Weak integral conditions for BMO

Abstract:

We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $\mathbb {R}^n$ and $h:[0,\infty )\to [0,\infty )$ is such that $h(t)\to \infty$ as $t\to \infty ,$ then \[ \sup _{J~\text {subcube}~Q}\frac 1{|J|}\int _J h\big (\big |\varphi -\textstyle {\frac 1{|J|} \int _J\varphi } \big |\big )<\infty \quad \Longrightarrow \quad \varphi \in \mathrm {BMO}(Q). \] Under some additional assumptions on $h$ we obtain estimates on $\|\varphi \|_{\mathrm {BMO}}$ in terms of the supremum above. We also show that even though the limit condition on $h$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic $\mathrm {BMO}$.