Another characterization of BMO

Abstract

The following characterization of functions of bounded mean oscillation (BMO) is proved. f is in BMO if and only if \[ f = α log ⁡ g ∗ − β log ⁡ h ∗ + b f = \alpha \log {g^ \ast } - \beta \log {h^ \ast } + b \] where g ∗ , ( h ∗ ) {g^ \ast },({h^ \ast }) is the Hardy-Littlewood maximal function of g, (h), respectively, b is bounded and ‖ f ‖ BMO ⩽ c ( α + β + ‖ b ‖ ∞ ) {\left \| f \right \|_{{\text {BMO}}}} \leqslant c(\alpha + \beta + {\left \| b \right \|_\infty }) .