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: