on various function spaces. Throughout, $\|\,\|_\theta$ denotes the operator norm on either $F_\theta(Y)$ or $F_\theta(\tilde Y)$. Similarly, $\|\,\|_\infty$ denotes the operator norm on either $L^\infty(Y)$ or $L^\infty(\tilde Y)$. Whether the space is $Y$ or $\tilde Y$ should be clear from the context.

\[part-infinite\]

In this part of the paper, we prove our main results in the infinite measure context. Throughout, we assume the setup from Section \[sec-main\], so $F:Y\to Y$ is a Gibbs-Markov map and $\varphi:Y\to{{\mathbb Z}}^+$ is a roof function satisfying assumptions (A1) and (A2). In addition, we make the standing assumption throughout this part of the paper that $\varphi$ is nonintegrable and $\mu(\varphi>t)\sim \ell(t)t^{-\beta}$ where $\beta\in(\frac12,1]$ and $\ell(t)$ is slowly varying.

Section \[sec-funct\] contains various operator-theoretic estimates. Our result on first order asymptotics (mixing) stated in Theorem \[thm-infinite\](a) is proved in Section \[sec-mixing\]. Our result on second order asymptotics and rates of mixing, Theorem \[thm-infinite\](b), is proved in Section \[sec-rates\].

Functional analytic estimates {#sec-funct}
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In this section, we carry out various operator-theoretic estimates. Most of these are fairly straightforward generalisations of the estimates in [@MT12] which built upon [@AaronsonDenker01; @GarsiaLamperti62]. However, the estimates in Lemma \[lem-R\] are considerably more complicated than in the discrete time case.

Estimates 