Let
d
>
1
d > 1
. It will be shown that the maximal operator
S
∗
{S^*}
of spherical means
S
R
,
R
>
0
{S_R},R > 0
, is bounded on
L
p
(
R
d
)
{L^p}({{\mathbf {R}}^d})
radial functions when
2
d
/
(
d
+
1
)
>
p
>
2
d
/
(
d
−
1
)
2d/(d + 1) > p > 2d/(d - 1)
, and it implies that, for every
L
p
(
R
d
)
{L^p}({{\mathbf {R}}^d})
radial function
f
(
t
)
,
S
R
f
(
t
)
f(t),{S_R}f(t)
converges to
f
(
t
)
f(t)
for a.e.
t
∈
R
d
t \in {{\mathbf {R}}^d}
when
2
d
/
(
d
+
1
)
>
p
≤
2
2d/(d + 1) > p \leq 2
. Also, it will be proved that there is an
L
2
d
/
(
d
+
1
)
(
R
d
)
{L^{2d/(d + 1)}}({R^d})
radial function
f
(
t
)
f(t)
with compact support such that
S
R
f
(
t
)
{S_R}f(t)
diverges for a.e.
t
∈
R
d
t \in {R^d}
.