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

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