p (D) or A p f (D) subject to a norm constraint. Preliminary constructive results a…
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.
On the Bergman Metric near Exponentially Flat Infinite Type Boundary Points
研究具有指数平坦无限型边界点的伪凸域的伯格曼度量，证明其不可能是爱因斯坦度量，为程 - 丘成桐猜想提供支持。
On Bergman type projection in some new analytic function spaces in bounded strongly pseudoconvex domains
本文将单位圆盘上Bergman型投影的有界性结果推广到具有光滑边界的有界强拟凸域中的两类解析函数空间。
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。
Compactness of composition operators on the Bergman space of the bidisc
本文研究双圆盘Bergman空间上复合算子的紧性，证明了在闭包上Lipschitz连续的全纯自映射φ所诱导的复合算子C_φ为紧算子的充分必要条件，用两个边界交集为空的条件刻画了紧性。
We obtain new two-sided norm estimates for the family of Bergman-type projections arising from the standard weights $(1-|z|^2)^α$ where $α>-1$. As $α\to -1$, the lower bound is sharp in the sense that it asymptotically agrees with the norm of the Riesz projection. The upper bound is estimated in terms of the maximal Bergman projection, whose exact operator norm we calculate. The results provide evidence towards a conjecture that was posed very recently by the first author.
We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints -- some of them necessary for a non-trivial Bergman space. However, these are mild constraints: unlike most earlier works on this subject, we are able to make estimates for non-convex pseudoconvex models as well. In fact, the domains we can analyse range from being mildly infinite-type to very flat at infinite-type boundary points.
Let $ϕ$ be an analytic self-map and $u$ be a fixed analytic function on the open unit disk $D$ in the complex plane $\CC.$ The weighted composition operator is defined\break by \begin{equation*} uC_ϕf =u \cdot (f\circ ϕ), f \in H(D). \end{equation*} Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.
Stević-Sharma type operators between Bergman spaces induced by doubling weights
本文利用 Khinchin 不等式、Geršgorin 定理和原子分解，刻画倍增权加权 Bergman 空间上 Stević-Sharma 型及加权微分复合算子和的范数与本质范数，并获得插值定理。
Let $Ω$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $Ω$ has the lower bound $\tilde g(δ_Ω(z)^{-1})$ where $δ_Ω(z)$ is the distance from $z$ to the boundary $\partialΩ$ and $\tilde g$ is a specific function defined by $f$. This refines Khanh-Zampieri's work in \cite{KZ12} with reducing the smoothness assumption of the boundary.
Asymptotic behaviour of the Bergman invariant and Kobayashi metric on exponentially flat infinite type domains
本文研究了指数平坦无限型边界点处Bergman不变量、Ricci曲率、标量曲率及Kobayashi度量的非切向渐近极限，通过局部化与伸缩方法给出证明。
We give a Hörmander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $δ_Ω^{-α}dV$, $α<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function theory of weighed Bergman spaces $A^2_α(Ω)$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $A^2_α(Ω)$ by proving an analogue of the Levi problem for $A^2_α(Ω)$ and giving an optimal Gehring type estimate for functions in $A^2_α(Ω)$. A vanishing theorem for $A^2_1…
We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly s…
Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels. The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric associated to the fixed metric, as well as in terms of the Monge-Ampere measure of the fixed metric itself on a certain support set. It is also shown that the …
On Bergman projections and sharp decomposition theorems in tubular and related domains in $C^n$
本文在管状域乘积中定义新的Bergman型积分算子与解析混合范数空间，证明算子有界性，并给出Bergman空间的锐分解定理，扩展了先前结果。
Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type
本文证明满足 Bell-Ligocka 条件 R 且具有良好各向异性伸缩的伪凸域上，Bergman 投影具备局部 $L^p$-Sobolev 与 Hölder 正则性，并说明 $h$-可延拓域适用该结论。
A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D'Angelo.
We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials, investigate the properties of the solution and provide a bound for the solution. This problem is an equivalent formulation of B. Korenblum's conjecture, also known as Korenblum's Maximum Principle: for $f$, $g\in \mathcal{A}^2(\mathbb{D})$, there is a constant $c$, $0<c<1$ such that if $|f(z)|\leq |g(z)|$ for all $z$ such that $c<|z|<1$, then $\|f\|_2\leq \|g\|_2$. Th…
We consider a class of domains, generalizing the upper half-plane, and admitting rotational, translational and scaling symmetries, analogous to the half-plane. We prove Paley-Wiener type representations of functions in Bergman spaces of such domains with respect to each of these three groups of symmetries. The Fourier series, Fourier integral and Mellin integral representations so obtained may be used to give representations of the Bergman kernels of these domains.
