Improved $L^p$ bounds for the strong spherical maximal operator

TL;DR

This paper establishes new $L^p$ bounds for the strong spherical maximal operator in higher dimensions, improving previous results by combining incidence geometry estimates with Fourier analysis.

Contribution

It introduces a novel incidence geometry estimate for ellipsoid neighborhoods and extends $L^p$ bounds to all dimensions $n \\geq 3$ for $p > 2$, approaching the conjectured sharp range.

Findings

Boundedness of the operator for $p > 2$ in all $n \\geq 3$

Improved bounds over previous work of Lee, Lee, and Oh

New incidence geometry estimate for ellipsoid neighborhoods

Abstract

We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n \geq 3$. This matches the conjectured sharp range $p>(n+1)/(n-1)$ when $n=3$. For $n=2$ the analogous estimate was recently proved by Chen, Guo and Yang. Our result builds upon and improves an earlier bound of Lee, Lee and Oh. The main novelty is an estimate in discretised incidence geometry that bounds the volume of the intersection of thin neighbourhoods of axis-parallel ellipsoids. This estimate is then interpolated with the Fourier analytic $L^p$-Sobolev estimates of Lee, Lee and Oh.