Mean ergodic theorems in $L^r(\hat{I}¼)$ and $H^r(\mathbb{T})$, $0<r<1$

Abstract

Let $T$ be the Koopman operator of a measure preserving transformation $\hat{I}¸$ of a probability space $(X,\hat{I}\pounds ,\hat{I}¼)$. We study the convergence properties of the averages $M_{n}f:=\frac{1}{n}{\sum }_{k=0}^{n-1}{T}^{k}f$ when $f\in L^r(\hat{I}¼)$, $0<r<1$. We prove that if $\int |M_{n}f{|}^{r}d\hat{I}¼\to 0$, then $f\in \overline{(I-T)L^r}$, and show that the converse fails whenever $\hat{I}¸$ is ergodic aperiodic. When $\hat{I}¸$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r\in (I-T)L^r$ for which $M_n{f}_r$ does not converge a.e. (although $\int |M_{n}f{|}^{r}d\hat{I}¼\to 0$). We further establish that for $1\le p<\frac{1}{r},$ there is a dense $G_{\hat{I}}´$ subset $\mathcal{F}\subset L^p(X,\hat{I}¼)$ such that ${\mathrm{limsup}}_n\frac{|T^{n}h|}{n^r}=\infty$ a.e. for any $h\in \mathcal{F}$.

18 pages, 25 references, 3 Lemmas, 6 Theorems, 12 Propositions and 7 Remarks