The Hardy space H1 on non-homogeneous metric spaces

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