Weak integral conditions for BMO

الأصل · EN

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)t→∞∞, then subcube Q 1|J|∫ⱼ h(|φ-1|J| ∫ⱼφ|)<∞ φ∈ BMO(Q). Under some additional assumptions on h we obtain estimates on φBMO in terms of the supremum above. We also show that even though the condition h(t)t→∞∞ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic BMO.