  Hardy space theory has been studied on manifolds or metric measure spaces
equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However,
there are natural examples where one finds a mix of both behaviour (locally
Gaussian and at infinity sub-Gaussian) in which case the previous theory
doesn't apply. Still we define molecular and square function Hardy spaces using
appropriate scaling, and we show that they agree with Lebesgue spaces in some
range. Besides, counterexamples are given in this setting that the $H^p$ space
corresponding to Gaussian estimates may not coincide with $L ^p$. As a
motivation for this theory, we show that the Riesz transform maps our Hardy
space $H^1$ into $L^1$ .
