Another characterization of BLO

Abstract

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