Factorization of Operators and Weighted Norm Inequalities

Abstract

The purpose of these lectures is to show how the theory of factorization of operators developed by B. Maurey in the 1970’s can be applied to obtain very interesting results about weighted norm inequalities. The idea to carry out this program is due to Jose Luis Rubio de Francia. He constructed a beautiful theory, which culminates in the extrapolation theorem. This theory is presented in chapter VI of our book [8] in the context of LP spaces. Here we have chosen to work in a more general class of Bausch function spaces, an approach that Jose Luis Rubio also adopted in some later works [20], [21], [22]. There are two reasons to do this. First of all, the presentation of the main results becomes much clearer, and besides there are very nice applications to Banach lattices to be discussed in Section 5. There are several approaches to extrapolation, giving rise to different results. We have chosen the original approach of Jose Luis Rubio de Francia, but we have completed the theory so that all the known results become part of it.