Given
p
p
and
q
q
satisfying
1
>
p
⩽
q
>
∞
1 > p \leqslant q > \infty
, sufficient conditions on nonnegative pairs of functions
U
,
V
U,V
are given to imply
\[
[
∫
R
n
|
f
^
(
x
)
|
q
U
(
x
)
d
x
]
1
/
q
⩽
c
[
∫
R
n
|
f
(
x
)
|
p
V
(
x
)
d
x
]
1
/
p
,
{\left [ {\int _{{R^n}}^{} {|\hat f(x){|^q}U(x)\,dx}} \right ]^{1/q}} \leqslant c{\left [ {\int _{{R^n}}^{} {|f(x){|^p}V(x)\,dx}} \right ]^{1/p}},
\]
where
f
^
\hat f
denotes the Fourier transform of
f
f
, and
c
c
is independent of
f
f
. For the case
q
=
p
′
q = p’
the sufficient condition is that for all positive
r
r
,
\[
[
∫
U
(
x
)
>
B
r
U
(
x
)
d
x
]
[
∫
V
(
x
)
>
r
p
−
1
V
(
x
)
−
1
/
(
p
−
1
)
d
x
]
⩽
A
,
\left [ {\int _{U(x) > Br} {U(x)\;dx}} \right ]\left [ {\int _{V(x) > {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right ] \leqslant A,
\]
where
A
A
and
B
B
are positive and independent of
r
r
. For
q
≠
p
′
q \ne p’
the condition is more complicated but also is invariant under rearrangements of
U
U
and
V
V
. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of
U
U
and
V
V
. Examples are given to show that the sufficient condition is not necessary for a pair
U
,
V
U,V
if the norm inequality is assumed only for that pair.