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 …
Hypercontractive inequalities and Nikol'ski\uı-type inequalities on weighted Bergman spaces
本研究得到了不同加权Bergman空间间的超压缩不等式，并建立了带最优常数的该空间Nikol'skiĭ型不等式。
We use the so called resolvent transform method to study the cyclicity of the one point mass singular inner function in weighted Bergman type spaces.
Recent advances in Bergman type projections in bounded pseudoconvex and tubular domains
本文为综述，汇总多复变中有界伪凸域、对称锥上管型域等场景下不同维数函数空间间Bergman型投影的最新成果，推广经典结论，提出新研究问题并展望其应用场景。
We first prove a Cauchy's integral theorem and Cauchy type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.
We formulate the Bergman-type interpolation problem on finite open Riemann surfaces covered by the unit disk. Our version of the interpolation problem generalizes Bergman-type interpolation problems previously studied by Seip, Berntsson, Ortega Cerdà, and a number of other authors. We then prove necessary and sufficient conditions for interpolation, and also some sufficient conditions under even weaker hypotheses. The results extend work of Ortega Cerdà, who resolved the case in which the boundary of the surface is pure $1$-dimensional. Our version of the interpolation problem effectively chan…
In this paper, we prove that the original Littlewood-Paley $g$-functions can be used to characterize Bergman spaces as well.
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb H$. In this work we deals with a well-known $(θ, u)-$hyperholomorphic $\mathbb H-$valued functions class related to elements of the kernel of the Helmholtz operator with a parameter $u \in\mathbb H$, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Given a domain $Ω\subset\mathbb H\cong \mathbb C^2$, we define and study a Bergman spaces th…
For a real number $α$ the Hilbert spaces $\mathscr{D}_α$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^α< \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_α$ to the case $α>0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p <2$.
We obtain some necessary and sufficient conditions for the boundedness of a family of positive operators defined on symmetric cones, we then deduce off-diagonal boundedness of associated Bergman-type operators in tube domains over symmetric cones.
This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symm…
We consider Volterra-type integration operators $T_g$ between Bergman spaces induced by weights $ω$ satisfying a doubling property. We derive estimates for the operator norms, essential and weak essential norms of $T_g: A_ω^p \to A_ω^q$, $0<p\leq q<\infty$. In particular, the operator $T_g: A_ω^1\to A_ω^1$ is weakly compact if and only if it is compact.
We characterize the $L^p-L^q$ boundedness of Bergman-type operators over the Siegel upper half-space. This extends a recent result of Cheng et. al. (Trans. Amer. Math. Soc. 369:8643--8662, 2017) to higher dimensions.
We use the author's previous work on atomic decompositions of Besov spaces with spectrum on symmetric cones, to derive new atomic decompositions for Bergman spaces on tube type domains. It is related to work by Ricci and Taibleson who derived decompositions for classical Besov spaces from atomic decompositions of Bergman spaces on the upper half plane. Moreover, for this class of domains our method is an alternative to classical results by Coifman and Rochberg, and it works for a larger range of Bergman spaces.
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.
Restricted type estimates on the Bergman projection of some singular domains
本文对单项式多面体上的Bergman投影建立加权限制型估计，由此重得$L^p$有界性，并发现某些情形下弱型端点失效。
Boundedness of Harmonic Conjugation on Weighted Bergman Spaces
本文证明了在满足Bekollé-Bonami权条件及一个依赖p的简单条件下，调和共轭算子在调和Bergman空间上有界，部分证明借助特殊的好lambda不等式。
It is well known that a function is in a Bergman space of the unit ball if and only if it satisfies some Hardy-type inequalities. We extend this fact to Bergman-Orlicz spaces. As applications, we obtain Gustavsson-Peetre interpolation of two Bergman-Orlicz spaces and we completely characterize symbols of bounded or compact Cesàro-type operators on Bergman-Orlicz spaces, extending known results for classical weighted Bergman spaces.
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。
A Characterization of Pluriharmonic Functions via $p$-Bergman Kernels and A Forelli-type Result
本文通过加权 $p$-Bergman 核的尖锐下界等号条件刻画多次调和函数，并证明一个 Forelli 型切片调和性结果。
Higher-order Volterra-type integral operator on Hardy and Bergman spaces
本文研究了高阶Volterra型积分算子T_{g,n}在单位圆盘上的性质，探讨了其在Hardy空间和加权Bergman空间上的有界性和紧性，并通过Carleson测度条件确定了算子的范数和本质范数估计。
We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Cesàro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Cesàro-type operators on weighted Hilbert-Bergman spaces.
We establish a version of a statement attributed to Kazhdan by Yau. As a corollary, we obtain a more transparent form of our uniformization theorem in complex algebraic geometry.
In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characteriz…
Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_α,K_α^+$ and the $L^p$-$L^q$ compactness of $K_α$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_α^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_α,K_α^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class…
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