Let
$w$
be either in the Muckenhoupt class of
${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$
weights or in the class of
$QC\left( {{\mathbb{R}}^{n}} \right)$
weights, and let
${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla \right)$
be the degenerate elliptic operator on the Euclidean space
${{\mathbb{R}}^{n}}$
,
$n\,\ge \,2$
. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space
$H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$
associated with
${{L}_{w}}$
for
$p\,\in \,(0,\,1]$
, and when
$p\,\in \,(\frac{n}{n+1},\,1]$
and
$w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$
with
${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$
, the authors prove that the associated Riesz transform
$\nabla L_{w}^{-1/2}$
is bounded from
$H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$
to the weighted classical Hardy space
$H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$
.