  Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and
Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce
the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with
$p\in (0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato
space ${\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ and the localized
Morrey-Campanato-BLO space $\widetilde{\mathcal E}^{\alpha, p}_{\mathcal
D}({\mathcal X})$ with $\az\in{\mathbb R}$ and $p\in(0, \infty)$ and establish
their basic properties including $H^{p, q}_{\mathcal D}({\mathcal X})=H^{p,
\infty}_{\mathcal D}({\mathcal X})$ and several equivalent characterizations
for ${\mathcal E}^{\alpha, p}_{\mathcal D}({\mathcal X})$ and $\wz{\mathcal
E}^{\alpha, p}_{\mathcal D}({\mathcal X})$. Especially, the authors prove that
when $p\in(0,1]$, the dual space of $H^{p, \infty}_{\mathcal D}({\mathcal X})$
is ${\mathcal E}^{1/p-1, 1}_{\mathcal D}({\mathcal X})$. Let $\rho$ be an
admissible function modeled on the known auxiliary function determined by the
Schr\"odinger operator. Denote the spaces ${\mathcal E}^{\alpha, p}_{\mathcal
D}({\mathcal X})$ and $\widetilde{\mathcal E}^{\alpha, p}_{\mathcal
D}({\mathcal X})$, respectively, by ${\mathcal E}^{\alpha, p}_{\rho}({\mathcal
X})$ and $\widetilde{\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$, when
${\mathcal D}$ is determined by $\rho$. The authors then obtain the boundedness
from ${\mathcal E}^{\alpha, p}_{\rho}({\mathcal X})$ to $\widetilde{\mathcal
E}^{\alpha, p}_{\rho}({\mathcal X})$ of the radial and the Poisson semigroup
maximal functions and the Littlewood-Paley $g$-function which are defined via
kernels modeled on the semigroup generated by the Schr\"odinger operator.
