A famous result of J. Bourgain says that for a probability measure preserving system $(X,\beta,\mu,T)$, with $T_1$ and $T_2$ powers of $T$, we have that for $f_1$, $f_2\in L^{\infty}(\mu)$,$$\frac{1}{N}\sum_{n=1}^{N}T_1^nf_1(x)\cdot T_2^nf_2(x)$$ converges almost surely. My question is that if we loose the condition on $f_1$ and $f_2$ to belong to $L^p$, $p\geq 1$, what will happen?
For the case of single transformation, it is known that $$\frac{1}{N}\sum_{n=1}^{N}T^{p(n)}f(x),\ p(n)\text{ a polynomial with integer coefficients}$$ converges almost surely for $f\in L^r$, $r>1$. This was proved by Bourgain, if my memory is correct. For $r=1$, this is true for $p(n)=n$, but false for $p(n)=n^k$, as shown in the papers of Z. Buczolich & R. Daniel Mauldin, and P. LaVictoire. So I am interested that for double recurrence, whether the behaviors are similar? I was trying to find some reference, not have not succeeded. Maybe someone more experienced can remind me of some results. Thanks for any ideas.
