Discrete multilinear maximal functions and number theory

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.