On the restriction of functions of bounded mean oscillation to the lower dimensional space
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.
spaces of continuous, Hölder continuous and k-times continuously differentiable functions
space of functions of bounded mean oscillation
scientific article; zbMATH DE number 3939980