Weighted Composition Operators from the Bloch Space to Weighted Banach Spaces on Bounded Homogeneous Domains

We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide ＂computable＂ estimates on the operator norm.

EXTENDED CESRO OPERATORS ON THE BLOCH SPACE IN THE UNIT BALL OF C^n

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn asWhere is the radial derivative of g. In this paper, the author characterizes g for which Tg is bounded (or compact) on the Bloch space B and the little Bloch space Bo-

PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES

For analytic functions u,ψin the unit disk D in the complex plane and an analytic self-mapφof D,we describe in this paper the boundedness and compactness of product type operators T_(u,ψ,φ)f(z)=u(z)f(φ(z))+ψ(z)f'(φ(z)),z∈D,acting between weighted Bergman spaces induced by a doubling weight and a Bloch type space with a radial weight.

A DERIVATIVE-HILBERT OPERATOR ACTING FROM LOGARITHMIC BLOCH SPACES TO BERGMAN SPACES

Letμbe a positive Borel measure on the interval[0,1).The Hankel matrix■with entriesμn,k=μn+k,whereμn=⨜[0,1)tndμ(t),induces,formally,the operator■where■is an analytic function in.We characterize the measuresμfor which■is bounded(resp.,compact)operator from the logarithmic Bloch space■into the Bergman space■,where 0≤α<∞,0<p<∞.We also characterize the measuresμfor which■is bounded(resp.,compact)operator from the logarithmic Bloch space■into the classical Bloch space■.

CLOSURE OF ANALYTIC FUNCTIONS OF THE BOUNDED MEAN OSCILLATION IN LOGARITHMIC BLOCH SPACES

For anyα∈R,the logarithmic Bloch space BLαconsists of those functions f which are analytic in the unit disk D with.■In this paper,we characterize the closure of the analytic functions of bounded mean oscillation BMOA in the logarithmic Bloch space BLαfor allα∈R.

Compactness of Composition Operators from the Bloch Space β to Q_K Spaces

Suppose that φ is an analytic self-map of the unit disk Δ. We consider compactness of the composition operator Cφ from the Bloch space B into the spaces QK defined by a nonnegative, nondecreasing function K（r） for 0 ≤ r 〈 Cφ. Our compactness condition depends only on Φ which can be considered as a slight improvement of the known results. The compactness of Cφ from the Dirichlet space D into the spaces QK is also investigated,

Boundedness from below of composition operators on the Bloch spaces

This paper proves that a necessary condition for a composition operator on the Bloch space of the unit disk to be bounded below given by Ghatage, Yan and Zheng is also sufficient. Given furthermore are a sufficient condition and a necessary condition of the boundedness from below for a composition operator on the Bloch space of a unit ball with dimensions bigger than one.

COMPOSITION OPERATORS FROM p-BLOCH SPACES TO LITTLE q-BLOCH SPACES ON THE UNIT BALL OF C^n

For all 0 〈 p, q 〈 ∞, let Cφ denote the composition operator from q-Bloch spaces βp to little p-Bloch spaces β0q on the unit ball of C^n. In this article, necessary and sufficient conditions for Cφ to be a bounded or compact operator are given.

COMPOSITION OPERATORS BETWEEN BLOCH TYPE SPACES IN THE UNIT BALL

In this paper, we obtain some new necessary and sufficient conditions for the boundedness and compactness of composition operators Cφ between Bloch type spaces in the unit ball Bn.

COMPOSITION TYPE OPERATORS FROM HARDY SPACES TO μ-BLOCH SPACES ON THE UNIT BALL

Let φ be a holomorphic self-map of Bn and ψ ∈ H（Hn）. A composition type operator is defined by Tψ,φ（f） = ψf o φ for f ∈ H（Bn）, which is a generalization of the multiplication operator and the composition operator. In this article, the necessary and sufficient conditions are given for the composition type operator Tψ,φ to be bounded or compact from Hardy space HP（Bn） to μ-Bloch space Bμ（Bn）. The conditions are some supremums concerned with ψ,φ, their derivatives and Bergman metric of Bn. At the same time, two corollaries are obtained.

