1. INTRODUCTION

5

where k is the derivative of the functional K. Note that both jK{t,a\A) and

fc(t, a, A) are decreasing functions. Therefore the Bp weights control the equivalence

between the K and k methods of interpolation (cf. §4 below).

Let us point out that it is only in this generalized setting that we can formulate

an analogue of Rubio de Francia's extrapolation theorem. To see better the role

that generalized weights play let us recall that by reiteration,

{Ae0lpQ-K,Ae1^p1-K)e,q = (Ao, A i ) ^ ^ ,

where rj — (1 — 0)6o + 66\. Thus, the second index is not important for reiteration.

Consequently for power weights, that is in the classical setting of the Lions-Peetre

spaces, reiteration plays the role of extrapolation in the sense of Rubio de Francia

and thus it is not of interest in this context. However, in the setting of the gen-

eralized Cp -weights, the following extrapolation result holds (see §3 Theorem 3.10

below.)

THEOREM.

Let A, B two compatible pairs of Banach spaces, and let T be a

linear operator bounded from AW^K into BW^K for some p, 1 p oo, and for

all w G Cp, with norm that depends only upon the Cp-constant for w. Then T is

also bounded from AV^K into BV^K for any q, 1 q oo, and for all v G Cq with

norm that depends only upon the Cq-constant for v.

As an important corollary of our presentation, we obtain a new result even for

classical functional parameter in terms of extrapolation (see Corollary 3.11).

We shall also show in §5 that if we have a family of estimates for a bounded

operator on the classical Lions-Peetre scale and if these estimates do not 'blow up'

too fast then an extension of the previous extrapolation theorem holds and can be

obtained using the extrapolation method of Jawerth and Milman (cf. [JM]). This

provides us with a simple and effective method to prove weighted Lorentz estimates

of the type considered by Arifio and Muckenhoupt (cf. [AM]).

Real interpolation spaces are constructed by means of decomposing effectively

their elements. Given a bounded operator T between two Banach pairs A and

B, these decompositions can be applied before and after applying T. These con-

siderations lead to the construction of operators that are based on these optimal

decompositions, we shall call them QK ^ and QK g and lead to the study of the

commutator

[T,ilK]a={TQKiA-nKiBT)a,

where a G Aw,p.K (cf. [RW], [JRW]).

For specific pairs these commutators are some of the classical commutators

studied in classical analysis. In the second part of the paper we consider the role that

weights play in the theory. In particular we consider commutators with operators

O which are constructed using these weights, as well as norm estimates of these

operations.

One concrete application of our results provides new weighted norm inequalities

for commutators of singular integrals with multiplications with BMO functions.

Moreover, as indicated above, these estimates, when combined with the results

described in [Ch], provide new applications to the regularity theory of quasilinear

elliptic equations.

We also remark that our methods also apply to other operators including frac-

tional integrals, and given the generality of our assumptions, our methods also