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.