  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 $L^{1, \infty}(\mu)$, if $T$ maps all $(p,
\gamma)$-atomic blocks with $p\in(1, \infty)$ 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.
