Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ for any $a \in L^1, b \in L^{\infty}$.
Assume $\phi_1 \in L^{\infty}(X), \int \phi_1 d\mu =0, \int |P^n \phi_1| d\mu \le \frac{1}{n^{1+\epsilon}}, \epsilon \in [0,1) $ for any $n \ge 1$. And assume coboundary condition: $0=\int \phi_1^2+2\cdot \sum_{i \ge 1} \phi_1\cdot P^i \phi_1 d\mu$.
Can we show: there is $\xi\in L^1$, such that $\phi_1=\xi-\xi \circ T$?
The present result I know so far is: when $\epsilon \ge 1$, we can use coboundary condition to prove $\sup_n \int \left(\sum_{i \le n} \phi_1 \circ T^i\right)^2 d\mu \le n^{1-\epsilon} \le 1$, then $\sum_{i \le n} \phi_1 \circ T^i \to \xi$ weakly in $L^2$, and we can show that $\phi_1=\xi-\xi \circ T$ by pass the limit to $\xi \in L^2$. But do not know how to deal with the case of slower mixing. I guess $\xi = -\sum_{i \ge 0} P^i \phi_1 \in L^1$?
