12 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
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 —
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
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).
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
(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
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.
We consider in detail the case where T is assumed to be bounded
for any weight w G Cr. Let w G Cp, and for u G
u 0, with
— 1? 1/s = 1 — r/p, let v G
be a function associated to u according
to lemma 2.6.
Then, we have the following chain of inequalities
/ roo \ r/p ,.00
/»o o /»oo
/ \Tf(x)\rv(x)w(x)dx C \f(x)\rv(x)w(x)dx
/»oo \ r/p / /.oo
- • • -
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. •
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
Then T is also bounded on
with norm depending only on that of S.
Therefore, if S is the Calderon operator we have: