We generalize the A p extrapolation theorem of Rubio de Francia to A ∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving L p norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator. © 2003 Elsevier Inc. All rights reserved.
Cruz-Uribe, D., Martell, J. M., & Pérez, C. (2004). Extrapolation from A ∞ weights and applications. Journal of Functional Analysis, 213(2), 412–439. https://doi.org/10.1016/j.jfa.2003.09.002