Characterizations of bounded mean oscillation

Abstract:

Recall that an integrable function $f$ on a cube ${Q_0}$ in ${{\mathbf {R}}^n}$ is said to be of bounded mean oscillation if there is a constant $K$ such that for every parallel subcube $Q$ of ${Q_0}$ there exists a constant ${a_Q}$ such that $\int _Q {|f - {a_Q}| \leq K|Q|}$, where $|Q|$ denotes the volume of $Q$. We prove here that if there is an integer $d$ and a constant $K$ such that for every parallel subcube $Q$ of ${Q_0}$ there exists a polynomial ${p_Q}$ of degree $\leq d$ such that $\int _Q {|f - {p_Q}| \leq K|Q|}$, then $f$ is of bounded mean oscillation.