Abstract

AbstractIn this chapter, we introduce and study a class of metric measure spaces \((\mathcal{X},d,\nu )\), which include both Euclidean spaces with nonnegative Radon measures satisfying the polynomial growth condition and spaces of homogeneous type as special cases. We also introduce the BMO-type space \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and the atomic Hardy space \({H}^{1}(\mathcal{X},\,\nu )\) in this setting, establish the John–Nirenberg inequality for \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and some equivalent characterizations of \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and \({H}^{1}(\mathcal{X},\,\nu )\), respectively, and show that the dual space of \({H}^{1}(\mathcal{X},\,\nu )\) is \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\).KeywordsMeasure SpaceHomogeneous TypeDoubling ConditionDoubling MeasureEquivalent CharacterizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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