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

12 pages; sharp one-dimensional theorem added in v2