Two Matrix Weighted Inequalities for Commutators with Fractional Integral Operators

TL;DR

This paper establishes two new matrix weighted inequalities for commutators involving fractional integral operators, extending previous scalar and Calderon-Zygmund results to a matrix setting with applications to fractional Bloom theory.

Contribution

It extends matrix weighted norm inequalities for commutators from Calderon-Zygmund operators to fractional integral operators, advancing the fractional Bloom theory in a matrix context.

Findings

Proved two matrix weighted norm inequalities for fractional integral commutators.

Extended scalar and Calderon-Zygmund results to fractional integral operators in a matrix setting.

Enhanced the fractional Bloom theory to accommodate matrix weights.

Abstract

In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix