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

Let $\mu$ be a non-negative Radon measure on ${\mathbb{R}}^d$ which only satisfies the polynomial growth condition. Let $\mathcal{Y}$ be a Banach space and $H^1(\mu )$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(\mu )$ to $\mathcal{Y}$ if and only if $T$ maps all $(p,\gamma )$-atomic blocks into uniformly bounded elements of $\mathcal{Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(\mu )$ to $L^{1,\mathrm{\infty }}(\mu )$, if $T$ maps all $(p, \gamma)$-atomicblockswith$p\in(1, \infty)$and$\gamma\in{\mathbb N}$ 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 $h^1(\mu )$, corresponding results are also presented. Finally, these results are applied to Calder'on-Zygmund operators, Riesz potentials and multilinear commutators generated by Calder'on-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.