Extrapolation from A∞ weights and applications

We generalize the Ap 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 Lp 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 Coifmanʹs 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.