PIP
14 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
PROOF.
Suppose that the weight w satisfies Mp and M1. Then
tP-1
j°° ^~dx
Ctp~x
(f
w{x)~p'^pdx
= c(- f w(x)-p'/pdx\
1
il
C- w(x)dx Cw(t),
t Jo
where we have used Jensen's inequality.
In order to conclude the proof we will prove that (2.12) implies the conditions
Mp and
Mp
(note that M
1
is trivially satisfied).
Using arguments similar to the ones used in the proof of the reverse Holder
inequality for the C\ weights, it is not difficult to prove that condition (2.12) also
implies a reverse Holder inequality. Actually, there exists a constant C 0 and
e 0 such that the following inequality holds
- /
x-ew(x)dx^-tp-1
r
°^^-dxCtew(t).
t Jo Jt xP
Hence
where in the last inequality we have used the fact that
r ^idx t- r ^
Jt
XP
- Jt
XP
X I
1/p'
1/P / roo „,e„„{„.\ \ -1/ P
c,
Consequently, the condition Mp holds. Now we shall prove that
Mp
holds:
IoW{x)dX)
( /
W(X)-P'/P
-ax
XP
/P
/ /»oc /
r
x \ -p'/p \ 1'P
/ rt \
11/P
I poo / px \~
P 'P
C( w(x)dx\ / aj-
1
-
e p
'
/ p
f /
y-ew(y)dy\
dx
Ct-'l* (j w(x)dx) (J
y-ew(y)dy]
C,
the estimate
/ y~ew(y)dy t~e / w(y)dy
Jo Jo
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