Mean ergodic theorems in $L^r(\mu)$ and $H^r(\mathbb T)$, $0<r<1$

Let $T$ be the Koopman operator of a measure preserving transformation $\theta$ of a probability space $(X,\Sigma,\mu)$. We study the convergence properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in L^r(\mu)$, $0<r<1$. We prove that if $\int |M_nf|^r d\mu \to 0$, then $f \in \overline{(I-T)L^r}$, and show that the converse fails whenever $\theta$ is ergodic aperiodic. When $\theta$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\int |M_nf|^r d\mu \to 0$). We further establish that for $1 \leq p <\frac{1}{r},$ there is a dense $G_\delta$ subset ${\mathcal F}\subset L^p(X,\mu)$ such that $\limsup_n \frac{|T^nh|}{n^r}=\infty$ a.e. for any $h \in {\mathcal F}$.