Convergence and divergence of averages along subsequences in certain Orlicz 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 $\{L{Log}^{\beta }L:\beta >0\}$.