On the restriction of functions of bounded mean oscillation to the lower dimensional space
type: Item
instance of: scholarly article
title: On the restriction of functions of bounded mean oscillation to the lower dimensional space
zbMATH Open document ID: 0586.46021
DOI: 10.1007/BF01190955
author: Eiichi Nakai
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: The \(L^ p\)-space has the property that if \(f\in L^ p({\mathbb{R}}^{m+n})\) then \(f(\cdot,y)\in L^ p({\mathbb{R}}_ x^ m)\) a.e. \(y\in {\mathbb{R}}^ n\). This property also holds for the spaces of continuous, Hölder continuous and k-times continuously differentiable functions, respectively. However, we point out that the space of functions of bounded mean oscillation does not have this property. More precisely, \(BMO_{\phi}\), the space of functions f such that \(\sup_{I}MO(f,I)/\phi (r)<\infty\) where I is a cube of side length r, MO(f,I) is the mean oscillation of f on I and \(\phi\) is a positive nondecreasing concave function, has this property if and only if \(\phi (r)^{-1}\int^{r}_{0}\phi (t)t^{-1}dt\) is bounded for \(r>0\). Furthermore, \(BMO_{\phi}\) is equivalent to the space of \(\phi\)-Hölder continuous functions only in this case.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 3939980
zbMATH Keywords: spaces of continuous, Hölder continuous and k-times continuously differentiable functions
zbMATH Keywords: space of functions of bounded mean oscillation
MaRDI profile type: Publication
cites work: Q4157125
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description: scientific article; zbMATH DE number 3939980