Another characterization of BLO

It is shown that a locally integrable function $f$ on ${{\mathbf{R}}^n}$ has bounded lower oscillation $(f \in {\text{BLO}})$ if and only if $f = MF + h$, where $F$ has bounded mean oscillation $(F \in {\text{BMO}})$ and $MF < \infty$ a.e., and $h$ is bounded. Here, $MF$ is a variant of the familiar Hardy-Littlewood maximal function: $MF = {\text{sup}_{Q\backepsilon x}}Q(F)$ (no absolute values), where $Q(F)$ is the mean value of $F$ over the cube $Q$.