Pointwise convergence of certain continuous-time double ergodic averages

TL;DR

This paper proves almost everywhere convergence of certain continuous-time quadratic averages involving two commuting actions, using advanced techniques from multilinear singular integrals and the triangular Hilbert transform.

Contribution

It establishes the pointwise convergence of quadratic averages for two commuting -actions, extending ergodic theory methods with new analytical tools.

Findings

Proves a.e. convergence of quadratic averages with -actions

Utilizes multilinear singular integrals and the triangular Hilbert transform

Advances understanding of continuous-time ergodic averages

Abstract

We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.