12 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
PROOF.
Let us consider first the case r p. Then, if we let t G (0,1] be
such that r = (1 t)p + t, we have, - = 1
L
and s = p'jt. We define the
sublinear operator N(u) (S(\u\lj/tw)w~1) , where S is the Calderon operator.
This operator is bounded on LP ^(w) since
w1~p
=
w~~p lp
G Cp. We can apply
Rubio de Francia's algorithm ([GR], lemma 5.1) and given u 0 we obtain v 0
satisfying a), b) and N(v) Cv, that is S(v1^tw)w~1 v1^ which means that
vl^w G C\. Finally, by lemma 2.5, we have (v^^wYw1"* G Ct+pri-t) = Cr.
lip r the same argument works (duality of the Cp classes) exactly as in [GR],
lemma 5.18.
The previous lemma combined with Holder's inequality and duality lead us to
the following extrapolation result (see [GR] theorem 5.19, for details).
PROPOSITION
2.7. Let T be a sublinear operator acting on functions defined
on (0, -hoc). Let 1 r +oo, 1 p oo. Suppose that T is bounded on
Lr(w)
(respectively, T is of weak type (r,r)), for every weight w G Cr with norm that
depends only upon the Cr-constant for w, then T is bounded on
Lp(w)
(respectively,
T is of weak type (p,p))7 for all weights w G Cp with a norm that depends only upon
the Cp-constant for w.
PROOF.
We consider in detail the case where T is assumed to be bounded
on
Lr(w)
for any weight w G Cr. Let w G Cp, and for u G
Ls(w),
u 0, with
||
W
IIL
S

1? 1/s = 1 r/p, let v G
Ls(w)
be a function associated to u according
to lemma 2.6.
Then, we have the following chain of inequalities
/ roo \ r/p ,.00
( /
\Tf(x)\pw(x)dx)
= /
\Tf(x)\ru(x)w(x)dx
/•CO
/»o o /»oo
/ \Tf(x)\rv(x)w(x)dx C \f(x)\rv(x)w(x)dx
Jo Jo
/»oo \ r/p / /.oo
/
\f(x)\•pw(x)dx\-
I
v(x)ip/r)'w(x)dw------
(p/r)'
c
;;;
- -
= C\\v\\Ls{v)\\f\\lP{w)C\\f\\lP(w),
where we have used duality, Holder's inverse inequality and the hypothesis. The
remaining cases can be obtained in a similar fashion adapting the arguments of
[GR], Chapter IV, Theorem 5.19, and we shall therefore omit the details.
REMARK.
It is also possible to extrapolate in the case r = oo, using the fol-
lowing extrapolation theorem by Garcia-Cuerva ([GC]):
Let S be a positive sublinear operator and let T be a mapping satisfying the
following condition: Every time that S is bounded on L°°(v), for some v 0, T is
also bounded on L°°(v), with norm depending only on that of S. Let 1 p oo
and w 0. Suppose that S is bounded on
Lp(w).
Then T is also bounded on
Lp(w)
with norm depending only on that of S.
Therefore, if S is the Calderon operator we have:
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