Timothy Ferguson
We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ and the related operator $f \mapsto f - f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_1(r,u-u(0))$ in terms of $\|u\|_{h^1}$, where $M_1$ is the integral mean with $p=1$.
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In this paper, the boundedness and compactness of the difference of composition-differentiation operators $D_φ-D_ψ$ acting from Hardy spaces $H^p$ to weighted Bergman spaces $A^q_α$ are completely characterize for all $0<p,q<\infty$.
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.
In the present work, we are interested in compact integration operators $I_g f(z) = \int_0^z f(ζ)g'(ζ)dζ$ acting on the Hardy space $H^2$ and on the weighted Bergman spaces $\mathcal{A}^2_α$. We give upper and lower estimates for the singular values of $I_g$.
The Hardy number and the Bergman number of a planar domain are equal
本文证明任意平面区域的Hardy数与Bergman数相等，从而改进Karafyllia的定理，揭示预设值域函数在Hardy空间与加权Bergman空间中的隶属关系可相互关联。
Zero Products of Toeplitz operators on the Hardy and Bergman spaces over an annulus
本文研究环域上Hardy与Bergman空间中Toeplitz算子的零积问题，在符号傅里叶展开条件下证明无零因子，并借助Abrahamse约化定理刻画紧Hankel算子，得出零积结果。
On the Hardy number and the Bergman number of a planar domain
本文研究了具有预定值域的函数在 Hardy 和加权 Bergman 空间中的包含关系，利用 Green 函数给出区域 Hardy 数的新刻画，并探讨 Hardy 数与 Bergman 数相等的条件及一般不相等性。
Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy Spaces
本文研究无限维解析函数Hilbert空间中采样序列与多项式子空间Marcinkiewicz-Zygmund不等式的关系，重点比较单变量Hardy空间与Bergman空间中截然不同的行为。
Let $\mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $\mathbb{D}$. We characterize bounded and compact Volterra type integration operators \[ J_{g}(f)(z)=\int_{0}^{z}f(λ)g'(λ)dλ\] between weighted Bergman spaces $L_{a}^{p}(ω)$ induced by $\mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.
We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_α$ to the Hardy spaces $H^q$ of the unit ball of $\mathbb{C}^n$ for all $0<p,q<\infty$. A partial solution to the case $n=1$ was previously obtained by Z. Wu in \cite{Wu}. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension $n$. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as tec…
Spectrum of Hausdorff operators on weighted Bergman and Hardy spaces of the upper half-plane
本文研究了上半平面加权伯格曼和幂加权硬叶空间上豪斯多夫算子的谱特性。
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 Generators of the Hardy and the Bergman Spaces
本文通过同时对生成元与 z 的乘法不变的子空间刻画 Hardy 空间和 Bergman 空间的生成元，并研究其游荡性及有界解析函数的稠密性。
We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the Bergman number of a simply connected domain in terms of extremal distance.
In thius paper we introduce the Hardy and Bergman spaces on hyperconvex domains relative to a acontinuous exhaustion function. We prove their basic properties and study their composition operators induced by holomorphic mappings between domains.
In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.
In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.
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.
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。
We show that the weighted Bergman-Orlicz space $A\_α^ψ$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $ψ$ satisfies the so-called $Δ^{2}$--condition. In addition we prove that this condition characterizes those $A\_α^ψ$ on which every composition operator is bounded or order bounded into the Orlicz space $L\_α^ψ$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $ψ$ satisfies the $Δ^{2}$--condition, a composition operator is compact on $A\_α^ψ$ if and only …
Hardy-Littlewood theorems and the Bergman distance
本文在非欧几里得框架下推广了Hardy-Littlewood定理，研究了单位圆盘上解析映射边界函数的平滑性，并引入了Bergman距离作为关键工具。
We study the question of when two weighted variable exponent Bergman spaces or Hardy spaces are equivalent. As an application, we show that variable exponent Hardy spaces have a close relation to classical Hardy spaces when the exponent is log-Hölder continuous and has bounded harmonic conjugate (when extended from its boundary values to be harmonic in the disc). We use this to characterize Carleson measures for these variable exponent Hardy spaces. We also prove under certain conditions an analogue of Littlewood subordination and a result on the boundedness of composition operators.
Let $\mathcal{W}$ be the corresponding wandering subspace of an invariant subspace of the Bergman shift. By identifying the Bergman space with $H^2(\mathbb{D}^2)\ominus[z-w]$, a sufficient and necessary conditions of a closed subspace of $H^2(\mathbb{D}^2)\ominus[z-w]$ to be a wandering subspace of an invariant subspace is given also, and a functional charaterization and a coefficient characterization for a function in a wandering subspace are given. As a byproduct, we proved that for two invariant subspaces $\mathcal{M}$, $\mathcal{N}$ with $\mathcal{M}\supsetneq\mathcal{N}$ and $dim(\mathcal…
We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions f…
We study a characterization of slice Carleson measures and of Carleson measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces $\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudohyperbolic balls is not possible.
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.
Octonionic monogenic and slice monogenic Hardy and Bergman spaces
本文研究八元数单演与切片单演 Hardy 和 Bergman 空间，通过引入平行内积克服非结合性困难，并给出单位球、半空间及条形域上切片再生核的显式公式与序列刻画。
It is known that, generically, Taylor series of functions holomorphic in the unit disc turn out to be universal series outside of the unit disc and in particular on the unit circle. Due to classical and recent results on the boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman spaces the situation is essentially different. In this paper it is shown that in many respects these results are sharp in the sense that universality generically appears on maximal exceptional sets. As a main tool it is proved that the Taylor (backward) shift on certain Bergman spaces is mixing.
