Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

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Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights (2006, methodology): “f T is bounded on some Lp(w) for some ﬁxed p and for all w ∈ Wp p0,q0 , then the same happens for every q ∈ (p0,q0) and the corresponding class of weights. Using ideas on extrapolation from [CMP] and [CGMP], we obtain vector-valued inequalities automatically again for limited ranges of p (Section 4.”
Weighted weak-type inequalities and a conjecture of Sawyer (2005, unknown): “11, we first note that in [8] it was shown that given 1 < q < ∞, for all 0 < p < ∞ and w ∈ A∞, ∥∥‖{Mfj}j‖`q∥∥Lp(w) ≤ C ∥∥M(‖{fj}j‖`q)∥∥Lp(w).”