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.     

The harmonic Bergman kernels for punctured domains

We obtain the explicit formulae for the harmonic Bergman kernels of Bn／{0} and Rn／Bn and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel.We also get the explicit formula for the weighted harmonic Bergman kernel of Bn／{0} with the weight 1/｜x｜4.     

Harmonic Bergman functions as radial derivatives of Bergman functions

In the setting of the half-space of the euclidean -space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.

     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

The rank of Hankel operators on harmonic Bergman spaces

We show that on the harmonic Bergman spaces, the Hankel operators with nonconstant harmonic symbol cannot be of finite rank.

     

The space of bounded analytic functions and weighted Bergman spaces

In this paper, we construct a class of weighted Bergman spaces such that they can generate $H^{\infty }$ in the sense of Möbius invariance. $H^{\infty }$ is also equal to the intersection of these spaces. Applying the relation between $H^{\infty }$ and these spaces, we show that the characterization via higher order derivatives does not hold for some of these weighted Bergman spaces.     

调和Bergman度量

We have constructed the positive definite metric matrixes for the bounded domains of Rn and proved an inequality which is about the Jacobi matrix of a harmonic mapping on a bounded domain of Rn and the metric matrix of the same bounded domain.     

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

Algebraic properties of Toeplitz and small Hankel operators on the harmonic Bergman space

In this paper,we discuss some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasihomogeneous symbols on the harmonic Bergman space of the unit disk in the complex plane C.We solve the product problem of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.Meanwhile,we characterize the commutativity of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.     

Generic vanishing for harmonic spinors of twisted Dirac operators

In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections.

     

加权调和Bergman核及其诱导的度量矩阵

在Rn中的有界域上建立加权调和Bergman核,并得出单位球的加权调和Bergman核的表达式;利用加权调和Bergman核在Rn的有界域上构造度量矩阵;得到关于调和映射的Jacobi矩阵与度量矩阵之间的一个不等式.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

A ``deformation estimate" for the Toeplitz operators on harmonic Bergman spaces

Let denote the open unit ball in for and the Lebesgue volume measure on . For , the (weighted) harmonic Bergman space is the space of all harmonic functions which are in . For , the Toeplitz operator is defined on by , where is the orthogonal projection of onto . In this note, we prove that for radial, .

     

Positive Toeplitz Operators Between the Harmonic Bergman Spaces

On the setting of the upper half space we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact positive Toeplitz operators taking a harmonic Bergman space b ^p into another b ^q for 1<p<, 1<q<. The case p=1 or q=1 seems more intriguing and is left open for further investigation. Also, we give criteria for positive Toeplitz operators acting on b ² to be in the Schatten classes. Some applications are also included.     

Quantities equivalent to the norm of a weighted Bergman space

Let 0?α<∞, 0<p<∞, and p−α>−2. If f is holomorphic in the unit disc D and if ω is a radial weight function of secure type, then the followings are equivalent: