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The Hardy Space $H^1$ on Non-homogeneous Metric Spaces

Let $({\mathcal X}, d, μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(μ)$ and prove that its dual space is the known space ${\rm RBMO}(μ)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(μ)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón--Zygmund operators from $H^1(μ)$ to $L^1(μ)$.