The principal problem considered is the determination of all nonnegative functions,
V
(
x
)
V(x)
, such that
‖
T
γ
f
(
x
)
V
(
x
)
‖
q
≤
C
‖
f
(
x
)
V
(
x
)
‖
p
\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p
where the functions are defined on
R
n
,
0
>
γ
>
n
,
1
>
p
>
n
/
γ
,
1
/
q
=
1
/
p
−
γ
/
n
{R^n},0 > \gamma > n,1 > p > n/\gamma ,1/q = 1/p - \gamma /n
, C is a constant independent of f and
T
γ
f
(
x
)
=
∫
f
(
x
−
y
)
|
y
|
γ
−
n
d
y
{T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy
. The main result is that
V
(
x
)
V(x)
is such a function if and only if
\[
(
1
|
Q
|
∫
Q
[
V
(
x
)
]
q
d
x
)
1
/
q
(
1
|
Q
|
∫
Q
[
V
(
x
)
]
−
p
′
d
x
)
1
/
p
′
≤
K
{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K
\]
where Q is any n dimensional cube,
|
Q
|
|Q|
denotes the measure of Q,
p
′
=
p
/
(
p
−
1
)
p’ = p/(p - 1)
and K is a constant independent of Q. Substitute results for the cases
p
=
1
p = 1
and
q
=
∞
q = \infty
and a weighted version of the Sobolev imbedding theorem are also proved.