A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let
T
α
f
(
x
)
=
∫
f
(
y
)
|
x
−
y
|
α
−
d
d
y
{T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy
be given for functions defined in
R
d
{{\mathbf {R}}^d}
. Let
υ
\upsilon
be a weight function which satisfies
\[
(
|
Q
|
−
1
∫
Q
[
υ
(
x
)
]
q
d
x
)
1
/
q
(
|
Q
|
−
1
∫
Q
[
υ
(
x
)
]
−
p
′
d
x
)
1
/
p
′
≤
K
(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} }
\]
for each cube,
Q
Q
, with sides parallel to a standard system of axes and
|
Q
|
|Q|
is the measure of such a cube. Suppose
1
/
q
=
1
/
p
−
α
/
d
1/q = 1/p - \alpha /d
and
0
>
α
>
d
,
1
>
p
>
d
/
α
0 > \alpha > d,1 > p > d/\alpha
. Then there exists a constant such that
|
|
(
T
α
f
)
υ
|
|
q
≤
C
|
|
f
υ
|
|
p
||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p}
. Certain results for
p
=
1
p = 1
and
q
=
∞
q = \infty
are also given.