Iterates of a Berezin-type transform

Let B be the open unit ball of Rn and dV denote the Lebesgue measure on Rn normalized so that the measure of B equals 1. Suppose f∈L¹(B,dV). The Berezin-type transform of f is defined by

     

The Berezin transform and radial operators

We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial symbol.

     

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator (E+Λ), where Λ=(−Δ)^1/2, Δ is the Laplacian and E is the identity operator.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

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.     

Berezin transform on harmonic Bergman spaces on the real ball

We provide the full asymptotic expansion of the harmonic Berezin transform on the unit ball in Rⁿ${\mathbb{R}}^n$ purely by means of transformations of hypergeometric functions and function?s “hypergeometrization”.     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z|$|z|$ on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

On the derivatives of the Berezin transform

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator on the Segal-Bargmann space, the Berezin transform of is a function whose partial derivatives of all orders are bounded. Similarly, if is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives' of any order of the Berezin transform of are bounded. Further generalizations are also discussed.

     

Poisson transform on Boehmians

Convolution theorem for Poisson transform on compactly supported distributions is obtained. Applying the convolution theorem, Poisson transform is extended as a linear continuous map from a suitable Boehmian space into another Boehmian space satisfying the convolution theorem.     

Canonical and Boundary Representations on the Lobachevsky Plane

Canonical representations on Hermitian symmetric spaces G/K were introduced by Vershik, Gelfand and Graev and Berezin. They are unitary. We study canonical representations in a wider sense. In this paper we restrict ourselves to a crucial example – the Lobachevsky plane: G=SU(1,1), K=U(1). Canonical representations are labelled by the complex parameter (Vershik–Gelfand–Graev's representations correspond to –3/2<<0). We decompose the canonical representations into irreducible components. The decomposition includes boundary representations generated by the canonical representations. So we study these boundary representations themselves. The decomposition of boundary representations is closely connected with the meromorphic structure of Poisson and Fourier transforms associated with canonical representations. In particular, second-order poles give second-order Jordan blocks. Finally, we give a full decomposition of the Berezin transform using generalized powers (Pochhammer symbols) instead of usual powers of .     

Berezin transform on the harmonic Fock space

We show that the Berezin transform associated to the harmonic Fock (Segal-Bargmann) space on Cn has an asymptotic expansion analogously as in the holomorphic case. The proof involves a computation of the reproducing kernel, which turns out to be given by one of Horn's hypergeometric functions of two variables, and an ad hoc determination of the asymptotic behaviour of the resulting integrals, to which the ordinary stationary phase method is not directly applicable.     

A mean value theorem on bounded symmetric domains

Let be a Cartan domain of rank and genus and , , the Berezin transform on ; the number can be interpreted as a certain invariant-mean-value of a function around . We show that a Lebesgue integrable function satisfying , , must be -harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex -space , but with the role of radius played by the quantity .

     

非均匀法向荷载下半空间的二阶弹性效应问题

本文提供各向同性弹性半空间，在非均匀分布法向荷载下，二阶弹性效应的一个封闭形式解，运用积分变换方法，讨论了按Ｈｅｒｔｚ规律分布的荷载情形；导出了不可压缩各向同性弹性材料的极限解；算出了上述二阶弹性材料问题在ｚ方向的位移和法向应力数值。我们发现，与线弹性情形相比较，在二阶弹性材料中相应位移增大而法向应力减小。     

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.     

The finite integral transform method for a parabolic differential equation on a graph

The finite integral transform method is set forth and justified for solving a mixed problem for a parabolic differential equation posed on a graph.     

Toeplitz operators with BMO symbols on the weighted Bergman space of the unit ball

In this paper we prove that the boundedness and compactness of Toeplitz operator with a BMO _α ¹ symbol on the weighted Bergman space A _α ²(B _n ) of the unit ball is completely determined by the behavior of its Berezin transform, where α > −1 and n ≥ 1.