Biparameter $\text{BMO}$ under the action of a rotation

Abstract

In this work, we aim to study the action of composing by a rotation on the biparameter $\text{BMO}$ space in ${\mathbb{R}}^2$. This $\text{BMO}$ space is not preserved by a rotation since it relies on the structure of axis-parallel rectangles. We will quantify this fact by interpolation inequalities. One straightforward application of the interpolation inequalities is a boundedness property of directional Hilbert transforms.