Functional calculus on \(BMO\) and related spaces
type: Item
instance of: scholarly article
title: Functional calculus on \(BMO\) and related spaces
zbMATH Open document ID: 1007.47028
author: Gérard Bourdaud
author: Massimo Lanza de Cristoforis
author: Winfried Sickel
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: It is considered a nonlinear autonomous superposition operator \[ {\mathbf T}_f [g] = f\circ g \] on a certain subspace \(X\) of the space \(BMO({\mathbb R}^n)\). The question is to characterise those Borel measurable functions \(f\) for which the operator \({\mathbf T}_f\) is continuous or differentiable on \(X\). The following subspaces \(X\) are considered: \(BMO({\mathbb R}^n)\), \(VMO({\mathbb R}^n)\) and \(CMO({\mathbb R}^n)\) (closure in \(BMO\)-norm of \(C_c^{\infty}({\mathbb R}^n)\) as well as their respective inhomogeneous counterparts. Namely, \(X = bmo({\mathbb R}^n)\) (a linear subspace of those functions having finite mean on the unit cube in \({\mathbb R}^n\)), \(X = cmo({\mathbb R}^n)\) (closure in \(bmo\)-norm of \(C_c^{\infty}({\mathbb R}^n)\)), \(X = vmo({\mathbb R}^n) = VMO({\mathbb R}^n) \cap bmo({\mathbb R}^n)\). The results by Fominykh-Chevalier, Brezis-Nirenberg and Marcus-Mizel are extended.
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1797579
zbMATH Keywords: autonomous superposition operator
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: vanishing mean oscillation
reviewed by: Sergei V. Rogosin
MaRDI profile type: Publication
full work available at URL: jfan.2001.3847
OpenAlex ID: W2021139051
cites work: Fonctions qui opèrent sur les espaces de Besov et de Triebel. (Functions which operate on Besov and Triebel spaces.)
cites work: Degree theory of BMO. I: Compact manifolds without boundaries
cites work: Quelles Sont Les Fonctions Qui Opèrent De Bmo Dans Bmo Ou De Bmo Dans L∞¯?
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Q4273944
cites work: On the transformation of functions of bounded mean oscillation
cites work: On functions with conditions on the mean oscillation
cites work: Q5556050
cites work: Every superposition operator mapping one Sobolev space into another is continuous
cites work: Functions of Vanishing Mean Oscillation
cites work: Bounded Toeplitz Operators on H 1 and Applications of the Duality Between H 1 and the Functions of Bounded Mean Oscillation
cites work: Q3142876
DOI: 10.1006/JFAN.2001.3847
description: scientific article; zbMATH DE number 1797579