On a Biparameter Maximal Multilinear Operator

TL;DR

This paper introduces a novel bi-parameter maximal multilinear operator related to ergodic averages, providing new estimates and insights into pointwise convergence problems in harmonic analysis.

Contribution

It develops a new bi-parameter maximal multilinear operator and establishes non-trivial estimates, advancing understanding beyond previous techniques.

Findings

Developed a bi-parameter maximal multilinear operator

Produced Hölder-type estimates for key terms

Linked the operator to pointwise convergence in ergodic theory

Abstract

It is well-known that estimates for maximal operators and questions of pointwise convergence are strongly connected. In recent years, convergence properties of so-called `non-conventional ergodic averages' have been studied by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In particular, much is known regarding convergence in $L^2$ of these averages, but little is known about pointwise convergence. In this spirit, we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator (over $\mathbb{R}$, thanks to a transference principle). Lacey and Demeter, Tao, and Thiele have studied maximal multilinear operators previously; however, the maximal operator we develop has a novel bi-parameter structure which has not been previously encountered and cannot be estimated using their techniques. We will carve this bi-parameter maximal multilinear operator using a certain Taylor series and produce non-trivial H\"{o}lder-type estimates for one of the two "main" terms by treating it as a singular integrals whose symbol's singular set is similar to that of the Biest operator studied by Muscalu, Tao, and Thiele.