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.
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