Zygmund-type Spaces Research Articles

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

On sum of weighted differentiation composition operators from Bergman spaces with admissible weights to Zygmund type spaces

On sum of weighted differentiation composition operators from Bergman spaces with admissible weights to Zygmund type spaces

Weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces

Abstract A family of Zygmund-type spaces, called Dirichlet–Zygmund-type spaces, are introduced. The boundedness, compactness and the essential norm of weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces are also investigated in this paper.

Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces

<abstract><p>Let $ u, v $ be two analytic functions on the open unit disk $ {\mathbb D} $ in the complex plane, $ \varphi $ an analytic self-map of $ {\mathbb D} $, and $ m, n $ nonnegative integers such that $ m < n $. In this paper, we consider the generalized Stević-Sharma type operator $ T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z)) $ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness.</p></abstract>

Product-Type Operators from Hardy Spaces to Bloch-Type Spaces and Zygmund-Type Spaces

The boundedness and compactness of a class of product-type operators from Hardy spaces to (little) Bloch-type space and (little) Zygmund-type space are characterized in this article. Meanwhile, we also estimate the norm of

On Stević-Sharma Operators from General Class of Analytic Function Spaces into Zygmund-Type Spaces

A Stevi c ′ -Sharma operator denoted by T ψ 1 , ψ 2 , φ is a generalization product of multiplication, differentiation, and composition operators. In this paper, we characterize the bounded and compact Stevi c ′ -Sharma operator T ψ 1 , ψ 2 , φ from a general class X of Banach function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. Using several restrictive terms, we show that all bounded operators T ψ 1 , ψ 2 , φ from X into the little Zygmund-type spaces are compact. As an application, we show that our results hold up for some other domain spaces of T ψ 1 , ψ 2 , φ , such as the Hardy space and the weighted Bergman space.

Weighted Composition Operators from Dirichlet-Zygmund Spaces into Zygmund-Type Spaces and Bloch-Type Spaces

The boundedness, compactness, and essential norm of weighted composition operators from Dirichlet-Zygmund spaces into Zygmund-type spaces and Bloch-type spaces are investigated in this paper.

Product-Type Operators on the Space of Fractional Cauchy Transforms

The space of fractional Cauchy transforms plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory. In this paper, we study the boundedness and compactness of product-type operators from the space of fractional Cauchy transforms to the Zygmund-type space in terms of the function theoretic characterization of Julia–Carathéodory type.

Product-type operators acting between Dirichlet and Zygmund-type spaces

LetD = {z ? C : |z| < 1} be the open unit disk in the complex plane C. By H(D), denote the space of all holomorphic functions on D. For an analytic self map ? on D and u, v ? H(D), we have a product type operator Tu,v,? defined by Tu,v,? f (z) = u(z) f (?(z)) + v(z) f ?(?(z)), f ? H(D), z ? D, This operator is basically a combination of three other operators namely composition operator, multiplication operator and differentiation operator. We study the boundedness and compactness of this operator from Dirichlet-type spaces to Zygmund-type spaces.

On Stević-Sharma operators from the minimal Möbius invariant space into Zygmund-type spaces

On Stević-Sharma operators from the minimal Möbius invariant space into Zygmund-type spaces

On Stevic-Sharma operator from Qk(p,q) space to Zygmund-type space

The aim of this paper is to investigate the boundedness and compactness of Stevic-Sharma operator T?1,?2,? from QK(p, q) and QK,0(p,q) spaces to Zygmund-type space and little Zygmund-type space. We also give the upper and lower estimations for the norm of T?1,?2,?.

Integral Superposition-Type Operators on Some Analytic Function Spaces

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-called n -generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.

New Characterizations for the Essential Norms of Generalized Weighted Composition Operators between Zygmund-Type Spaces

We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund-type spaces. Consequently, we obtain new characterizations for the compactness of such operators.

Product-Type Operators from the Bloch Space into Zygmund-Type Spaces

Some characterizations for the boundedness, compactness and essential norm of a class of product-type operators $$T^n_{u,v,\varphi }$$ from the Bloch space and little Bloch space into Zygmund-type spaces are given in this paper.

BSE Properties of Some Banach Function Algebras

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (X, d) and Dales-Davie algebras D(X, M) are BSE-algebras for certain underlying plane sets X. Moreover, we investigate BSE properties of certain subalgebras of Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces $${{\cal B}_\alpha}$$ and Zygmund type spaces Zα are also investigated in different cases of α ∈ ℝ.

On Stević-Sharma operator from the mixed-norm spaces to Zygmund-type spaces

On Stević-Sharma operator from the mixed-norm spaces to Zygmund-type spaces

Operator Norms and Essential Norms of Weighted Composition Operators from Analytic Function Spaces into Zygmund-type Spaces

In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk into Zygmund-type spaces. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little Zygmund-type space are compact and characterize such operators. We apply our results to the cases when X is the Hardy space $$H^p$$ for $$1\le p\le \infty $$ and the weighted Bergman space $$A_\alpha ^p$$ for $$\alpha >-1$$ and $$1\le p<\infty $$ .

Weighted Composition Operators from Besov Zygmund-Type Spaces into Zygmund-Type Spaces

The boundedness, compactness, and essential norm of weighted composition operators from Besov Zygmund-type spaces into Zygmund-type spaces are investigated in this paper.

The Norm of a Composition Operator from the Space of Cauchy Transforms into Zygmund-Type Spaces

The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm of the composition operator acting between the space of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting between these spaces.

Essential Norm of Integral Type Operators Mapping into Certain Banach Spaces of Analytic Functions

We study certain integral type operators acting on a large class of Banach spaces of analytic functions on the open unit disc. The operators map into weighted Banach spaces of analytic functions, Bloch type spaces or Zygmund type spaces. Besides characterizing boundedness, we give essential norm estimates of these operators. In order to investigate these operators, we also study weighted differentiation composition operators between these spaces.