Maximal Ergodic Inequalities for Banach Function Spaces

TL;DR

This paper extends the Transfer Principle to broader group actions, enabling the derivation of weak type maximal inequalities on various Banach function spaces and establishing almost sure convergence of ergodic averages.

Contribution

It generalizes the Transfer Principle to $\sigma$-compact locally compact Hausdorff groups, linking space properties to maximal inequalities and ergodic convergence.

Findings

Extended Transfer Principle to broader group actions

Derived weak type maximal inequalities for various Banach spaces

Proved almost sure convergence of ergodic averages

Abstract

We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of $\sigma$-compact locally compact Hausdorff groups acting measure-preservingly on $\sigma$-finite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on different Banach function spaces, and how the properties of these function spaces influence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages.