Variational inequalities for bilinear averages

TL;DR

This paper establishes variational inequalities for bilinear averages, extending previous results, and applies these to prove almost everywhere convergence of ergodic averages along cubes in dynamical systems.

Contribution

It introduces new variational inequalities for bilinear averages, generalizing prior work, and demonstrates their application to ergodic theory convergence problems.

Findings

Established variational inequalities for bilinear averages.

Proved almost everywhere convergence of ergodic averages along cubes.

Extended previous inequalities to broader classes of averages.

Abstract

We obtain variational inequalities for some classes of bilinear averages of one variable, generalizing the variational inequalities for averages of R. Jones {\it et al}. As an application we get almost everywhere convergence for the ergodic averages along cubes on a dynamical system.