Let
M
g
+
M_g^ +
be the maximal operator defined by
\[
M
g
+
f
(
x
)
=
sup
h
>
0
(
∫
x
x
+
h
|
f
(
t
)
|
g
(
t
)
d
t
)
(
∫
x
x
+
h
g
(
t
)
d
t
)
−
1
,
M_g^ + f(x) = \sup \limits _{h > 0} \left ( {\int _x^{x + h} {|f(t)|g(t)dt} } \right ){\left ( {\int _x^{x + h} {g(t)dt} } \right )^{ - 1}},
\]
where
g
g
is a positive locally integrable function on
R
{\mathbf {R}}
. We characterize the pairs of nonnegative functions
(
u
,
v
)
(u,v)
for which
M
g
+
M_g^ +
applies
L
p
(
v
)
{L^p}(v)
in
L
p
(
u
)
{L^p}(u)
or in weak-
L
p
(
u
)
{L^p}(u)
. Our results generalize Sawyer’s (case
g
=
1
g = 1
) but our proofs are different and we do not use Hardy’s inequalities, which makes the proofs of the inequalities self-contained.