Dimension-free Maximal Inequalities for Spherical Means in the Hypercube

TL;DR

This paper extends dimension-free $L^p$ bounds for spherical maximal functions in the hypercube to all $p > 1$, using spectral techniques, and shows no such bounds exist at the endpoint for $p=1$.

Contribution

It generalizes previous $L^2$ results to all $p > 1$ and introduces spectral methods inspired by ergodic theory to analyze hypercube spherical means.

Findings

Dimension-free $L^p$ bounds established for all $p > 1$

No dimension-free weak-type (1-1) bound at the endpoint

Spectral techniques effectively analyze spherical maximal functions

Abstract

We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique developed by Stein and Nevo and Stein in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type (1-1) bound exists at the endpoint.