Maximal ergodic inequalities for Banach function spaces

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. Next we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. In closing we briefly indicate the utility of these results for Statistical Physics.