Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B$ and $ψ(z)$ a holomorphic function on $B$, where $B$ is the unit ball of ${\Bbbb C}^n$. Let $0<p,s<+\infty, -n-1<q<+\infty, q+s>-1$ and $α\geq 0,$ this paper gives some necessary and sufficient conditions for the weighted composition operator $\wco$ induced by $ϕ$ and $ψ$ to be bounded and compact between the space $\Fs$ and $α$-{\sl Bloch} space $β^α.$
The boundedness and compactness of weighted composition operators from $H^\infty$ to the Bloch space in the unit ball of Cn are investigated in this paper. In particular, some new characterizations for the boundedness and the essential norm of weighted composition operators are given.
When 0<p<1, it is known that the p-Bloch and (1-p)-Lipschitz spaces of the unit ball in n-dimensional complex Eucllidean space are equal as sets. We prove that these spaces are additionally norm-equivalent, thus extending known results for n=1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of composition operators between p- and q-Lipschitz spaces of the ball.
Let $g$ be a holomorphic map of $B$, where $B$ is the unit ball of ${C}^n$. Let $0<p<+\infty, -n-1<q<+\infty$, $q>-1$ and $α>0$. This paper gives some necessary and sufficient conditions for the Extended Ces$\acute{a}$ro Operators induced by $g$ to be bounded or compact between generalized Besov space $B(p,q)$ and $α$- Bloch space ${\mathcal B}^α.$
We give some new estimates for the norm and essential norm of a weighted composition operator on the Bloch space. As corollaries, we obtain some new characterizations of the boundedness and compactness of a weighted composition operator on the Bloch space.
In this paper, we give two new characterizations for the boundedness and compactness of the difference of two weighted composition operators acting from $H^\infty$ to the Bloch space.
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 this article, we initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.
We found several new equivalent characterizations for the boundedness and compactness of the differences of weighted differentiation composition operators from Bloch-type space to weighted-type space. Especially, we estimated its essential norm in terms of the $n$-th power of the induced analytic self-maps on the unit disk.
In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in $\mathbb{C}^n$ with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space $H^\infty$.
On the space of bounded analytic functions and the Bloch space on the unit disk, we study the compact intertwining relations for composition operators, whose intertwining operators are Volterra type operators. Further, we consider the compact intertwining relations, which are between the whole collection of composition operators and some Volterra operator, and the whole collection of bounded Volterra operators and some composition operator.
Let $\mathbb{B}$ be the unit ball of a complex Banach space $X$. In this paper, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $\mathbb{B}$ by using the radial derivative. Next, we define an extended Cesàro operator $T_φ$ with holomorphic symbol $φ$ and characterize those $φ$ for which $T_φ$ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those $φ$ for which $T_φ$ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol $φ$. When $\mat…
Let p,q>0. We extend to the n-polydisk previous one-variable characterization results of K. Madigan on the $p$-Lipschitz space and K. Madigan/A. Matheson on the Bloch space by obtaining function-theoretic conditions on a holomorphic self-map of the polydisk such that the induced composition operator is bounded or compact between p- and q-Bloch spaces of the polydisk. These conditions turn out to be different in the cases when p is in (0,1) and when p is at least 1. We also obtain corresponding characterization results for composition operators between generalized little p- and q-Bloch spaces o…
Let $ϕ$ and $ψ$ be holomorphic self-maps of the unit polydisc $U^n$ in the $n$-dimensional complex space, and denote by $C_ϕ$ and $C_ψ$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_ϕ-C_ψ$ from Bloch space to bounded holomorphic function space in the unit polydisc. Moreover the compactness of the difference is also characterized.
Let $λ_i (i=1,...,k)$ be any nonzero complex scalars and $φ_i (i=1,..,k)$ be any analytic self-maps of the unit disk $\mathbb{D}$. We show that the operator $\sum_{i=1}^kλ_iC_{φ_i}$ is compact on the Bloch space $\mathcal{B}$ if and only if $$\lim_{n\to\infty}\|λ_1φ_1^n+λ_2φ_2^n+...+λ_kφ_k^n\|_{\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.
Makarov's principle relates three characteristics of Bloch functions that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, we show that the universal bounds agree if we take the supremum over the Bloch unit ball. For the supremum (of either of these quantities), we give the estimate $Σ^2_{\mathcal B} < \min(0.9, Σ^2)$, where $Σ^2$ is the analogous quantity associated to the unit ball in the $L^\infty$ norm on t…
In this paper, we study the boundedness and compactness of the differences of two weighted composition operators acting from $α$-Bloch space to $β$-Bloch space on the open unit disk. This study has a relationship to the topological structure of weighted composition from $α$-Bloch space to $β$-Bloch space.
Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…
In this paper, we study the boundedness and essential norms of the differences of two generalized composition operators acting from $α$-Bloch space to $β$-Bloch space on the open unit disk. From essential norms, we get the compactness of the differences of two generalized composition operators. This study has a relationship to the topological structure of generalized composition operators acting from $α$-Bloch space to $β$-Bloch space.
