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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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