# Is there a version of the Return Times Theorem for Dunford-Schwartz operators?

Bourgain's "Return Times Theorem" establishes that if $(\Omega_{j},\mathcal{F}_{j},\mathbb{P}_{j},T_{j})$ ($j=1,2$) are measure-preserving Dynamical systems (i.e. $(\Omega_{j},\mathcal{F}_{j},\mathbb{P}_{j})$ are probability spaces and $T_{j}:\Omega_{j}\to\Omega_{j}$ are measure-preserving transformations), then for every $p_{1}>1$ and every $X\in L^{p_{1}}_{\mathbb{P}_{1}}$, there exists $\Omega_{1,X}$ (depending only on $X$) with $\mathbb{P}_{1}\Omega_{1,X}=1$ such that, if $p_{2}=p_{1}/(p_{1}-1)$ is the conjugate of $p_{1}$, then for every $Y\in L^{p_{2}}_{\mathbb{P}_{2}}$ and every $(\omega_{1},\omega_{2})\in \Omega_{1,X}\times\Omega_{2}$ $$\frac{1}{n}\sum_{k=0}^{n-1}T_{1}^{k}X(\omega_{1})T_{2}^{k}Y(\omega_{2}) \,\,\,\,\,\,\,\,(1)$$ is convergent, where $T^{k}_{j}Z=Z \circ T^{k}_{j}$ is the $k-$th iterate of the "Koopman" operator $Z\mapsto Z\circ T_{j}$ in $L^{p_{j}}_{\mathbb{P}_{j}}$ (the reason for this pedantic presentation will be cleared below). See for instance this entry of Tao's blog and the references therein for a more detailed discussion.

As indicated by Tao in his post, the Return Times Theorem comes as an answer to a natural question arising from the "automatic" extension of Birkhoff's Ergodic theorem to double recurrence.

Now, Birkhoff's ergodic theorem is a particular instance of the following satement: if $T$ is a positive contraction in $L^p_{\mathbb{P}}$ for every $p\geq 1$ (a positive Dunford-Schwartz operator), then for every $p\geq 1$ and every $Z\in L^{p}_{\mathbb{P}}$ $$\frac{1}{n}\sum_{k=0}^{n-1}T^{k}Z$$ converges $\mathbb{P}-$a.s. The proof of this theorem can be found for instance in the recent book by Eisner et.al.

Question: Is there a version of the Return Times Theorem for (positive) Dunford-Schwartz operators?

In other words. Can we prove the existence of $\Omega_{1,X}$ as above such that (1) holds in $\Omega_{1,X}\times\Omega_{2}$ if we replace $T_{1}$ and $T_{2}$ by positive Dunford-Schwartz operators in the corresponding $L^{p}-$spaces?

Motivation: (Besides natural curiosity) I want to prove that for a certain Dunford-Schwartz operator $T$ and a given $Z\in L^{2}_{\mathbb{P}}$, the averages $$\frac{1}{n}\sum_{k=0}^{n-1}|T^{k}Z|^{2} \,\,\,\,(2)$$ are $\mathbb{P}-$a.s. convergent. This claim would follow from an affirmative answer to the question by considering $T_{1}=T_{2}=T$ and $Z=X=\bar{Y}$ (the complex conjugate of $Y$).

As a matter of fact, (2) can be proved, using results by Assani, under the assumptions that $T$ and $T^{*}$ preserve the constant function $\mathbf{1}$ (which is actually verified by my particular operator) and that $Z\in L^{p}_{\mathbb{P}}$ for some $p>2$, but I haven't found so far an argument for the case $p=2$ (if you can give one please do so).

Thank you!