I am reading a paper and it seems to be giving me more questions than answers. I will briefly explain the setup, as well as my attempts to make sense of them.
Suppose $(X,\mathcal{F},\mu)$ is a probability space and $F:X\to X$ is a measure preserving transformation. Suppose that $p\geq 2$ and $h\in L^{p-1}(X)$. It is assumed that $\max_{0\leq k\leq n}|h\circ F^k|=o(n^{1/p})$ a.s.
However, does the previous statement not imply $h\in L^p(X)$? Because certainly we have $|h|\leq \max_{0\leq k\leq n}|h\circ F^k|\leq n^{1/p}$ a.s for $n$ sufficiently large, and so on taking the $L^p$ norm, $\|h\|_p\leq n^{1/p}$. Or am I mistaken here?
Another question, which is related; Does $\max_{0\leq k\leq n}|h\circ F^k|=o(n^{1/p})$ a.s. imply that $\|\max_{0\leq k\leq n}|h\circ F^k|\|_p=o(n^{1/p})$? My guess is yes, because for any $C>0$, we can take $n$ sufficiently large so that
$\frac{\max_{0\leq k\leq n}|h\circ F^k|}{n^{1/p}}\leq C$. Applying the $L^p$-norm should give the result.
Does my logic appear to be flawed here? I am pretty new to big O/little O notation so any comments would be helpful!
The information that $\max_{0\leqslant k\leqslant n}|h\circ F^k|=o\left(n^{1/p}\right)$ is equivalent to $$\tag{*}\lim_{n\to \infty}\frac 1 {n^{1/p }} \max_{0\leqslant k\leqslant n}\left|h\circ F^k\right|=0\mbox{ a.s.}.$$ The convergence $(*)$ holds automatically if $F$ is invariant, that is, if $h\circ F=h$ almost surely hence one can find a source of inspiration in nonergodic dynamical systems to find an example of function satisfying $(*)$ and which does not belong to $\mathbb L^p$. But it is true that $f\in\mathbb L^p$ implies (*), as a consequence of the Borel-Cantelli lemma.
The argument in the opening post uses the fact that $f/n$ is almost surely bounded, but the chosen $n$may depend on$\omega$.