2. CALDERON WEIGHTS 11

PROPOSITION

2.4. IfweCp then there exists e 0 such that

x~epw(x)

G

Mp

and xepw(x) G Mp.

PROOF.

According to proposition 2.2, w = wow1~p and there exist constants

C 0 and e 0 such that

Jo

xe

Jt

\xj

x€ te

for all £ 0, i = 0,1. Hence, by an argument similar to the one in the converse

part of the proof of proposition 2.2 we see that the weight

x~epw(x)

G

Mp

and the

weight

xepw(x)

G Mp. •

REMARK.

The last proposition ensures that a weight w in the class Cp supports

stronger integrability conditions at 0 and at oo.

We are now going to study analogs of Rubio de Francia's extrapolation theorem

in the context of Cp weights.

First, we need the following convexity property.

LEMMA

2.5. Let 1 p oo. IfueCi,ve Cp and 0 s 1 then ^v1'3 e

^s-\-p(l — s) •

PROOF.

In order to check the condition Ms+p(!_s) we use Holder's inequality

with exponents 1/s and 1/(1 — s) in the first factor, and in the second one the fact

that if u € C\ and (3 0 then for x t,

u(xf

c( T

»(»»*y.

y

Therefore, we have

(jf

"ixJlt-7d*)

Of

»M-»M1-r-*-fc)'",,""

which is bounded by the hypothesis.

The condition M s + p ( 1 _ s ) is checked in a similar way. •

The next lemma is the crucial step for the extrapolation result. We follow the

same ideas as in ^4p-theory (see [GR], chapter IV]).

LEMMA

2.6. Let l p o o , 1 r oo; r ^ p. Let s be such that ^ — |1— -\.

Let w G Cp. Then, \/u 0 in Ls{w) there exists v 0 in Ls(w) such that:

a) u(x) v(x) for a.e. x G (0, +oo).

b) \\V\\L°(W) C\\u\\Ls{w).

c) Ifrp,vw£Cr. If p r,

v~1w

G Cr.