# The sharp square function estimate with matrix weight

## TL;DR

This paper proves the matrix A2 conjecture for the dyadic square function, establishing sharp linear bounds and mixed estimates involving matrix weights, using sparse domination techniques.

## Contribution

It provides the first sharp linear dependence result for the matrix A2 conjecture in the context of dyadic square functions and introduces a novel sparse domination approach.

## Key findings

- Established sharp linear A2 bounds for matrix weighted dyadic square functions.
- Derived mixed estimates involving A2 and A∞ matrix weight constants.
- Introduced a sparse domination method inspired by the integrated form of the matrix-weighted square function.

## Abstract

We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{\infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.