been proved only for a very thin set of Anosov flows (those that possess a contact structure or have exceptionally smooth stable and unstable foliations), see [@Dolgopyat98a; @Liverani04]. On the other hand, superpolynomial decay of correlations holds for “typical” uniformly hyperbolic flows [@Dolgopyat98b; @FMT07] for observables that are sufficiently regular. The typical set of flows includes those with a pair of periodic points whose ratio of periods is Diophantine [@Dolgopyat98b] and also includes an open and dense set of flows [@FMT07]. Results on superpolynomial decay were extended by [@M07] to nonuniformly hyperbolic flows whose Poincaré map is within the class considered in [@Young98]. For flows whose Poincaré map lies in the class considered in [@Young99], it was shown in [@M09] that typically polynomial decay holds for sufficiently regular observables.

In the current paper, we develop a continuous time operator renewal theory, and thereby obtain results on sharp lower bounds for finite measure semiflows with polynomial decay of correlations, and mixing (as well as higher order asymptotics) for infinite measure semiflows, extending the discrete time results of [@Gouezel-sharp; @Sarig02; @MT12].

Illustrative examples
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To describe the main results, we consider (mainly for convenience) the family of Pomeau-Manneville intermittent maps [@PomeauManneville80] considered 