For
1
>
p
⩽
q
>
∞
1 > p \leqslant q > \infty
and
w
(
x
)
w(x)
,
v
(
x
)
v(x)
nonnegative functions on
R
n
{{\mathbf {R}}^n}
, we show that the weighted inequality
\[
(
∫
|
T
f
|
q
w
)
1
/
q
⩽
C
(
∫
f
p
v
)
1
/
p
{\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}}
\]
holds for all
f
⩾
0
f \geqslant 0
if and only if both
\[
∫
[
T
(
χ
Q
v
1
−
p
′
)
]
q
w
⩽
C
1
(
∫
Q
v
1
−
p
′
)
q
/
p
>
∞
\int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} > \infty }
\]
and
\[
∫
[
T
(
χ
Q
w
)
]
p
′
v
1
−
p
′
⩽
C
2
(
∫
Q
w
)
p
′
/
q
′
>
∞
{\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} > \infty
\]
hold for all dyadic cubes
Q
Q
. Here
T
T
denotes a fractional integral or, more generally, a convolution operator whose kernel
K
K
is a positive lower semicontinuous radial function decreasing in
|
x
|
|x|
and satisfying
K
(
x
)
⩽
C
K
(
2
x
)
K(x) \leqslant CK(2x)
,
x
∈
R
n
x \in {{\mathbf {R}}^n}
. Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights
v
v
on
R
n
{{\mathbf {R}}^n}
and
w
w
on
R
+
n
+
1
{\mathbf {R}}_ + ^{n + 1}
for which the Poisson operator is bounded from
L
p
(
v
)
{L^p}(v)
to
L
q
(
w
)
{L^q}(w)
is given.
