4

JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ

Let w be a weight and let 1 p oo. We define Aw^p-x as the class of vectors

a G Ao + A\ for which the function t~lK(t, a; A) G Lp(w). For a G AWJP;K we let

MU _ = (f(^^)'H'"-

If we consider the J-method of interpolation, we define AW:P;j as the class of el-

ements a G AQ + A\ for which there exists a representation a — J0°° ^-dt with

a(t) G AQ H A\ satisfying t~1J(t,a(t); A) G Lp(w). For this class we consider the

corresponding norms

M

^ . M (£(«*&*)'wf

where the inf runs over all possible representations of a.

The classical scales of real interpolation spaces of Lions-Peetre correspond to the

power weights w = tp~p0~1. Apart from this case, the most studied classes of inter-

polation spaces are the so called "functional parameter" and the "quasipower" cases.

These spaces are defined by a slightly more general class of weights than powers.

If v is a quasipower weight, the interpolation scales associated with the functional

parameter are classically defined by (Ao,Ai)ViP]K = {a G AQ + A\\ \\CL\\VIP]K 00}

where

f°° - dt

\\a\\v,p,K= (K(t,a;A)v)PT.

Jo l

The study of interpolation spaces defined using these classes of weights was initiated

in [K] and continued in [G] and many other papers, cf. [BK]. In our notation we

have

{AO,AI)V^P]K = AW:P;K

with w = vptp~1.

The point of view advocated in this paper is related to work by Sagher [Sg],

where Calderon type of weights are used to extend classical interpolation theorems

and also with a question raised by E. Hernandez and J. Soria. In [HS] the authors

showed that weights of the form w = vptp~1^ v a quasipower weight, are in the class

Cp (see Theorem 4.1 in [HS]) and asked for a general theory of real interpolation

method with weights in Cp, which "would be more general than the existing ones."

In this paper we have developed such a theory in detail. Furthermore we have

proved that for p = 1 the scales introduced by this method can be represented

using quasipower weights (see Proposition 3.8 below). More generally, for p 1,

we indicate the relationship between our interpolation scales with those derived

using the classical functional parameter approach (see Proposition 3.9 below).

One can also consider other classes of weights and their corresponding asso-

ciated interpolation methods. In this paper we also develop, in some detail, the

theory associated with the Bp classes introduced in [AM] in the study of the Hardy

operator P acting on weighted Lp(w) spaces but restricted to non-increasing func-

tions. Their direct import in interpolation theory can be seen from the fact that,

if Ao fl Ai is dense in AQ, we have (cf. [BS])

1 - 1 /•* -

-K(t,a;A) = - / k(x,a,A)dx = P(k(-,a;A))(t),

t t

Jo