1. Introduction

Many of the problems we study in analysis can be rephrased in terms of the

study of the action of operators in function spaces. The more we understand about

the action of a given operator in as many as possible function spaces, the more we

understand about the nature of the problem under consideration. Interpolation is

a very useful tool for this purpose as it provides us with methods to obtain new

estimates from old ones.

In recent years a theory of weighted norm inequalities for classical operators

has been developed. It has led to a deeper understanding of many important prob-

lems in analysis. Moreover, it has led to the discovery of unexpected relationships

between different areas of analysis.

At the basis of these developments it is the celebrated theory of Muckenhoupt

for the maximal operator of Hardy-Littlewood M (see [Mul]). Let 1 p oo,

then we have M :

Lp(w)

— »

Lp(w)

if and only if the weight w belongs to the class

Ap. The Ap classes of weights admit a concrete description and their properties

have been intensively investigated. Muckenhoupt's results led to an extensive study

of weighted norm inequalities for classical operators (singular integral, multipliers,

square functions, ...). This has also uncovered deep connections between classes of

weights and function spaces, like Ap and BMO.

A beautiful connection between the theory of weighted norm inequalities and

the theory of factorization of operators on Banach spaces is given by Rubio de

Francia's extrapolation theorem. A simple version of this result states that if T is

a bounded linear operator on L2(w), for all w G A2, then T is also bounded on

Lp(w), for all 1 p 00 and for all w G Ap.

There is a close connection between interpolation theory and weighted norm

inequalities. In particular, interpolation theory provides methods to obtain re-

arrangement inequalities for operators to which one can then apply weighted norm

inequalities. While interpolation has been useful in the study of weighted norm in-

equalities it seems to us that a deeper study of the connection between interpolation

theory and weighted norm inequalities is still to be developed.

For example, what would be the analog of Rubio's theorem for interpolation

scales?, what would be the natural classes of weights that we should consider in a

general theory of real interpolation scales?

In the first part of this paper we consider these questions and establish an

extrapolation theorem for operators acting on weighted real interpolation spaces.

In order to even formulate these results we are forced to generalize the classical

theory of Lions-Peetre through the consideration of weighted Lp spaces. In this

Received by the editor November 20, 1996; and in revised form December 28, 1999.

The first author was partially supported by DGES.

The third author was partially supported by DGES.

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