A note on the off-diagonal Muckenhoupt-Wheeden conjecture
Abstract
We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given
1<p<q<∞
and a pair of weights
(u,v)
, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities:
M:
L
p
(v)→
L
q
(u)andM:
L
q
′
(
u
1−
q
′
)→
L
p
′
(
v
1−
p
′
),
then any Calderón-Zygmund operator
T
and its associated truncated maximal operator
T
⋆
are bounded from
L
p
(v)
to
L
q
(u)
. Additionally, assuming only the second estimate for
M
then
T
and
T
⋆
map continuously
L
p
(v)
into
L
q,∞
(u)
. We also consider the case of generalized Haar shift operators and show that their off-diagonal two weight estimates are governed by the corresponding estimates for the dyadic Hardy-Littlewood maximal function.