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The Hardy space \(H^1\) on non-homogeneous spaces and its applications – a survey. (English) Zbl 1277.42002

Summary: Let \((\mathcal X,d,\mu)\) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. 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 \(\widetilde{\text{RBMO}}(\mu)\), the regularized BLO spaces RBLO(\(\mu\)) and \(\widetilde{\text{RBLO}}(\mu)\), the Hardy spaces \(H^1(\mu)\) and \(\widetilde H^1(\mu)\), the behaviour of the Calderoń-Zygmund operator and its maximal operator on Hardy spaces and Lebesgue spaces, a weighted norm inequality for the multilinear Calderoń-Zygmund operator, the boundedness on Orlicz spaces and the endpoint estimate for the commutator generated by the Calderoń-Zygmund operator or the generalized fractional integral with any RBMO(\(\mu\)) function or any \(\widetilde{\text{RBMO}}(\mu)\) function, and equivalent characterizations for the boundedness of the generalized fractional integral or the Marcinkiewicz integral, respectively.

MSC:

42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30L99 Analysis on metric spaces
Full Text: MNR