1. INTRODUCTION 3

If / is a measurable function defined on the interval (0, oo), the Calderon op-

erator S is defined by

Sf(t) = / min{l/x, l/t}f(x)dx

Jo

=- [ f(x)dx+ r f~^dx

t Jo Jt x

= P(/)(t) + Q(/)(*).

P is called the Hardy operator and Q is its adjoint. The Calderon operator plays

an important role in interpolation theory. In particular it controls the relationship

between the if-method and the J-method of interpolation, a result due to Brundyi

and Krugljak (see [BK]). Indeed, given a compatible couple of Banach spaces A =

(AQ,AI),

if a G

AQ

+ A\ is an element for which there exists a representation

a = J0°° ^p-dt, with a(t) e A0nA1 then

K{t,a;A)

s

rj(x,a(x);A)\

where K(t, a; A) is the classical if-functional of interpolation, i.e.

K(t,a;A) = mf{\\a0\\Ao+t\\a1\\Al},

where the inf runs over all possible decompositions a = CLQ + a\ with a^ G i i , and

J is the classical J-functional given by

J(t,a;A) =max{||a||A0Jt||a|Ui}

for the elements a G A0 fl A\.

In the first part of the paper we show that the class of weights that controls

the weighted LP estimates for the Calderon operator can be used develop a rich

theory of interpolation which includes some novel features that are not present in

the classical spaces of Lions-Peetre.

Heuristically the Calderon operator is 'the only' operator we should consider

in order to obtain a universal class of weights that is suitable for interpolation

theory, since it majorizes in a suitable sense all other operations in a given inter-

polation segment. This should be compared with the corresponding theory of Ap

Muckenhoupt weights, based on the maximal operator of Hardy and Littlewood.

Let u be a weight on (0, oo), i.e., w is a measurable function, w 0 a.e. with

respect to the.Lebesgue measure. We denote by

Lp(w),

1 p oo, the classes of

Lebesgue measurable functions / defined on the interval (0, oo) such that

r /«oo \ l / p

J

\f(t)\pw(t)dt\

+oo.

For p = oo the corresponding space, L°°(w), is defined using the norm

H/Hoo,™ = H / H l o o + ° ° -

In [GC] these spaces are denoted by w~1L°° and we shall also adopt this notation,

when convenient, in what follows.