Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures

Let mu be a non-negative Radon measure on R-d which satisfies only the polynomial growth condition. Let Y be a Banach space and H-1(mu) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H-1 (mu) to Y if and only if T maps all (p, gamma)-atomic blocks into uniformly bounded elements of Y; moreover, the authors prove that for a sublinear operator T bounded from L-1 (mu) to L-1,L-infinity(mu), if T maps all (p, gamma)-atomic blocks with p is an element of (1, infinity) and gamma is an element of N into uniformly bounded elements of L-1 (mu), then T extends to a bounded sublinear operator from H-1 (mu) to L-1 (mu). For the localized atomic Hardy space h(1) (mu), the corresponding results are also presented. Finally, these results are applied to Calderon-Zygmund operators, Riesz potentials and multilinear commutators generated by Calderon-Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.