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Hardy spaces on metric measure spaces with generalized sub-gaussian heat kernel estimates

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 $L^1$ .