Discrete maximal operators over surfaces of higher codimension

TL;DR

This paper studies discrete maximal operators over higher codimension surfaces, uniting harmonic analysis, discrete geometry, and number theory, and introduces multilinear techniques to obtain near-optimal bounds.

Contribution

It introduces a novel approach using multilinearity to analyze discrete maximal operators over higher codimension surfaces, achieving near-optimal bounds.

Findings

Established bounds for discrete maximal operators over triangular configurations

Developed a multilinear method for nontrivial $ ext{ell}^1$-estimates

Unified themes from harmonic analysis, geometry, and number theory

Abstract

Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research. Here we unite these themes to study discrete analogues of operators involving higher (intermediate) codimensional integration. We consider a maximal operator that averages over triangular configurations and prove several bounds that are close to optimal. A distinct feature of our approach is the use of multilinearity to obtain nontrivial $\ell^1$-estimates by a rather general idea that is likely to be applicable to other problems.