Weighted Hardy spaces associated with elliptic operators. Part II : characterizations of h1 L (w)

Given a Muckenhoupt weight w and a second order divergence form elliptic operator L, we consider different versions of the weighted Hardy space H1 L (w)defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by L. We show that all of them are isomorphic and also that H1 L (w) admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.