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Given a Muckenhoupt weight $w$ and a second order divergence form ellipticoperator $L$, we consider different versions of the weighted Hardy space$H^1_L(w)$ defined by conical square functions and non-tangential maximalfunctions associated with the heat and Poisson semigroups generated by $L$. Weshow that all of them are isomorphic and also that $H^1_L(w)$ admits amolecular characterization. One of the advantages of our methods is that ourassumptions extend naturally the unweighted theory developed by S. Hofmann andS. Mayboroda and we can immediately recover the unweighted case. Some of ourtools consist in establishing weighted norm inequalities for the non-tangentialmaximal functions, as well as comparing them with some conical square functionsin weighted Lebesgue spaces.

Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$