On the integrability of the maximal ergodic function

Abstract:

Let $G = {{\mathbf {R}}^d}$ or ${{\mathbf {Z}}^d}$ and consider an ergodic measure-preserving action of G on a probability space $(X,\mathfrak {A},P)$, let $f \in {L^1}(X,P)$ and Mf be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: if $|f|{\log ^ + }|f|$ is integrable then Mf is integrable. For the particular case $G = {\mathbf {Z}}$ this result was already obtained by D. Ornstein whose proof is based on induced transformations and seems to be specific to Z, our proof is based on a result of E. M. Stein on the Hardy-Littlewood maximal function on ${{\mathbf {R}}^d}$ and its analogue on ${{\mathbf {Z}}^d}$.