Weak integral conditions for BMO

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 $h:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is such that $h(t)\mathrm{\infty },$ then $$\underset{J\text{subcube}Q}{\sup }\frac{1}{\mathrm{\mid }J\mathrm{\mid }}{\int }_{J}h(\mathrm{\mid }\phi -\frac{1}{\mathrm{\mid }J\mathrm{\mid }}{\int }_J\phi \mathrm{\mid })<\mathrm{\infty }⟹\phi \in BMO(Q)\mathrm{.}$$ Under some additional assumptions on $h$ we obtain estimates on $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)\mathrm{\infty }$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic BMO.