Maximal inequality for high-dimensional cubes

TL;DR

This paper establishes that the best constant in the weak (1,1) maximal inequality for high-dimensional cubes grows faster than a logarithmic rate as the dimension increases, revealing new asymptotic behavior.

Contribution

It provides simplified lower bounds for the maximal inequality constant in high dimensions, using probabilistic techniques involving Brownian bridges.

Findings

The constant grows faster than $( ext{log } n)^{1-o(1)}$ as $n o

The approach simplifies previous methods by Aldaz.

Application of Donsker's theorem links the distribution of cube coordinates to Brownian bridges.

Abstract

We present lower estimates for the best constant appearing in the weak $(1,1)$ maximal inequality in the space $(\R^n,\|\cdot\|_{\iy})$. We show that this constant grows to infinity faster than $(\log n)^{1-o(1)}$ when $n$ tends to infinity. To this end, we follow and simplify the approach used by J.M. Aldaz. The new part of the argument relies on Donsker's theorem identifying the Brownian bridge as the limit object describing the statistical distribution of the coordinates of a point randomly chosen in the unit cube $[0,1]^n$ ($n$ large).