Weighted and vector-valued variational estimates for ergodic averages

TL;DR

This paper establishes new weighted and vector-valued variational bounds for ergodic averages in Euclidean spaces, utilizing advanced inequalities to connect ergodic averages with dyadic martingales.

Contribution

It introduces novel weighted and vector-valued variational estimates for ergodic averages, employing an $ ext{ell}^r$ reverse H"older inequality for variation seminorms.

Findings

Weighted square function estimate for ergodic averages

Connection between ergodic averages and dyadic martingales

Use of $ ext{ell}^r$ reverse H"older inequality for variation seminorms

Abstract

We prove weighted and vector-valued variational estimates for ergodic averages on $\mathbb{R}^d$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an $\ell^r$ version of a reverse H\"older inequality for variation seminorms.