# Sharp matrix weighted strong type inequalities for the dyadic square function

## TL;DR

This paper improves the understanding of matrix weighted dyadic square functions by establishing sharp inequalities and pointwise sparse domination, extending previous results to more general settings.

## Contribution

It refines sparse domination techniques to prove sharp two matrix weighted inequalities for dyadic square functions for 1 < p ≤ 2.

## Key findings

- Established sharp two matrix weighted strong type inequalities.
- Proved pointwise sparse domination for general matrix weighted dyadic square functions.
- Extended previous results to broader matrix weight classes.

## Abstract

In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$.