For
1
>
p
⩽
q
>
∞
,
0
>
α
>
n
1 > p \leqslant q > \infty ,0 > \alpha > n
and
w
(
x
)
,
υ
(
x
)
w(x),\upsilon (x)
nonnegative weight functions on
R
n
{R^n}
we show that the weak type inequality
\[
∫
{
T
α
f
>
λ
}
w
(
x
)
d
x
⩽
A
λ
−
q
(
∫
|
f
(
x
)
|
p
υ
(
x
)
d
x
)
q
/
p
\int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}}
\]
holds for all
f
⩾
0
f \geqslant 0
if and only if
\[
∫
Q
[
T
α
(
χ
Q
w
)
(
x
)
]
p
′
υ
(
x
)
1
−
p
′
d
x
⩽
B
(
∫
Q
w
)
p
′
/
q
′
>
∞
\int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty
\]
for all cubes
Q
Q
in
R
n
{R^n}
. Here
T
α
{T_\alpha }
denotes the fractional integral of order
α
,
T
α
f
(
x
)
=
∫
|
x
−
y
|
α
−
n
f
(
y
)
d
y
\alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy
. More generally we can replace
T
α
{T_\alpha }
by any suitable convolution operator with radial kernel decreasing in
|
x
|
|x|
.
