Improved $L^p$ bounds for the strong spherical maximal operator
Jonathan Hickman, Joshua Zahl
TL;DR
This work proves $L^p$-boundedness of the strong spherical maximal operator $\mathcal{M}_{\mathrm{st}}$ for all $n\ge 3$ and $p>2$, improving previous multiparameter bounds and matching the sharp range in dimension $3$. The authors introduce a discretised local maximal operator $M^{\delta}$ and establish a sharp-looking $L^2$ bound with a small loss $\delta^{-\varepsilon}$, then reduce the problem to volume bounds for intersections of refined axis-parallel ellipsoidal annuli via slicing and duality. The crux is an incidence-geometry argument that controls how often refined ellipsoids can be simultaneously tangential, together with a meticulous removal of exceptional sets to avoid degenerate configurations. By coupling these geometric bounds with Fourier-analytic $L^p$–Sobolev estimates (via interpolation with results from prior work) and a multiparameter Littlewood–Paley framework, the paper elevates the local estimates to the global $L^p$-boundedness result. The methods offer a pathway toward the stronger conjectured range $p>(n+1)/(n-1)$ and advance the understanding of multiparameter maximal operators through discretised geometric analysis.
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.