We characterize the analytic self-maps $ϕ$ of the unit disk ${\Bbb D}$ in ${\Bbb C}$ that induce continuous composition operators $C_ϕ$ from the log-Bloch space $\mathcal{B}^{\log}({\Bbb D})$ to $μ$-Bloch spaces ${\mathcal B}^μ({\Bbb D})$ in terms of the sequence of quotients of the $μ$-Bloch semi-norm of the $n$th power of $ϕ$ and the log-Bloch semi-norm (norm) of the $n$th power $F_n$ of the identity function on ${\Bbb D}$, where $μ:{\Bbb D}\rightarrow (0,\infty)$ is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of $C_ϕ$ between these spaces, t…
Denote by $ B_X $ the unit ball of an infinite-dimensional complex Hilbert space $ X. $ Let $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X. $ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $ \mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X) $ (and $ \mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $ C_{ψ,φ} $ via the estimates for the restrictions of $ ψ$ …
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's rel…
Let $U^{n}$ be the unit polydisc of ${\Bbb C}^{n}$ and $ϕ=(ϕ_1, >..., ϕ_n)$ a holomorphic self-map of $U^{n}.$ By ${\cal B}^p(U^{n})$, ${\cal B}^p_{0}(U^{n})$ and ${\cal B}^p_{0*}(U^{n})$ denote the $p$-Bloch space, Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc $U^n$ respectively, where $p, q>0$. This paper gives the estimates of the essential norms of bounded composition operators $C_ϕ$ induced by $ϕ$ between ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) and ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As their applic…
In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk $\mathbb{D}$. Then we show that a harmonic Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic Bloch spa…
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.
The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.
Suppose $φ$ is a holomorphic self map of the unit disk and $C_φ$ is a composition operator with symbol $φ$ that fixes the origin and $0<|φ'(0)|<1$. This work explores sufficient conditions that ensure all holomorphic solutions of Schröder equation for the composition operator $C_φ$ belong to a Bloch-type space $\mathcal{B}_α$ for some $α>0$. The results from composition operators have been extended to weighted composition operators in the second part of this work.
Let $Ω$ be a bounded symmetric domain except the two exceptional domains of ${\Bbb C}^N$ and $ϕ$ a holomorphic self-map of $Ω.$ This paper gives a sufficient and necessary condition for the composition operator $C_ϕ$ induced by $ϕ$ to be compact on the Bloch space $β(Ω)$.
Let $ϕ(z)=(ϕ_1(z), ...,ϕ_n(z))$ be a holomorphic self-map of $U^n$ and $ψ(z)$ a holomorphic function on $U^n,$ where $U^n$ is the unit polydisk of ${\Bbb C}^n.$ Let $p\geq 0,$ $q\geq 0$, this paper gives some necessary and sufficient conditions for the weighted composition operator $W_{ψ,ϕ}$ induced by $ψ$ and $ϕ$ to be bounded and compact between $p$-Bloch space ${\cal B}^p(U^n)$ and $q$-Bloch space ${\cal B}^q(U^n).$
In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of $\mathbb{C}^n$.
We give embedding theorems for weighted Bergman-Orlicz spaces on the ball and then apply our results to the study of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman-Orlicz spaces, different from $H^{\infinity}$, on which every composition operator is bounded.
Hölder continuity and composition operators in pluriharmonic Bloch spaces
Huaqiao University（华侨大学）；Guangdong Technion - Israel Institute of Technology（广东以色列理工学院）；Shantou University（汕头大学）；Technion - Israel Institute of Technology（以色列理工学院）
We obtain some estimates for norm and essential norm of the difference of two composition operators between weighted Bergman spaces $A^p_α$ and $A^q_β$ on the unit ball. In particular, we completely characterize the boundedness and compactness of $C_φ-C_ψ: A^p_α\to A^q_β$ for full range $0<p,q<\infty$, $-1<α,β<\infty$.
In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $\mathbb{B}$ in quaternions $\mathbb{H}$ through the slice regular rotation $\widetilde{f}_{u}$ of each slice regular function $f:\overline{\mathbb{B}}\rightarrow \mathbb{H}$ with $f'(0)=1$ for some $u\in \partial\mathbb{B}$.
We find a lower bound for the essential norm of the difference of two composition operators acting on $H^2(B_N)$ or $A^2_s(B_N)$ ($s>-1$). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.
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 $A_α^2(\mathbb{D})$.
This paper deals with the the norm of the weighted Bergman projection operator $P_α:L^\infty\to \mathcal{B}$ where $α>-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space $\mathbf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.
Let μbe any weight function defined on the unit disk $\Bbb D$ and let $ϕ$ be an analytic self-map of $\Bbb D$. In the present paper we show that the essential norm of composition operator $C_ϕ$ mapping from the $α$-Bloch space, with $α>0$, to $μ$-Bloch space $\mathcal{B}^μ$ is comparable to $$ \limsup_{|a|\to 1^-}\|σ_a\circϕ\|_{\mathcal{B}^μ}, $$ where, for $a\in\Bbb D$, $σ_a$ is a certain special function in $α$-Bloch space.
In this paper, we investigate some properties of pluriharmonic mappings defined in the unit ball. First, we discuss some geometric univalence criteria on pluriharmonic mappings, and then establish a Landau-Bloch theorem for a class of pluriharmonic mappings.
Let $\mathcal{B}^ω(B_d)$ denote the $ω$-weighted Bloch space in the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. We show that the quadratic integral $$ \int_x^1 \frac{ω^2(t)}{t}\, dt,\quad 0<x<1, $$ governs the radial divergence and integral reverse estimates in $\mathcal{B}^ω(B_d)$.
Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B_n$ and $ψ(z)$ a holomorphic function on $B_n$, and $H(B_n)$ the class of all holomorphic functions on $B_n$, where $B_n$ is the unit ball of $C^n$, the weight composition operator $W_{ψ,ϕ}$ is defined by $W_{ψ,ϕ}=ψf(ϕ)$ for $f\in H(B_n)$. In this paper we estimate the essential norm for the weighted composition operator $W_{ψ,ϕ}$ acting from the Hardy space $H^p$ to $H^q$ ($0<p,q\leq \infty$). When $p=\infty$ and $q=2$, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necess…
In this paper we give some results concerning Frechet differentiable mappings between domains in normed spaces with controlled growth. The results are mainly motivated by Pavlovic's equality for the Bloch semi-norm of continuously differentiable mappings in the Bloch class on the unit ball of the Euclidean space as well as the very recent Jocic's generalization of this result.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.
This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers acting on M${\rm \ddot{o}}$bius invariant spaces $Q_p$, which unify BMOA and Bloch space in the scale of $p$. The boundedness and compactness of these operators on $Q_p$ spaces are determined by means of an embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic tent type spaces $T_q^\infty$.
In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $α$-Bloch spaces.
We consider a class of two-parameter weighted integral operators induced by harmonic Bergman-Besov kernels on the unit ball of $\mathbb{R}^{n}$ and characterize precisely those that are bounded from Lebesgue spaces $L^{p}_α$ into Harmonic Bergman-Besov $b^{q}_β$ or weighted Bloch Spaces $b^{\infty}_β $, for $1\leq p\leq\infty$, $1\leq q< \infty$ and $α,β\in \mathbb{R}$. These operators can be viewed as generalizations of the harmonic Bergman-Besov projections. Also, our results remove the disturbing conditions $β>-1$ when $q<\infty$ and $β\geq 0$ when $q=\infty$ of Doğan (A Class of Integral O…
In this paper, we investigate some properties on harmonic functions and solutions to Poisson equations. First, we will discuss the Lipschitz type spaces on harmonic functions. Secondly, we establish the Schwarz-Pick lemma for harmonic functions in the unit ball $\IB^n$ of $\IR^n$, and then we apply it to obtain a Bloch theorem for harmonic functions in Hardy spaces. At last, we use a normal family argument to extend the Landau-Bloch type theorem to functions which are solutions to Poisson equations.
We consider weighted Bergman projection $P_α: L^{\infty}(\Bbb B) \rightarrow {\cal B} $ where $α>-1$ and $\cal B$ is the Bloch space of the unit ball $\Bbb B$ of the complex space $\Bbb C^n.$ We obtain the exact norm of the operator $P_α$ where the Bloch space is observed as a space with norm (and semi-norm) induced from the Besov space $B_{p},0<p<\infty,(B_{\infty}=\cal B).$ Our work contains, as a special case, the main results from \cite{Kalaj} and \cite{Ant}.
Let $H(\mathbb{B})$ denote the space of all holomorphic functions on the unit ball $\mathbb{B}\in \mathbb{C}^n$. In this paper we investigate the boundedness and compactness of the products of radial derivative operator and the following integral-type operator $$ I_ϕ^g f(z)=\int_0^1 \Re f(ϕ(tz))g(tz)\frac{dt}{t},\ z\in\mathbb{B} $$ where $g\in H(\mathbb{B}), g(0)=0$, $ϕ$ is a holomorphic self-map of $\mathbb{B}$,\ between Zygmund spaces and Bloch spaces.
The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc $\mathbb{D}$ under a holomorphic function $f$ (such that $f(0)=0$ and $f'(0)=1$) always contains an open disc with radius larger than a universal constant. In this paper we prove a Bloch-Landau type Theorem for slice regular functions over the skew field $\mathbb{H}$ of quaternions. If $f$ is a regular function on the open unit ball $\mathbb{B}\subset \mathbb{H}$, then for every $w \in \mathbb{B}$ we define the regular translation $\tilde f_…
