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.