Norm-variation of ergodic averages with respect to two commuting transformations

TL;DR

This paper proves a sharp quantitative convergence result for double ergodic averages with two commuting transformations using advanced harmonic analysis techniques, including bounds for multilinear singular integrals.

Contribution

It introduces a novel approach to analyze ergodic averages via multilinear harmonic analysis, providing new bounds and convergence results.

Findings

Established sharp convergence rates for double ergodic averages

Developed bounds for multilinear singular integrals with entangled structure

Connected ergodic theory with harmonic analysis techniques

Abstract

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.