Sharp matrix weighted weak and strong type inequalities for the dyadic square function

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 quantitative two matrix weighted estimates for the matrix weighted dyadic square function and along the way prove quantitative two matrix weighted estimates for the matrix weighted maximal function. In particular, we prove sharp two matrix weighted weak and strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$ and prove sharp two matrix weighted weak type inequalities for the matrix weighted maximal function.