Generalized Hölder spaces of holomorphic functions in domains in the complex plane
type: Item
instance of: scholarly article
title: Generalized Hölder spaces of holomorphic functions in domains in the complex plane
zbMATH Open document ID: 1411.30039
published in: Mediterranean Journal of Mathematics
publication date: 2018-12-28T00:00:00Z
review text: The paper is concerned with the study of some nonstandard Banach spaces of analytic functions that enjoy a kind of smoothness property up to the boundary. Let \(\omega : [0,d]\to \mathbb{R}^+\) satisfies the following conditions: {\parindent=6mm \begin{itemize}\item[1)] \(\omega\) is continuous and \(\omega(0) = 0\), \item[2)] \(\omega\) is almost increasing, in the sense that there is a constant \(C\geq 1\) such that \(x \leq y\) implies \(\omega(x) \leq C\,\omega(y)\), \item[3)] \(\omega(h)/h\) is almost decreasing (there exists \(C\geq 1\) such that \(x \leq y\) implies \(\omega (y) \leq C\, \omega (x)\)). \end{itemize}} Such a function \(\omega\) is called a modulus of continuity. Now, let \(\omega : [0,2]\to \mathbb{R}^+\) be a modulus of continuity, and define \(A^\omega(D)\) as the space of all analytic functions in the unit disk \(D\) such that \[ |f(z)-f(w)|\leq C\, \omega (|z-w|),\quad z,w \in D, \] where \(C\) is a constant. The norm of a function \(f \in A^\omega(D)\) is given by \[ \| f\|_{A^\omega(D)}= \sup_{ z,w\in D}\frac{ |f(z)-f(w)|}{ \omega (|z-w|)} +\| f\| _{L^\infty (D)}. \] Similarly, define \(B^\omega(D)\) as the space of all analytic functions in the unit disk that satisfy \[ |f^\prime(z)|\leq C\frac{\omega (1-|z|)}{ 1-|z|},\quad z\in D, \] where \(C\) is a constant independent of \(z\) and \(w\). The norm of \(f\in B^\omega (D)\) is given by \[ \| f\| _{B^\omega(D)} = \sup_{ z\in D}\frac{ (1-|z|)|f^\prime (z)|}{\omega (1-|z|)} +\|f\|_{L^\infty(D)}. \] The first main result of the paper under review says that if \(\omega\) satisfies \[ \int_t^d \frac{\omega (s)}{s^2} ds \leq C \frac{\omega(t)}{t},\quad 0 < t < d, \] then \(A^\omega(D)\) is continuously embedded in \(B^\omega(D)\); that is, the inclusion \(A^\omega(D) \hookrightarrow B^\omega(D)\) is continuous. On the other hand, if \(\omega\) satisfies \[ \int_0^t \frac{\omega (s)}{s} ds \leq C \omega(t),\quad 0 < t < d, \] then the imbedding \(B^\omega(D) \hookrightarrow A^\omega(D)\) is co…
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 30H99
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6997105
zbMATH Keywords: spaces of analytic functions
zbMATH Keywords: variable exponent spaces
zbMATH Keywords: modulus of continuity
author: Alexey N. Karapetyants
author: S. G. Samko
MaRDI profile type: Publication
full work available at URL: s00009-018-1272-z
OpenAlex ID: W2900562478
cites work: Variable exponent Bergman spaces
cites work: Toeplitz operators on variable exponent Bergman spaces
cites work: Variable Lebesgue spaces. Foundations and harmonic analysis
cites work: Lebesgue and Sobolev spaces with variable exponents
cites work: Q4458484
cites work: Q4490700
cites work: Spaces BMOp(·)(𝔻) of a variable exponent p(z)
cites work: Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces
cites work: On boundedness of Bergman projection operators in Banach spaces of holomorphic functions in half-plane and harmonic functions in half-space
cites work: On mixed norm Bergman-Orlicz-Morrey spaces
cites work: Mixed norm Bergman-Morrey-type spaces on the unit disc
cites work: Mixed norm variable exponent Bergman space on the unit disc
cites work: Fractional integrals and singular integrals in the holder classes of variable order
cites work: Integral operators in non-standard function spaces. Volume 1: Variable exponent Lebesgue and amalgam spaces
cites work: Fractional integration operator of variable order in the Hölder spaces \(H^{\lambda (x)}\)
cites work: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators
cites work: Q4945090
cites work: Q3447431
DOI: 10.1007/S00009-018-1272-Z
description: scientific article; zbMATH DE number 6997105