We discuss the notion of an inner function for spaces of analytic functions in multiply connected domains in $\mathbb{C}$, giving a historical overview and comparing several possible definitions. We explore connections between inner functions, zero-divisors for Hardy spaces and Bergman spaces, and weighted reproducing kernels. After recording some obstructions and negative results, we suggest avenues for further research and point out several open problems.
Polynomial Null Solutions to Bosonic Laplacians, Bosonic Bergman and Hardy Spaces
本文研究了玻色子拉普拉斯算子的多项式零解性质，提出玻色子伯格曼空间并探讨其再生核与投影，进一步分析了玻色子哈代空间的结构与相关结果。
Minimal $L^2$ integrals for the Hardy spaces and the Bergman spaces
本文研究了Hardy空间和Bergman空间的最小$L^2$积分及其关系，给出了Saitoh猜想有限点版本的解，并提出了Hardy空间的最优$L^2$延拓定理及等式成立的刻画。
Embedding and compact embedding between Bergman and Hardy spaces
本文确定了${\rm{\bf C}}^n$开单位球上Hardy空间与加权Bergman空间间嵌入及紧嵌入的精确条件，覆盖所有可能情况，引入“紧拟合”概念并提出相关猜想。
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…
Let $D$ be a domain in the complex plane $\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\infty]$ such that $f$ belongs to the classical Hardy space $H^p (\mathbb{D})$ whenever $0<p<h(D)$ and $f$ is holomorphic on the unit disk $\mathbb{D}$ with values in $D$. As an analogue notion to the Hardy number of a domain $D$ in $\mathbb{C}$, we introduce the Bergman number of $D$ and we denote it by $b(D)$. Our main result is that, if $D$ is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply con…
Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-α}$, $0<α<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.
On the flatness of the Bergman space as a module over the Hardy algebra
研究复平面开单位圆盘上伯格曼空间作为哈代代数模的平坦性问题，通过特定运算证明其非平坦，且存在极大理想使\(\mathfrak{m}A^2 = A^2\)。
We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to prove a converse to an extension of Ryabykh's theorem about Hardy space regularity for Bergman space extremal functions. We also prove that the $H^q$ norm of the Szegö projection of $F^{p/2} \overline{F}^{(p/2)-1}$ cannot be too small if $F$ is analytic, for certain values of $p$ and $q$. We apply this to show that the best analytic approximation in $L^p$ o…
This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbolβ}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space $A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply in finite-variable setting.
In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, ψ, ϕ}$ on weighted Bergman spaces $\mathcal{A}^2_α$ with respect to the conjugation $C_{μ, η}$ for $μ, η\in \{z\in \mathbb{C}:|z|=1\}$. We obtain explicit conditions for which the operator $D_{n, ψ, ϕ}$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces.
Trace ideal criteria for generalized integration operators
本文研究了作用于单位圆盘Hardy与Bergman空间的广义积分算子类$T_{g,a}$，完全刻画了使其属于Schatten-von Neumann理想的符号$g$的空间。
We construct an example to show that no condition of slow decrease of the modulus of a function is sufficient to make it cyclic in the Hardy space of the bidisc. This is similar to what is well known in the case of the Hardy space of the disc, but in contrast to the case of the Bergman space of the disc.
We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy spac…
This paper is based on the course \lq\lq Weighted Hardy-Bergman spaces\rq\rq\, I delivered in the Summer School \lq\lq Complex and Harmonic Analysis and Related Topics\rq\rq at the Mekrijärvi research station of University of Eastern Finland, June $2014$. The main purpose of this survey is to present recent progress on the theory of Bergman spaces $A^p_ω$, induced by radial weights $ω$ satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$.
Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces
本文刻画了使广义Hilbert算子$\mathcal{H}_{\mu,\alpha}$在Hardy空间上有定义并满足积分表示，且从$H^p$到加权Bergman空间$A_{\alpha-2}^q$有界或紧的测度条件。
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。
On harmonic quasiregular mappings in Bergman spaces
本文证明拟正则调和函数满足 Hardy–Littlewood 型 Bergman 空间结论，并给出单叶调和映射及其偏导数属于 a^p 的非平凡指数范围，推广至调和拟共形映射。
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions
本文定义了基于 Dunkl 算子的 $λ$-解析函数，并研究了相关 Bergman 和 Hardy 空间的投影有界性、函数增长、稠密性、完备性、对偶空间及刻画与插值等问题。
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…
We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we…
A sequence which is a finite union of interpolating sequences for $H^\infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f \mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also…
Hausdorff operators on weighted Bergman and Hardy spaces
本文研究上半平面加权Bergman空间与幂加权Hardy空间上的Hausdorff算子，给出其定义并探讨该算子实版本的相关应用，属于复分析与调和分析领域的算子理论研究。
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
A generalization of the conjugate Hardy $H^2$ spaces
本文研究了共轭Hardy $H^2$ 空间的一个推广，探讨了其最小范数性质及与最小 $L^2$ 积分的关系，并应用于平面区域上共轭Hardy $H^2$ 核与Bergman 核的单调性及相互关系研究。
Composition operators on function spaces on the halfplane: spectra and semigroups
本文研究了右半平面 Zen 空间上的复合算子，推广了关于范数、本质范数、（本质）谱及半群的已有工作，并通过 Hardy--Bergman 空间实例说明结果。
In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obt…
We study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…
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 the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.
We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $φ$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_α$. We investigate the optimal $β$ such that the composition operator $\mathcal{C}_φ$ maps $\mathcal{D}_α$ boundedly into $\mathcal{D}_β$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and …
The Bergman and the growth numbers of domains and hyperbolic geometry
研究刻画一般区域伯格曼数的方法，通过在增长空间定义增长数，证明伯格曼数与增长数的等式，给出特定区域示例解决前人问题，还将类似思路用于布洛赫型空间。
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