Sparse Bounds for the Discrete Spherical Maximal Function

TL;DR

This paper establishes sparse bounds for the discrete spherical maximal operator, providing conjecturally sharp estimates with an endpoint, using a novel, efficient method inspired by harmonic analysis techniques.

Contribution

Introduces a new, efficient method to prove sparse bounds for the discrete spherical maximal function, incorporating the Hardy-Littlewood Circle method for the first time in this context.

Findings

Sparse bounds are proven for the operator.

Bounds are conjecturally sharp and include an endpoint.

Method simplifies the proof process using a single estimate per decomposition element.

Abstract

We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.