Zygmund Spaces Research Articles

On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

We study regularity of solutions u u to ∂ ¯ u = f \overline \partial u=f on a relatively compact C 2 C^2 domain D D in a complex manifold of dimension n n , where f f is a ( 0 , q ) (0,q) form. Assume that there are either ( q + 1 ) (q+1) negative or ( n − q ) (n-q) positive Levi eigenvalues at each point of boundary ∂ D \partial D . Under the necessary condition that a locally L 2 L^2 solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1 / 2 1/2 derivative when q = 1 q=1 and f f is in the Hölder–Zygmund space Λ r ( D ) \Lambda ^r( D) with r > 1 r>1 . For q > 1 q>1 , the same regularity for the solutions is achieved when ∂ D \partial D is either sufficiently smooth or of ( n − q ) (n-q) positive Levi eigenvalues everywhere on ∂ D \partial D .

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.

The finite Hilbert transform acting in the Zygmund space LlogL

The finite Hilbert transform acting in the Zygmund space LlogL

Measure theoretic aspects of the finite Hilbert transform

AbstractThe finite Hilbert transform , when acting in the classical Zygmund space (over ), was intensively studied in [8]. In this note, an integral representation of is established via the ‐valued measure for each Borel set . This integral representation, together with various non‐trivial properties of , allows the use of measure theoretic methods (not available in [8]) to establish new properties of . For instance, as an operator between Banach function spaces is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for plays a crucial role.

On the spaces introduced by N. K. Karapetyants and B. S. Rubin, and their connection with Lorentz–Zygmund spaces

We consider spaces introduced by N. K. Karapetyants and B. S. Rubin in 1982, to characterize, in particular, the image of the fractional integral Riemann–Liouville operator. These spaces lie near L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{\\infty }$$\\end{document}. We show that they coincide with well-known Lorentz–Zygmund spaces. This allows us to reformulate one result from N. K. Karapetyants and B. S. Rubin dealing with Riemann–Liouville fractional integral operator J0+α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${J}_{0+}^{\\alpha }$$\\end{document} defined on Lp0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{p}\\left(\ ext{0,1}\\right)$$\\end{document} (1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty$$\\end{document}) in the borderline case α=1/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1/p$$\\end{document}. Using of the well-developed theory of Lorentz–Zygmund spaces leads to new results on the fractional integral Riemann–Liouville operator.

Composition Operators on Weighted Zygmund Spaces of the First Loo-keng Hua Domain

Let HEI denote the first Loo-keng Hua domain. In this paper, we obtain many elementary results on HEI by the continuous and careful discussions. In some applications, we obtain some necessary conditions or sufficient conditions for the boundedness and compactness of the composition operators on weighted Zygmund space defined on HEI.

Nice operators and nice spaces

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.

Volterra-Composition Operators Acting on $S^{p}$ Spaces and Weighted Zygmund Spaces

Let $\varphi$ be an analytic selfmap of the open unit disk $\mathbb{D}$ and $g$ be an analytic function on $\mathbb{D}$. The Volterra-type composition operators induced by the maps $g$ and $\varphi$ are defined as $$\left(I_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f^{\prime}(\varphi(\zeta)) g(\zeta) d\zeta \hspace{0.07in} \text{and} \hspace{0.07in} \left(T_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f(\varphi(\zeta)) g^{\prime}(\zeta) d\zeta .$$For $1\leq p&lt;\infty$, $S^p(\mathbb{D})$ is the space of all analytic functions on $\mathbb{D}$ whose first derivative $f^{\prime}$ lies in the Hardy space $H^p(\mathbb{D})$, endowed with the norm $\displaystyle \|f\|_{S^p}=|f(0)|+\|f^{\prime}\|_{H^p}$. Let $\mu: (0,1] \rightarrow (0, \infty)$ be a positive continuous function on $\mathbb{D}$ such that for $z\in\mathbb{D}$ we define $\mu(z) = \mu(|z|)$. The weighted Zygmund space $\mathcal{Z}_{\mu}(\mathbb{D})$ is the space of all analytic functions $f$ on $\mathbb{D}$ such that $\sup_{z\in \mathbb{D}} \mu(z) |f^{\prime\prime}(z)|$ is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between $S^p$ spaces and weighted Zygmund spaces.

ON SOLVABILITY OF A GENERAL ELLIPTIC BOUNDARY VALUE PROBLEM IN H¨OLDER – ZYGMUND SPACES WITH VARIABLE SMOOTHNESS

Fredholm solvability of an elliptic boundary value problem, corresponding to the Green operator from the L. Boutet deMonvel algebra on a smooth manifold with compact boundary is studied in the paper. The H¨older - Zygmund spaces with variable smoothness are considered as function spaces. Sufficient conditions are given for the Fredholm property of the Green operator from the considered algebra in such spaces.

Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space BHλ,0&lt;λ&lt;∞ to the weighted harmonic Zygmund space ZHβ,0&lt;β&lt;∞. The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between BHλ and ZHβ is bounded. Secondly, we will study the compactness case of the composition operator between BHλ and ZHβ. Finally, we will estimate the essential norms of the composition operator between BHλ and ZHβ.

Weighted composition operators from weighted Bergman-Orlicz spaces with doubling weights to weighted Zygmund spaces

Weighted composition operators from weighted Bergman-Orlicz spaces with doubling weights to weighted Zygmund spaces

Multiplication Operators on Weighted Zygmund Spaces of the First Cartan Domain

Inspired by some recent studies of the multiplication operators on holomorphic function spaces of the classical domains such as the open unit disk, the unit ball and the unit polydisk, the purpose of the present paper is to study just the operators that are defined on weighted Zygmund spaces of the first Cartan domain. We obtain some necessary conditions and sufficient conditions for the operators to be bounded and compact.

Surjective Isometries of Multiplication Operators and Weighted Composition Operators on nth Weighted-Type Banach Spaces of Analytic Functions

This paper is to characterize isometries of multiplication operators and weighted composition operators on the nth weighted-type Banach spaces {Vn : n ∈ N} of analytic functions on the open unit disk of which the Bloch space and the Zygmund space are particular cases at n = 1, 2. We give characterizations of the symbols ψ and φ for which the multiplication operator Mψ and the weighted composition operator Wψ,φ are surjective isometries. Moreover, we show that generalized weighted composition operators are not isometric on Vn.

Composition Operators From Harmonic Lipschitz Space Into Weighted Harmonic Zygmund Space

The paper investigate a necessary and sufficient condition for the composition operator from harmonic Lipschitz spaces LipHα, (0&lt;α&lt;1) into weighted harmonic Zygmund spaces ZHβ, (0&lt;β&lt;∞) to be bounded and compact on the open unit disk. As an application, it estimates the essential norms of such an operator from LipHα into ZHβ spaces.

Mappings of generalized finite distortion and continuity

AbstractWe study continuity properties of Sobolev mappings , , that satisfy the following generalized finite distortion inequality for almost every . Here and are measurable functions. Note that when , we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion , where a sharp condition for continuity is that is in the Zygmund space for some . We also show that one can slightly relax the boundedness assumption on to an exponential class with , and still obtain continuous solutions when with . On the other hand, for all with , we construct a discontinuous solution with and , including an example with and .

Full Double Hölder Regularity of the Pressure in Bounded Domains

Abstract We consider Hölder continuous weak solutions $u\in C^{\gamma }(\Omega )$, $u\cdot n|_{\partial \Omega }=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega \subset{\mathbb{R}}^{d}$. If $\Omega $ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2\gamma }_{*}(\Omega )$ in the case $\gamma \in (0,\frac{1}{2}]$, where $C^{2\gamma }_{*}$ is the Hölder–Zygmund space, which coincides with the usual Hölder space for $\gamma &amp;lt;\frac 12$. This result, together with our previous one in [ 11] covering the case $\gamma \in (\frac 12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma }_{*}$ estimate for the pressure on the whole space ${\mathbb{R}}^{d}$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma =\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood–Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step that plays a major role in our approach.

Convergence of a conjugate function in Zygmund space by almost Nörlund transform

Abstract The proposed work aims to study the degree of convergence of a function h ~ {\widetilde{h}} , conjugate to a Lebesgue integrable 2 ⁢ π {2\pi} -periodic function h, in the Zygmund class Z r ( ν ) {Z^{(\nu)}_{r}} , r ≥ 1 {r\geq 1} , by almost Nörlund transform. Our results give sharper estimates than the previous existing results.

Characterizations of Hankel type measures

In this paper, motivated by the definition of the Hankel measure and the well-known theory of the Carleson type measure, we introduce and characterize certain Hankel type measures. The characterizations of these measures involve Hardy spaces, Lipschitz spaces, the Zygmund space, Hankel operators, and the balayage of measures. Among positive Borel measures supported on [0,1), we also consider further the relation between Carleson type measures and these Hankel type measures.

Degree of Convergence of a Function in Generalized Zygmund Norm Using Karamata-Matrix (KλA) Product operator

In the present paper, we obtain the results on the degree of convergence of a function of Fourier series in generalized Zygmund space using Karamata-Matrix (KλA) product operator. We also study an application of our main result.

Composition operators from harmonic $ \mathcal{H}^{\infty} $ space into harmonic Zygmund space

&lt;abstract&gt;&lt;p&gt;This research paper sought to characterize the boundedness and compactness of composition operators from the space $ \mathcal{H}^{\infty} $ of bounded harmonic mappings into harmonic Zygmund space $ \mathcal{Z}_H $, on the open unit disk. Furthermore, we obtain an estimate of the essential norms of such an operator. These results extends the similar results that were proven for composition operators on analytic function spaces.&lt;/p&gt;&lt;/abstract&gt;