2. Calderon weights
Let P, Q be the operators defined in the Introduction. Results by Muckenhoupt
(see [Mu2], [Ma]), which extend Hardy's inequalities, ensure that:
A. Pf G
Lp(w)
for all f G
Lp(w)
(1 p oo) if and only if there exists a
constant C 0 such that for almost all t 0
(/°° ^
d x
)
l P
(I*
xrp,/Pdx) P
^
c (MP)
for 1 p oo, or
^-dx Cw(t) (Mx)
/
forp= 1,
and
B. Qf G
Lp(w)
for all f G
Lp(w)
(1 p oo) if and only if there exists a
constant C 0 such that for almost all t 0
/or 1 p oo, or
(^j\{x)dx^ " U°°
W{x)~j/Vdx\
C
(Mp)
i
rl
- / w(x)dx Cw(t)
(M1)
t Jo
for p = 1.
With the exception of the trivial case w = 0 a.e., the conditions above imply
that w 0 a.e. and all the integrals appearing in (Mp) and
(Mp)
are finite.
It will be important for us to relate the weighted norm inequalities for the
Calderon or Hardy operators in terms of the conditions on the weights. It follows
from [Mu2] that, for 1 p oo,
\H\Mp \\P\\LrM^LP(W)
P1/PP'1/P'\\W\\MP
4|H|M
P
, (2.1)
where
||W||MP
is the infimum of the constants C appearing in the definition of the
Mp condition. Likewise
H I MP
\\Q\\LP(W)^LP(W)
P1/PP'1/P'
\\W\\MP M\W\\MP
(2.2)
where ||iu||MP is the infimum of the constants C appearing in the definition of the
Mp
condition.
Note that
1 H I M P | H I M
P
. (2.3)
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