Reducibility and Unitarily Equivalence for a Class of Analytic Multipliers on the Dirichlet Space

In this paper, we first prove that if $\phi $ is a finite Blaschke product with $N=2,3$ zeros, then $M_\phi $ is reducible on the Dirichlet space if and only if $\phi $ is equivalent to $z^N$ . Also, we prove that $M_\phi $ is unitary equivalent to Dirichlet shift of multiplicity $N$ if and only if $\phi =\lambda z^N$ for some unimodular constant $\lambda $ .     

圆环上Dirichlet空间上的乘子

单位圆盘上的经典Dirichlet空间中的乘子比Hardy空间、Bergman空间复杂许多, 例如有界性问题是一个迄今悬而未决的问题, 很多基本问题都有待解决. 圆环作为一类典型的复连通区域, 其上的函数结构更复杂. 本文主要讨论圆环上的Dirichlet空间上乘子的可逆性与Fredholm 性质, 计算了具有洛朗多项式符号的乘子的谱与本性谱. 此外, 解决了文献[6]所提出的关于一般符号乘子的谱与本性谱问题.     

A Corona Theorem for Multipliers on Dirichlet Space

We prove a corona theorem for infinitely many functions from the multiplier algebra on Dirichlet space.     

Dirichlet型空间上的乘子

本文主要根据τ与μ的不同取值分三种情况刻划了Cn中单位球上Dirichlet 型空间上的点乘子空间M(Dτ,Dμ),并通过两个函数的构造表明:当τ≤n时,包含 关系M(Dτ)(?)Dτ和当τ>μ,τ>n-1时,包含关系M(Dτ)(?) M(Dμ)是严格的.     

Dirichlet空间上一类乘法算子的酉等价类

本文研究了Dirichlet空间D上由二阶Blaschke积φ定义的乘法算子M_φ的酉等价类.主要结果表明对于D的一个乘子ψ,乘法算子M_ψ酉等价于M_φ当且仅当存在常数θ,|θ|θ=1,使得ψ(z)=φ(θz).这一结果是完全不同于Hardy空间与Bergman空间上的相应结果.     

Multipliers on Dirichlet Type Spaces

In this paper, we characterize the pointwise multiplier space M (D _τ, D _μ) of Dirichlet type spaces in the unit ball of C ⁿ for the values of τ, μ in three cases: (i) τ < 0,μ < 0, (ii) τ < μ, (iii) τ≥μ, τ > n, and construct two functions to show that M(D _τ) ⊂D _τ properly if τ≤n and M(D _τ) ⊂M(D _μ) properly if τ > μ and τ > n− 1. Supported by the National Natural Science Foundation of China and the National Education Committee Doctoral Foundation of China     

On a Generalized Dirichlet Problem for Analytic Family

In this paper, we give the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron in Cn, and get a necessary and sufficient condition for a generalized Dirichlet problem to be solvable for an analytic family on a bounded holomorphic domain. Especially, we derive that this condition is just that the continuous real boundary value is prescribed on and only on the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron.     

Bergman空间上一类解析Toeplitz算子的强不可约性

令D为单位圆盘D={z∈C:|z|<1},L_a~2(D)为L~2(D)中解析函数构成的Bergman空间.设f(z)=a_0+a_1z+a_2z~2+…,用算子理论的技巧给出解析Toeplitz算子T_f为强不可约算子的一个充分条件.     

关于一些解析函数空间的系数乘子(英文)

In this paper, we give some new results of the coefficient multiplier of some analytic function spaces, characterize the coefficient multiplier spaces (Hα, p Hβ q) and (Ap,α), Aq,β) with 0 < p ≤ 1, p ≤ q ≤∞, 0 ≤α < β < ∞ and show (Hα∞, Hβ∞) = Hβ-α1 with 0 < α < β < ∞.     

Toeplitz Operators on the Dirichlet Space

In this paper a decomposition of Sobolev space is obtained. Then we prove that a Toeplitz operator on the Dirichlet space is compact only when it is the zero operator. For two Toeplitz operators on the Dirichlet space, we obtain the conditions for that they commute, their product is a Toeplitz operator, and their commutator or semi-commutator has finite rank, respectively.     

Sharp Spectral Multipliers for a New Class of Grushin Type Operators

We describe weighted restriction (Plancherel) type estimates and sharp Hebisch-Müller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting.     

Kato Space for Dirichlet Forms

We study the space of Kato measures relative to a Dirichlet form and we prove that a local solution of a problem relative to a Kato measure is locally continuous. Moreover if the measure of an intrinsic ball is equivalent to a power of the radius we prove also that the density of the form relative to a local solution is locally a Kato measure.     

Dirichlet Problem for a Class of Quasilinear Elliptic Equations

In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved.     

Reducibility of a Class of Operator Functions to Block-Diagonal Form

Mathematical Notes -     

Dirichlet空间上的紧算子

若S是Dirichlet空间上有限个Toeplitz算子乘积的有限和, S为紧算子的充要条件是: 当z→∂D时, S的Berezin型变换收敛到0; 若S是Dirichlet空间上Hankel算子, S为紧算子的充要条件是: 当z→ D时, S作用在类再生核上按范数收敛到0.     

On the Reducibility of a Class of Linear Almost Periodic Differential Equations

In this paper, we use KAM methods to prove that there are positive measure Cantor sets such that for small perturbation parameters in these Cantor sets a class of almost periodic linear differential equations are reducible.     

Bogomolov Multipliers for Some p-groups of Nilpotency Class 2

The Bogomolov multiplier B_0(G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G.The triviality of the Bogomolov multiplier is an obstruction to Noether's problem.We show that if G is a central product of G_1 and G_2,regarding K_i ≤ Z(G_i),i = 1,2,and θ:G_1→G_2 is a group homomorphism such that its restriction θ_(K_1):K_1→K_2 is an isomorphism,then the triviality of B_0(G_1/K_1),B_0(G_1) and B_0(G_2) implies the triviality of B_0(G).We give a positive answer to Noether's problem for all 2-generator p-groups of nilpotency class 2,and for one series of 4-generator p-groups of nilpotency class 2(with the usual requirement for the roots of unity).     

Quadratic Extremal Problems on the Dirichlet Space

A class of quadratic extremal problems over the unit ball of the Dirichlet space is studied. Existence of extremal functions is established. The boundary regularity of solutions is determined in terms of the regularity of the underlying functional. It is further shown that the extremal functions are entire functions in a range of cases.