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

We present an extrapolation theory that allows us to obtain, from weighted Lp inequalities on pairs of functions for p fixed and all A∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A∞ weights and also modular inequalities with A∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz–Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.