Spherical maximal estimates via geometry

TL;DR

This paper introduces a geometric method to analyze the $L^p$ boundedness of Stein's spherical maximal operator, avoiding Fourier analysis, and recovers a weak form of the classical theorem.

Contribution

It provides a new geometric approach to spherical maximal estimates, offering an alternative proof to Stein's theorem without Fourier transform techniques.

Findings

Recovered a weak form of Stein's spherical maximal theorem

Developed a simple geometric framework for $L^p$ boundedness

Avoided reliance on Fourier analysis in the proof

Abstract

We present a simple geometric approach to studying the $L^p$ boundedness properties of Stein's spherical maximal operator, which does not rely on the Fourier transform. Using this, we recover a weak form of Stein's spherical maximal theorem.