ON DIRICHLET TYPE SPACES AND α-BLOCH SPACES IN THE UNIT BALL OF C^n

In this paper, the authors study the inclusion relations between Dirichlet type spaces DΥT and α-Bloch spaces βα by means of higher radial derivative. The strictness and the best possibility of the inclusion relations are shown with constructive methods. Furthermore, they sharpen one of the results when Υ=n, which proves that a conjecture in [7] is true.

<i>Q_K</i>Type Spaces and Bloch Type Spaces on the Unit Ball

Different function spaces have certain inclusion or equivalence relations. In this paper, the author introduces a class of M&#246;bius-invariant Banach spaces QK,0 (p,q) of analytic function on the unit ball of Cn, where K:(0,∞)→[0,∞) are non-decreasing functions and 0P∞, p/2-n-1q∞, studies the inclusion relations between QK,0 (p,q) and a class of B0α spaces which was known before, and concludes that QK,0 (p,q) is a subspace of B0(q+n+1)/p, and the sufficient and necessary condition on kernel function K(r) such that QK,0 (p,q)= B0(q+n+1)/p.

EQUIVALENCES BETWEEN BLOCH TYPE SPACES AND BMO SPACES ON THE HYPERBOLIC DOmAINS IN C

For an analytic function f on the hyperbolic domain Ω in C,the following conclusions are obtained: (i)f∈B(Ω)=BMOA(Ω,m)if and only if Ref∈B(?)(Ω)=BMOH(Ω,m).(ii)QB_h(Ω)=B_h(Ω) (BMOH,(Ω,m)=BMOH(Ω,m)if and only if C(Ω)=inf{Z_o(z)·δ_o(z)·z≡Ω}>0,Also some applica- lions to automorphic function are considered.

DERIVATIVES OF HARMONIC MIXED NORM AND BLOCH-TYPE SPACES IN THE UNIT BALL OF R^n

Let H（B） be the set of all harmonic functions f on the unit ball B of Rn. For 0 p, q ≤ ∞ and a normal weight φ, the mixed norm space Hp,q, φ（B） consists of all functions f in H（B） for which the mixed norm p,q, φ ∞. In this article, we obtain some characterizations in terms of radial, tangential, and partial derivative norms in Hp,q, φ（B）. The parallel results for the Bloch-type space are also obtained. As an application, the analogous problems for polyharmonic functions are discussed.

BOUNDEDNESS FROM BELOW OF MULTIPLICATION OPERATORS BETWEEN α-BLOCH SPACES ON THE UNIT BALL

In this article, we characterize the boundedness from below of multiplication operators between α-Bloch spaces on the unit ball.

CRITERIA FOR AN ANALYTIC FUNCTION BELONGING TO THE Q_K SPACES

The author gives a mild integral condition in a nondecreasing function K : [0, ∞) → [0, ∞), which is sufficient and the best possible to ensure that f is a Bloch function if and only if f belongs to QK, a Mbius-invariant space of functions analytic in the unit disk. Their contributions are slight improvements of known results, and the proofs presented here are independently developed. The corresponding results for meromorphic case are also given.

On Subspaces of Bloch Space and Series with Hadamard Gaps

本文研究了由修改的Ｂｅｓｏｖ模和Ｃａｒｌｅｓｏｎ测度所定义的Ｂｌｏｃｈ空间的一类子空间．利用这类子空间与Ｈａｄａｍａｒｄ缺项级数之间的关系，证明了他们之间的包含关系．作为所得结果的一个简单推论，本文对Ｋ．Ｓｔｒｏｅｔｈｏｆ关于ＢＭＯＡ和ＶＭＯＡ的一个未解决的问题作出了否定的回答．

Composition Operators on the Bloch Space of Several Complex Variables

In this paper,we study the boundedness and compactness of composition operator C on the Bloch space β（Ω）,Ω being a bounded homogeneous domain.For Ω=Bn,we give the necessary and sufficient conditions for a composition operator C to be compact on β（Bn）or β0（Bn）.

On the composition operators on the Bloch space of several complex variables

In this paper we first look upon some known results on the composition operator as bounded or compact on the Bloch-type space in polydisk and ball, and then give a sufficient and necessary condition for the composition operator to be compact on the Bloch space in a bounded symmetric domain.

Compact composition operators on the Bloch space in polydiscs

Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for every ε > 0, there exists a δ > 0, such that $$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$ whenever dist(φ(z), ?U n )<δ.