Remarks on some subspaces of BMO(\({\mathbb R}^n\)) and of bmo(\({\mathbb R}^n\)).
type: Item
instance of: scholarly article
title: Remarks on some subspaces of BMO(\({\mathbb R}^n\)) and of bmo(\({\mathbb R}^n\)).
zbMATH Open document ID: 1061.46025
author: Gérard Bourdaud
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116007
review text: On the \(n\)-dimensional Euclidean space, the author introduces the function space VMO (resp., CMO) as the BMO-closure of all infinitely differentiable functions of bounded mean oscillation (resp., of all infinitely differentiable functions with compact support). Various characterizations of these spaces are provided. For example, \(f\in \text{VMO}\) iff \(f\) is the limit in BMO of a sequence of uniformly continuous BMO-functions iff \(f=f_0 + R_1 f_1+\cdots+R_n f_n\) where the \(f_j\) are bounded uniformly continuous functions and the \(R_j\) are the Riesz transformations iff \(\lim_{t\to 0}\sup_{| B| \leq t}| B| ^{-1}\int_B | f-f_B| =0\) (the last condition is the usual definition of VMO; throughout, \(B\) is a ball and \(f_B\) is the average of \(f\) over \(B\)), etc. Next, \(f\in \text{CMO}\) iff \(f\) is in the BMO-closure of the set of continuous functions vanishing at infinity iff \(f\) is in the intersection of VMO and the BMO-closure of the set of compactly supported continuous functions iff \(f=f_0+R_1 f_1 +\cdots+R_n f_n\) with all \(f_j\) continuous and vanishing at infinity, etc. The BMO-closure of the set of compactly supported BMO-functions is characterized in the same spirit. Also, a similar story is told about the closures of various natural function classes in the space bmo determined by the norm \[ \sup_{| B| <1} | B| ^{-1}\int_B | f-f_B| + \sup_{| B| =1}\int_B | f| . \] Various isolated facts of a similar nature can be found dispersed in the literature, but the paper under review seems to present the first systematic treatment of this range of problems. At the end, some mistakes of forerunners are indicated and the corresponding counterexamples are given.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 1794828
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: continuous mean oscillation
reviewed by: S. V. Kislyakov
MaRDI profile type: Publication
OpenAlex ID: W2322977084
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DOI: 10.5802/AIF.1915
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1794828