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

Abstract

Let \((X, d, \mu)\) 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 BMO spaces RBMO\((\mu)\) and \(\widetilde{\mathrm{RBMO}}(\mu)\), the regularized BLO spaces 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 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.