a constant $c>0$ (depending only on $\alpha$) such that $$n^{1-\beta} \int_X v \,w\circ f^n\to c \int_X v\int_X w,$$ for all observables $v,w$ supported in a compact subset of $(0,1]$ with $v$ Hölder and $w$ in $L^\infty$. The same result holds for $\alpha=1$ with $n^{1-\beta}$ replaced by $\log n$. For $\alpha\ge2$, such results cannot hold in the generality considered in [@MT12] but, using the extra structure of the maps , the corresponding results were obtained by [@Gouezel11] for all $\alpha>1$. In addition, rates of mixing and higher asymptotics for $\alpha\in(1,2)$ were obtained in [@MT12], improved upon in [@Terhesiu-app]. and extended to the case $\alpha\ge2$ in [@Terhesiu-sub].

#### Continuous time, finite measure

Now suppose that $\varphi_X:X\to{{\mathbb R}}^+$ is a Hölder roof function bounded away from zero. We form the suspension flow $f_t:\Lambda\to\Lambda$ in the usual way (see Section \[sec-main\] for definitions). This semiflow has an indifferent periodic orbit corresponding to the indifferent fixed point $0\in X$.

In the finite measure case $\alpha\in(0,1)$, it follows from [@M09] that typically $\int_\Lambda v \,w\circ f_t-\int_\Lambda v\int_\Lambda w$ decays at the rate $t^{-(\beta-1)}$ as $n\to\infty$ for observables $v,w:\Lambda\to{{\mathbb R}}$ where $v$ is Hölder and $w$ is sufficiently smooth in the flow direction.

Here we prove that 