Weighted Hardy spaces associated with elliptic operators. II: Characterizations of \(H^1_L(w)\) (Q1660534)

The present paper is the second of a series of three works developing the theory of weighted Hardy spaces associated to elliptic operators (see [the authors, Trans. Am. Math. Soc. 369, No. 6, 4193--4233 (2017; Zbl 1380.42019); J. Geom. Anal. 29, No. 1, 451--509 (2019; Zbl 07024076)]). In this particular item, given a weight \(w\) in the class of Muckenhoupt and a second order divergence form elliptic operator \(L\), different characterizations of the weighted Hardy spaces \(H_L^1(w)\) are considered. Starting from a characterization in terms of conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by \(L\) it is shown that all of them are isomorphic and a decomposition in terms of molecules of the space is obtained.