Let
M
+
f
(
x
)
=
sup
h
>
0
(
1
/
h
)
∫
x
x
+
h
|
f
(
t
)
|
d
t
{M^ + }f(x) = {\sup _{h > 0}}(1/h)\int _x^{x + h} {|f(t)|\,dt}
denote the one-sided maximal function of Hardy and Littlewood. For
w
(
x
)
⩾
0
w(x) \geqslant 0
on
R
R
and
1
>
p
>
∞
1 > p > \infty
, we show that
M
+
{M^ + }
is bounded on
L
p
(
w
)
{L^p}(w)
if and only if
w
w
satisfies the one-sided
A
p
{A_p}
condition:
\[
(
A
p
+
)
[
1
h
∫
a
−
h
a
w
(
x
)
d
x
]
[
1
h
∫
a
a
+
h
w
(
x
)
−
1
/
(
p
−
1
)
d
x
]
p
−
1
⩽
C
\left ( {A_p^ + } \right )\qquad \left [ {\frac {1} {h}\int _{a - h}^a {w(x)dx} } \right ]{\left [ {\frac {1} {h}\int _a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqslant C
\]
for all real
a
a
and positive
h
h
. If in addition
v
(
x
)
⩾
0
v(x) \geqslant 0
and
σ
=
v
−
1
/
(
p
−
1
)
\sigma = {v^{ - 1/(p - 1)}}
,then
M
+
{M^ + }
is bounded from
L
p
(
v
)
{L^p}(v)
to
L
p
(
w
)
{L^p}(w)
if and only if
\[
∫
I
[
M
+
(
χ
I
σ
)
]
p
w
⩽
C
∫
I
σ
>
∞
\int _I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int _I {\sigma > \infty } }
\]
for all intervals
I
=
(
a
,
b
)
I = (a,b)
such that
∫
−
∞
a
w
>
0
\int _{ - \infty }^a {w > 0}
. The corresponding weak type inequality is also characterized. Further properties of
A
p
+
A_p^ +
weights, such as
A
p
+
⇒
A
p
−
ε
+
A_p^ + \Rightarrow A_{p - \varepsilon }^ +
and
A
p
+
=
(
A
1
+
)
(
A
1
−
)
1
−
p
A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}
, are established.
