Affiliation:
1. Department of Mathematics and Physics, Shijiazhuang Tiedao University , Shijiazhuang 050043 , P. R. China
Abstract
Abstract
We mainly study the dimension-free
L
p
{L}^{p}
-inequality of the truncated maximal operator
M
n
a
f
(
x
)
=
sup
t
>
0
1
∣
B
a
1
∣
∫
B
a
1
f
(
x
−
t
y
)
d
y
,
{M}_{n}^{a}f\left(x)=\mathop{\sup }\limits_{t\gt 0}\frac{1}{| {B}_{a}^{1}| }\left|\mathop{\int }\limits_{{B}_{a}^{1}}f\left(x-ty){\rm{d}}y\hspace{-0.25em}\right|,
where
B
a
1
=
{
x
:
a
≤
∣
x
∣
≤
1
}
{B}_{a}^{1}=\left\{x:a\le | x| \le 1\right\}
. When
0
≤
a
<
1
0\le a\lt 1
, we prove that
‖
M
n
a
‖
L
p
(
R
n
)
≤
C
(
p
)
‖
f
‖
L
p
(
R
n
)
\Vert {M}_{n}^{a}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}
for
p
>
n
/
(
n
−
1
)
p\gt n\hspace{0.1em}\text{/}\hspace{0.1em}\left(n-1)
. When
a
=
1
a=1
, we prove that
‖
M
n
1
‖
L
p
(
R
n
)
≤
C
(
p
)
‖
f
‖
L
p
(
R
n
)
\Vert {M}_{n}^{1}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}
for
p
≥
2
p\ge 2
.
Reference20 articles.
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