JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
Indeed,
idxiI
1 = / dx %
(i;^(i;^\r)ir^r
DEFINITION
2.1. Let 1 p oo. We say that a weight w G Cp if it satisfies the
conditions Mp and
Mp
simultaneously. We say that a weight w G C^ if
w~1
G C\.
We also define the corresponding "norms" for the Cp weights (1 p oo) by
\\w\\Cp = \\w\\MP + \\w\\Mp. (2.4)
Since the Calderon operator is S P + Q, and P, Q are positive, the class
Cp is actually the class of weights for which 5 is bounded from
Lp(w)
into
Lp(w).
Then
^lhllcp||S|Up(ti,)-.LP(«,)4|h||Cp. (2.5)
For p = oo it is easy to see that S is bounded from L°°(w) into L°°(w) if and only
if S(w~1)(x) Cw~l(x), a.e. x, for some constant C 0, that is, W - 1 G Ci .
The operator S is positive and selfadjoint, and since the class C\ coincides with
the class of weights for which there exists a constant C 0 such that Sw Cw, we
can apply to these classes the strong machinery developed by Rubio de Francia and
prove for them similar properties to the ones satisfied by the classical Ap classes,
i.e., factorization and extrapolation. We also study reverse Holder type inequalities
for them.
PROPOSITION
2.2. Let 1 p oo. A weight w G Cp if and only if there exist
two weights Wo, w\ G C\ such that w WQW1~P.
PROOF. If w G Cp then the desired factorization can be obtained using Rubio
de Francia's factorization theorem (cf. [GR]). Indeed, we just need to recall that
S P + Q is selfadjoint and the following duality property holds: w G Cp if and
only if w~v lp G Cp. Let us also remark that [Mu2] has constructive factorizations
results of the same type for each of the classes Mp and Mp of weights.
The converse can be obtained by direct computation. Suppose that i^, i = 0,1,
are two C\-weights. Let w = WQW1
~p.
By definition we can assume that
t'1 I Wi(x)dx+ (-J Wi(x)dx Cwi(t) (2.6)
for all t 0, i = 0,1, and for some constant C 0. We will now prove that
condition (Mp) holds for w.
/ [°°
w0(x)Wl(x)1-^
\
1 / p
/ /•* ,/p
\1/P'
= 1 / —~^p
dx)
[
w°^ wl(x)dxJ
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