Multipliers between $ BMO $ Spaces on Open Unit Ball

and spaces consist of all those locally integrable functions on the open unit ball such that their mean oscillations on any Bergman ball are, respectively, bounded and vanishing to zero as the Bergman ball approaches the boundary of the unit ball. The harmonic (or analytic) versions of the and spaces are the Bloch space and the little Bloch space, respectively. The study of these spaces plays an important role in modern analysis, especially in the the area of operator theory and holomorphic spaces. In this paper, we characterize systematically the multipliers of and spaces. Submitted: January 9, 2001?Revised: October 16, 2002     

单位球加权Bergman空间上的紧算子

In this paper,we study the compact operators on weighted Bergman spaces of the unit ball.Extending Miao and Zheng’result in 2004,we obtain the necessary and sufficient conditions for the operator to be compact on weighted Bergman spaces of the unit ball under some integrable conditions.     

Commuting dual Toeplitz operators on weighted Bergman spaces of the unit ball

In this paper, we study the commutativity of dual Toeplitz operators on weighted Bergman spaces of the unit ball. We obtain the necessary and sufficient conditions for the commutativity, essential commutativity and essential semi-commutativity of dual Toeplitz operator on the weighted Bergman spaces of the unit ball.     

Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces

Recently, Schatten-Herz type Toeplitz operators have been studied on the Bergman spaces and the harmonic Bergman spaces. Motivated these results, we study characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman spaces.     

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Double integral characterizations of harmonic Bergman spaces

Recently Li et al. have characterized, except for a critical case, the weighted Bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions. In this paper we extend those characterizations to the case of weighted harmonic Bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case. We also investigate the possibility of extensions to the half-space setting. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the half-space.     

A reproducing kernel thesis for operators on Bergman-type function 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.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

A^p（φ）上Toeplitz算子的换位子

本文使用不变加权面积平均值性质刻画了单位圆盘内的调和函数．由此我们探讨了加权Bergman空间A^p(φ)上的Toeplitz算子，给出了两个具有界调和符号的Toeplitz算子交换或本质交换的一些充要条件．     

A Theorem of Nehari Type on the Bergman Space

In this paper we concern with the characterization of bounded linear operators S acting on the weighted Bergman spaces on the unit ball. It is shown that, if S satisfies the commutation relation STzi = Tzi（i =1,..,n）, where Tzi = zif and Tzi= P（zif） where P is the weighted Bergman projection, then S must be a Hankel operator.     

Compactness of the Hardy-Littlewood operator on some spaces of harmonic functions

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ? ⁿ such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMO _p , and Dirichlet spaces.     

Ap(ψ)上Toeplitz算子的换位子

本文使用不变加权面积平均值性质刻画了单位圆盘内的调和函数.由此我们探讨了加权Bergman空间Ap(ψ)上的Toeplitz算子,给出了两个具有界调和符号的Toeplitz算子交换或本质交换的一些充要条件.     

Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L¹_a and spaces of operators on them, as well as the space C(T)/A, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a sequence {f_n} in the unit ball of A that converges weakly to f but not in the norm.     

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball

We give a necessary and sufficient condition for Hankel operators H_f on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered.     

单位球上Hardy空间之间的加权复合算子

We consider the weighted composition operators between Hardy spaces on the unit ball, and obtain some sufficient and necessary conditions of bounded or compact weighted composition operators. We also prove that the operator from H1 to H1 is compact if and only if it is weakly compact. Meanwhile, we get the analogue on the Bergman spaces.     

Hardy–Carleson Measures and Their Dual Poisson–Szegö Transforms

Recently Choe et al. have introduced the notion of dual Berezin transforms and used it to obtain new characterizations of the Carleson measures for the weighted Bergman spaces over the unit ball in C ⁿ . Continuing our investigation on the Hardy spaces, we obtain new characterizations of the Carleson measures for the Hardy spaces by means of the dual Poisson–Szegö transforms introduced by Koosis. Compared with the results for the weighted Bergman spaces, our results for the Hardy spaces not only show an similarity, but also reveal a new characterization.     

Harmonic mappings in Bergman spaces

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk $\mathbb D $ of $\mathbb C $ . Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball $\mathbb B ^{n}$ of $\mathbb C ^{n}$ . The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on $\mathbb B ^{n}$ in terms of their complex gradients.     

Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. The research of the second author is partially supported by a Fulbright grant.     

AN UPPER BOUND OF THE ESSENTIAL NORM OF COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

In this paper, we define the generalized counting functions in the higher dimensional case and give an upper bound of the essential norms of composition operators between the weighted Bergman spaces on the unit ball in terms of these counting functions. The sufficient condition for such operators to be bounded or compact is also given.