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

TL;DR

This paper investigates the bounds of matrix-valued maximal inequalities in high-dimensional Euclidean spaces, showing that for large dimensions, the bounds remain stable and generalize known scalar results.

Contribution

It extends scalar maximal inequality bounds to matrix-valued cases, demonstrating dimension-independent bounds for large n.

Findings

$L_p$-bounds are independent of dimension $n$ for large $n$

Weak type $(1,1)$ bounds behave similarly to scalar case

Generalizes Stein and Str"omberg's results to matrix-valued inequalities

Abstract

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.