Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of H_L^(p)(w) and the weighted Hardy space associated with the Riesz transform

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 $q\in \mathcal{W}_w(q_-(L),q_+(L))$.