A decomposition of functions with vanishing mean oscillation (Q2767865)

Say that a locally integrable function \(f\) on \(\mathbb R^n\) belongs to BLO if NEWLINE\[NEWLINE \|f\|_{\text{BLO}}=\sup_{Q} \frac 1{|Q|} \int_{Q} \Bigl(f(x)-\inf_{Q}f\Bigr) dx<+\infty, NEWLINE\]NEWLINE where the supremum is taken over all cubes \(Q\subset \mathbb R^n\) and \(\inf_{Q}f\) denotes the essential infimum of \(f\) over \(Q\). It is easily seen that BLO is a strict subspace of BMO. NEWLINENEWLINENEWLINESay that \(f\) belongs to VLO if \(f\in \text{BMO}\) and NEWLINE\[NEWLINE f_{Q}-\inf_{Q}f=o(1)\;(l(Q)\to 0) NEWLINE\]NEWLINE where \(f_{Q}\) denotes the mean value of \(f\) over \(Q\). Note that VLO is a proper subspace of VMO. NEWLINENEWLINENEWLINEThe main theorem of this paper states that, for all function \(f\in \text{VMO}\), there exist functions \(F\) and \(G\) in VLO such that \(f=F-G\) and NEWLINE\[NEWLINE \|F\|_{\text{BLO}} + \|G\|_{\text{BLO}} \leq C\|f\|_{\text{BMO}}, NEWLINE\]NEWLINE where \(C>0\) only depends on \(n\). NEWLINENEWLINENEWLINEThis result is the VMO version of a theorem by \textit{R. R. Coifman} and \textit{R. Rochberg} [in Proc. Am. Math. Soc. 79, 249-254 (1980; Zbl 0432.42016)], which states that each BMO function may be written as the difference of two BLO functions.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00020].