# Sharp bounds and T1 theorem for Calder\'on-Zygmund operators with matrix kernel on matrix weighted spaces

## TL;DR

This paper introduces a new class of Calderón-Zygmund operators with matrix kernels on matrix weighted spaces, proves a T1 theorem, and establishes sharp bounds using dyadic decompositions and Bellman functions.

## Contribution

It defines W-Calderón-Zygmund matrix kernels, proves a T1 theorem for these operators, and provides sharp bounds in matrix weighted spaces, extending previous scalar results.

## Key findings

- Established a T1 theorem for matrix kernel operators
- Provided a representation theorem via dyadic W-Haar shifts and paraproducts
- Derived sharp bounds using Bellman function techniques

## Abstract

For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calder\'on-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of Hyt\"onen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.