Discrete multilinear maximal functions and number theory

TL;DR

This paper explores the discrete slicing method for multilinear maximal functions, establishing bounds and connections to number theory, aiming to expand its applications in discrete harmonic analysis.

Contribution

It generalizes the discrete slicing technique for multilinear operators, providing a unified framework and new insights with number theoretic links.

Findings

Established sharp bounds for multilinear discrete operators

Connected discrete slicing to number theory

Unified theory for discrete multilinear maximal functions

Abstract

Many multilinear discrete operators are primed for pointwise decomposition; such decompositions give structural information but also an essentially optimal range of bounds. We study the (continuous) slicing method of Jeong and Lee -- which when debuted instantly gave sharp multilinear operator bounds -- in the discrete setting. Via several examples, number theoretic connections, pointed commentary, and a unified theory we hope that this useful technique will lead to further applications. This work generalizes, and was inspired by, the author's work with Palsson on a special case.