The Hardy space $H^1$ on non-homogeneous spaces and its applications – a survey

Let $(\mathcal X,d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this article, the authors give a survey on the Hardy space $H^1$ on non-homogeneous spaces and its applications. These results include: the regularized $\mathrm{BMO}$ spaces $\mathrm{RBMO}(\mu)$ and $\widetilde{\mathrm{RBMO}}(\mu)$, the regularized $\mathrm{BLO}$ spaces $\mathrm{RBLO}(\mu)$ and $\widetilde{\mathrm{RBLO}}(\mu)$, the Hardy spaces $H^1(\mu)$ and $\widetilde H^1(\mu)$, the behaviour of the Calderón–Zygmund operator and its maximal operator on Hardy spaces and Lebesgue spaces, a weighted norm inequality for the multilinear Calderón–Zygmund operator, the boundedness on Orlicz spaces and the endpoint estimate for the commutator generated by the Calderón–Zygmund operator or the generalized fractional integral with any $\mathrm{RBMO}(\mu)$ function or any $\widetilde{\mathrm{RBMO}}(\mu)$ function, and equivalent characterizations for the boundedness of the generalized fractional integral or the Marcinkiewicz integral, respectively.