Maximal Ergodic Inequalities for Banach Function Spaces

TL;DR

The paper develops a unified Transfer Principle for maximal inequalities and almost-sure ergodic convergence that extends beyond $L^p$ to a broad class of rearrangement invariant Banach function spaces on actions of $\sigma$-compact locally compact groups. It defines transferable operators and the transferred operator $T^\#$, proves Kolmogorov-type and weak-type results in terms fundamental functions, and establishes a general three-step method to obtain pointwise ergodic theorems. The framework yields concrete weak-type transfers for Orlicz and related spaces, including cases where $G$ is amenable and the function spaces satisfy growth conditions like $\Delta_2$ or $\mathcal{U}/\mathcal{L}$, with corollaries that recover classical $L^p$ results as special cases. The authors then apply the theory to Orlicz spaces arising in Statistical Physics, notably $L^{\cosh-1}$ and $L\log(L+1)$, obtaining almost-sure convergence of ergodic averages for observables in these spaces and illustrating the practical impact on physical statistics.

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 $σ$-compact locally compact Hausdorff groups acting measure-preservingly on $σ$-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.

Theorems & Definitions (69)

• Definition 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6