Restricted weak-type endpoint estimates for discrete k-spherical maximal functions

TL;DR

This paper advances the understanding of discrete k-spherical maximal functions by establishing restricted weak-type endpoint estimates, introducing new approximation formulas, and leveraging recent progress on Vinogradov mean value conjectures.

Contribution

It refines boundedness results for discrete k-spherical maximal functions, introduces a novel approximation formula, and improves bounds along sparse subsequences using recent number theory progress.

Findings

Established restricted weak-type endpoint estimates.

Improved bounds for lacunary discrete k-spherical maximal functions.

Introduced a density-parameter analogous to Minkowski dimension.

Abstract

In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the l^p(Z^d)-boundedness results for discrete k- spherical maximal functions to a restricted weak-type result at the endpoint. Moreover we introduce a novel Approximation Formula for a single average; this allows us to improve our bounds for discrete k-spherical maximal functions along sparse subsequences of radii by exploiting recent progress of Wooley on the Vinogradov mean value conjectures. In particular we have improved bounds for lacunary discrete k-spherical maximal functions when k>2. We introduce a density-parameter, which may be viewed as a discrete version of Minkowski dimension used in related works on the conitnuous phenomena, to prove our results for sparse averages.