On the closures of Dirichlet type spaces in the Bloch space
type: Item
instance of: scholarly article
title: On the closures of Dirichlet type spaces in the Bloch space
zbMATH Open document ID: 1417.30053
author: Guanlong Bao
author: Nihat Gökhan Göğüş
published in: Complex Analysis and Operator Theory
publication date: 2019-03-06T00:00:00Z
review text: Let $\mathbb{D}$ be the open unit disk in the complex plane, and let $\mathrm{Hol}(\mathbb{D})$ denote the space of holomorphic functions on the unit disk. A function $f\in\mathrm{Hol}(\mathbb{D})$ is said to belong to the Bloch space $\mathcal{B}$ if \[ \|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbf{D}}(1-|z|^2)|f^\prime (z)|<\infty. \] Similarly, for $s>-1$, the space $\mathcal{D}_s$ consists of $f\in \mathrm{Hol}(\mathbf{D})$ for which \[ \|f\|_{\mathcal{D}_s}=|f(0)|+\left (\int_{\mathbb{D}}(1-|z|^2)^s|f^\prime (z)|^2dA(z)\right )^{1/2}<\infty, \] where $dA(z)$ denotes the normalized Lebesgue area measure in the unit disk. For $s=0$, the space $\mathcal{D}_s$ reduces to the classical Dirichlet space, and for $s=1$, it will become the well-known Hardy space $H^2$ on the unit disk. \par It follows from the maximum modulus principle, or the Schwarz-Pick lemma that the Bloch space contains $H^\infty$, the space of bounded analytic functions on the unit disk. It is still an open problem to describe the closure of $H^\infty$ in the Bloch space $\mathcal{B}$; this question was raised by \textit{J. M. Anderson} et al. [J. Reine Angew. Math. 270, 12--37 (1974; Zbl 0292.30030)]. However, in 2010 a description of the closure of $H^p\cap \mathcal{B}$, for $1< p< \infty$, in the Bloch norm was obtained by \textit{N. M. Galán} and \textit{A. Nicolau} [St. Petersbg. Math. J. 22, No. 1, 55--59 (2011; Zbl 1211.30064)]. Later on in 2015, this result was improved by \textit{P. Galanopoulos} et al. [J. Math. Anal. Appl. 429, No. 2, 1214--1221 (2015; Zbl 1327.32008)] to incorporate the case $0< p< \infty$ in the unit ball of $\mathbb{C}^n$. \par In the paper under review, the authors provide a description of the closure in the Bloch norm of $\mathcal{D}_s\cap \mathcal{B}$. The main result of this paper, Theorem 2.1, states that for $-1<s\le 1$, a positive integer $n$, and a function $f\in\mathcal{B}$, the following holds: $f$ belongs to the closure in the Bloch norm of $\mathcal{D}_s\c…
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 7032869
zbMATH Keywords: Bloch space
zbMATH Keywords: Dirichlet-type spaces
zbMATH Keywords: interpolating Blaschke product
MaRDI profile type: Publication
full work available at URL: s11785-017-0676-2
OpenAlex ID: W2607811583
cites work: Q4043323
cites work: Q3681112
cites work: Composition operators and closures of some Möbius invariant spaces in the Bloch space
cites work: Boundary multipliers of a family of Möbius invariant function spaces
cites work: Hankel matrices acting on Dirichlet spaces
cites work: The Libera operator on Dirichlet spaces
cites work: An Interpolation Problem for Bounded Analytic Functions
cites work: Interpolations by bounded analytic functions and the corona problem
cites work: Q5618011
cites work: Closure of Hardy spaces in the Bloch space
cites work: Q3922907
cites work: Analytic functions of bounded mean oscillation and the Bloch space
cites work: Carleson measures for the Bloch space
cites work: INTEGRABILITY OF THE DERIVATIVE OF A BLASCHKE PRODUCT
cites work: Weighted Bloch, Lipschitz, Zygmund, Bers, and growth spaces of the ball: Bergman projections and characterizations
cites work: Composition operator on Bloch type spaces
cites work: The closure of the Hardy space in the Bloch norm
cites work: Bounded functions in Möbius invariant Dirichlet spaces
cites work: On the zeros of functions in Dirichlet-type spaces
cites work: Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces
cites work: Forelli--Rudin estimates, Carleson measures and \(F(p,q,s)\)-functions
cites work: A new characterization of Dirichlet type spaces and applications
cites work: Multipliers of the Dirichlet space
cites work: Distance of a Bloch Function to the Little Bloch Space
cites work: Holomorphic \(Q\) classes
cites work: Q5489391
cites work: Q5447193
cites work: Bloch type spaces of analytic functions
cites work: Q3447431
DOI: 10.1007/S11785-017-0676-2
Recommended article: The closure of Dirichlet spaces in the Bloch space
Recommended article: Small composition operators on analytic vector-valued function spaces
Recommended article: Closure of Hardy spaces in the Bloch space
Recommended article: On interpolating sequences for Bloch type spaces
Recommended article: Composition operators from Bloch type spaces to F(p, q, s) spaces
Recommended article: Multiplicative Isometries and Isometric Zero-Divisors
Recommended article: Essential norm of weighted composition operators from \(H^{\infty}\) to the Zygmund space
Recommended article: Q4826223
Recommended article: Composition operators and closures of Dirichlet type spaces $\mathcal{D}_{\mu}$ in Bloch type spaces
Recommended article: WEIGHTED BLOCH SPACES, $H^p$ , AND ${\rm BMOA}$
description: scientific article; zbMATH DE number 7032869