10 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
Q ^ ) H (
S
) ^ ^ ^ ^ ^
(2
.10)
P^HW)(s) =
lJ\(X)l^^dX^
(2.11)
where P^ = Po .(* oP and Q^ = Qo .(* oQ.
Now, in order to prove proposition 2.3, pick a constant e 0 such that eC 1.
By (2.10) and (2.11), we have
and
/ _ -. JS X
*"Wfc
£«"**(,,).)_-£-/•!£*.
Moreover, by (2.8) and (2.9)
and
Qo
[J2ek-1Pk\
(w) C'w.
This means that
rt
and
t Jo x e Jo t Jt x
JQ
= - f w(x)dx+t-e r ^jdx C'w{t),
J0 x* Jt
s2-'
Jt x* Jx
s2-
as we wished to show.
REMARKS.
i) If a weight satisfies the corresponding inequality for some e 0 it also does for
any 0 S e.
ii) The method of proof of Proposition 2.3 also applies to the Bp weights (see
Definition 4.1 below) studied by Arino and Muckenhoupt (cf. [AM]). For
these weights it is known that w e Bp = w G 23p-e, for some e 0 (see [AM],
[Nl]). We can obtain a simple direct proof of this fact using arguments similar
to the ones in Proposition 2.3 (cf. [BR]).
As a consequence of propositions 2.2 and 2.3 we achieve a kind of reverse Holder
type inequality for the class Cp.
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