A matrix weighted $T1$ theorem for matrix kernelled CZOs and a matrix weighted John-Nirenberg theorem

TL;DR

This paper establishes a matrix weighted $T1$ theorem for certain Calderón-Zygmund operators with matrix kernels and extends the John-Nirenberg inequality to the matrix setting, advancing the understanding of matrix weighted harmonic analysis.

Contribution

It introduces a matrix weighted $T1$ theorem for matrix kernelled CZOs and extends the John-Nirenberg inequality to the matrix context, connecting to recent characterizations of matrix BMO.

Findings

Proves a matrix weighted $T1$ theorem for matrix kernelled CZOs.

Establishes a matrix weighted John-Nirenberg inequality.

Connects to recent characterizations of matrix BMO and two-weight boundedness.

Abstract

In this paper, we will prove a matrix weighted $T1$ theorem regarding the boundedness of certain matrix kernelled CZOs on matrix weighted $L^p(W)$ for matrix A${}_p$ weights $W$. Using some of the ideas from the proof, we will also establish a natural matrix weighted John-Nirenberg result that extends to the matrix setting (in the case when one of the weights is the identity) a very recent characterization of both S. Bloom's BMO space and the two weight boundedness of commutators by I. Holmes, M. Lacey, and B. Wick.