2. CALDERON WEIGHT S 9
which, by (2.6) is bounded by a constant times
Since t x implies
Wi WX
Jo
and t x implies
//o "
Jo
then
/
s~1wo(s)ds
/
s~1wo(s)d,
Jx Jt
Since the verification of the condition (Mp) can be obtained in a similar fashion
we shall skip the details.
REMARK.
The converse of Proposition (2.2) is also a consequence of the main
result in [Bl], since S is an operator defined by means of a positive kernel. Another
direct consequence of the result in [Bl] is the following factorization for Mp and
Mp
weights:
PROPOSITION
2.2'. Let 1 p oo. A weight w e Mp (resp.
Mp)
if and only
if there exist two weights wo G Mi (resp.
M1),
w\ G M
1
(resp. Mi) such that
W = WQW-L
The following result can be seen as a type of reverse Holder inequality for
Ci weights. Results of this type have applications in interpolation theory for re-
arrangement invariant Banach spaces that we shall pursue elsewhere (see [BR],
[BMR]).
PROPOSITION
2.3. If w e C\ then there exists e 0 such that, for some
constant C 0, w satisfies
/ (-) w(x)dx+ (-] w(x)dx Ctw(t)
for all t 0.
PROOF.
Let C be the constant such that P(w)(t) + Q(w)(t) Cw(t), Vt 0.
In order to prove the proposition we shall use the following facts which can be easily
proved by Fubini's theorem and by induction
p0Q = Q0p (2.7)
P o Q^ (^)
C
kw, for all k 1 (2.8)
Q o p(k\w) p~xCkw, for all k 1 (2.9)
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