An embedding relation for bounded mean oscillation on rectangles

Abstract

In the two-parameter setting, we say a function belongs to the mean little $BMO$, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the author in relation with the multiplier algebra of the product $BMO$ of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation $bmo(\mathbb{T}^N)$ is a strict subspace of the mean little $BMO$.