Boundedness of Linear Operators via Atoms on Hardy Spaces with Non-doubling Measures

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