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 Rn and h: [0,∞) → [0,∞) is such that h(t)→∞ as t→∞, then sup J subcube Q 1 |J|_ J h ϕ – 1 |J| J ϕ _< ∞ =⇒ ϕ ∈ BMO(Q). Under some additional assumptions on h we obtain estimates on _ϕ_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 BMO.