The Discrete Spherical Maximal Function: A new proof of $\ell^2$-boundedness

TL;DR

This paper presents a new direct proof demonstrating the $ ext{ell}^2$-boundedness of the Discrete Spherical Maximal Function, avoiding reliance on transference theorems or Fourier transform asymptotics.

Contribution

It introduces a novel proof technique for $ ext{ell}^2$-boundedness that is more direct and self-contained compared to previous methods.

Findings

Proves $ ext{ell}^2$-boundedness of the Discrete Spherical Maximal Function

Avoids use of transference theorems and Fourier asymptotics

Simplifies understanding of the maximal function's boundedness

Abstract

We provide a new direct proof of the $\ell^2$-boundedness of the Discrete Spherical Maximal Function that neither relies on abstract transference theorems (and hence Stein's Spherical Maximal Function Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres.