16 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
hi) Let / be an element in
Lq(w),
then
wtep
G Mp with constant
\\wtep\\Mv
n * ,
1 - ep\\w\\Mp
whenever ep||u||M 1. By Holder's inequality we have
\P{!){t)\\j\f(x)\dx
/1 ft \
P/I
/ 1 ft \ Q-p/q
( - / \f(x)\qlpxqlp-l-edx\ (- x(-l+€p'(q-p)dx\
/-, ft \ Pit
with C = I - - 1 . Therefore,
f
\P{f){t)\qw{t)tq-pdt Cq
J (P
(\f\q/pxqlp-1-^ (i))P tpcw{t)dt
/•oo /»oo ;
C" \
\f\qxq-p-epw(x)xepdx
= C" /
|/|9x«-pda;
Jo Jo
where
C" =
q-p\q~p(
M\w\\Mp
£P J \l-ep||H|M
p
Thus, if we once again take 2pe||?i;||Mp = 1 we obtain
\\wtq-p\\Mq
8||«;||Mp(g-p)
1
-
p / 9
and the result follows.
REMARK.
In order to complete the information we have about the Cp weights
it is worth to mention here (cf. the Remark after Proposition 3.6 bellow) that the
classes Cp are log-convex. This fact should be added to the results given in the
Lemma 2.5, quoted before.
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