A local Tb theorem for matrix weighted paraproducts

TL;DR

This paper establishes a local Tb theorem for vector-valued paraproducts with matrix weights, extending scalar and diagonal weight cases and introducing a new matrix reverse Hölder class.

Contribution

It generalizes previous scalar and diagonal weight results to matrix weights, introducing a new matrix reverse Hölder class for paraproduct analysis.

Findings

Proved a local Tb theorem for matrix weighted paraproducts.

Extended scalar weight results to matrix weights.

Introduced a new matrix reverse Hölder class.

Abstract

We prove a local $Tb$ theorem for paraproducts acting on vector valued functions, with matrix weighted averaging operators. The condition on the weight is that its square is in the $L_2$ associated matrix $A_\infty$ class. We also introduce and use a new matrix reverse H\"older class. This result generalizes the previously known case of scalar weights from the proof of the Kato square root problem, as well as the case of diagonal weights, recently used in the study of boundary value problems for degenerate elliptic equations.