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

TL;DR

This paper investigates how rotation affects the biparameter BMO space in , revealing it is not invariant under rotation and establishing interpolation inequalities that lead to boundedness results for directional Hilbert transforms.

Contribution

It introduces interpolation inequalities to quantify the non-invariance of biparameter BMO under rotation and applies these to prove boundedness of directional Hilbert transforms.

Findings

Biparameter BMO is not preserved under rotations.

Interpolation inequalities quantify the effect of rotation.

Boundedness of directional Hilbert transforms is established.

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.