The behavior of the bounds of matrix-valued maximal inequality in ${\mathbb{R}}^n$ for large $n$

In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in ${\mathbb{R}}^n$ for large $n$. The main result of this paper is that the $L_p$-bounds ($p>1$) can be taken to be independent of $n$, which is a generalization of Stein and Str"omberg’s resut in the scalar-valued case. We also show that the weak type $(1,1)$ bound has similar behavior as Stein and St"omberg’s.