Convergence and divergence of averages along subsequences in certain Orlicz spaces

TL;DR

This paper investigates the convergence and divergence behavior of averages along subsequences within Orlicz spaces, extending classical ergodic theorems to more general function spaces.

Contribution

It analyzes the convergence properties of subsequence averages in Orlicz spaces, revealing new phenomena beyond classical $L^p$ spaces.

Findings

Subsequence averages can converge in some Orlicz spaces but diverge in others.

The behavior of averages depends on the specific Orlicz space parameters.

The study extends classical ergodic theorems to a broader class of function spaces.

Abstract

The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\sum_{k=0}^{N-1} f(\sigma^k x)$ converges almost everywhere for $x\in X$ and $f\in L^{1}(X)$, where $\sigma$ is a measure preserving transformation of a probability measure space $X$. It was shown that there are operators of the form $T^N f(x)=(1/N)\sum_{k=0}^{N-1}f(\sigma^{n_k}x)$ for a subsequence $\{n_k\}$ of the positive integers that converge in some $L^p$ spaces while diverging in others. The topic of this talk will examine this phenomenon in the class of Orlicz spaces $\{L{Log}^\beta L:\beta>0\}$.