Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions

TL;DR

This paper advances the understanding of $ ext{ell}^p$-boundedness for integral $k$-spherical maximal functions, reducing the required dimension growth from cubic to quadratic, and applies these results to improve bounds in the ergodic Waring--Goldbach problem.

Contribution

It introduces new techniques that lower the dimension growth condition from cubic to quadratic for the boundedness of $k$-spherical maximal functions.

Findings

Reduced the dimension growth requirement from cubic to quadratic.

Established improved $ ext{ell}^p$-boundedness bounds for integral $k$-spherical maximal functions.

Applied results to enhance bounds in the ergodic Waring--Goldbach problem.

Abstract

We improve the range of $\ell^p(\mathbb Z^d)$-boundedness of the integral $k$-spherical maximal functions introduced by Magyar. The previously best known bounds for the full $k$-spherical maximal function require the dimension $d$ to grow at least cubicly with the degree $k$. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.