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

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.

18 pages. Published in Discrete Analysis Journal on 29 May 2018

References in corpus (2)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three

Cited by in corpus (3)

• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• Bounds for discrete multilinear spherical maximal functions in higher dimensions
• Supercritical discrete restriction estimates for forms in many variables