Wavelet characterization of weighted spaces

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New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators (2011, methodology): “classical Hardy-Orlicz spaces of Janson [34] (w≡ 1 in this context) and the classical weighted Hardy spaces of Garc´ıa-Cuerva [22], Stromberg and Torchinsky [57] (Φ(t) ≡ tp in this context), see also [46, 10, 23] . NEW HARDY SPACES OF MUSIELAK-ORLICZ TYPE 9 Next, to introduce the atomic Hardy spaces”
Commutators in the two-weight setting (2015, background): “x) ··= sup r>0 |f ∗ϕr(x)| is the maximal function. There are many equivalent ways to deﬁne the weighted Hardy spaces – in terms of the square function, or in terms of an atomic decomposition – see [12,39]. COMMUTATORS IN THE TWO-WEIGHT SETTING 23 Now let f ··= g1(Tg2)−(T∗g1)g2. Then for g1 ∈ Lq(λ”