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
=- [ 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 =
if a G
+ A\ is an element for which there exists a representation
a = J0°° ^p-dt, with a(t) e A0nA1 then
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
1 p oo, the classes of
Lebesgue measurable functions / defined on the interval (0, oo) such that
r /«oo \ l / p
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.
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