  We consider Muckenhoupt weights $w$, and define weighted Hardy spaces
$H^p_{\mathcal{T}}(w)$, where $\mathcal{T}$ denotes a conical square function
or a non-tangential maximal function defined via the heat or the Poisson
semigroup generated by a second order divergence form elliptic operator $L$. In
the range $0<p< 1$, we give a molecular characterization of these spaces.
Additionally, in the range $p\in \mathcal{W}_w(p_-(L),p_+(L))$ we see that
these spaces are isomorphic to the $L^p(w)$ spaces. We also consider the Riesz
transform $\nabla L^{-\frac{1}{2}}$, associated with $L$, and show that the
Hardy spaces $H^p_{\nabla L^{-1/2},q}(w)$ and
$H^p_{\mathcal{S}_{\mathrm{H}},q}(w)$ are isomorphic, in some range of $p'$s,
and $q\in \mathcal{W}_w(q_-(L),q_+(L))$.
