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

Martell, José María; Prisuelos-Arribas, Cruz

Publ. Mat., Barc. 62, No. 2, 475-535 (2018).

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.

Reviewer: Santiago Boza (Barcelona)

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

Keywords:

Hardy spaces; second order divergence form elliptic operators; heat and Poisson semigroups; conical square functions; non-tangential maximal functionss; molecular decomposition; Muckenhoupt weights; off-diagonal estimates