Characterization of two parameter matrix-valued BMO by commutator with the Hilbert transform

TL;DR

This paper characterizes two-parameter matrix-valued BMO functions through iterated commutators with the Hilbert transform, extending scalar results to the matrix-valued setting using dyadic shift representations.

Contribution

It provides a new characterization of matrix-valued BMO functions via commutators with the Hilbert transform, generalizing scalar results to the matrix case.

Findings

Equivalence between BMO norm and commutator operator norm.

Use of dyadic shift representation for the Hilbert transform.

Extension of scalar commutator bounds to matrix-valued functions.

Abstract

In this paper we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$\| B \|_{BMO} \lesssim \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R}^2;\mathbb{C}^d) \rightarrow L^2(\mathbb{R}^2;\mathbb{C}^d)} \lesssim \| B \|_{BMO}.$$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.