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Weighted Hardy spaces associated with elliptic operators. II: Characterizations of $$H^1_L(w)$$. (English) Zbl 1407.42011
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.

##### MSC:
 42B30 $$H^p$$-spaces 35J15 Second-order elliptic equations 42B37 Harmonic analysis and PDEs 42B25 Maximal functions, Littlewood-Paley theory 47D06 One-parameter semigroups and linear evolution equations 47G10 Integral operators
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