Let
{
S
k
}
k
≧
1
{\{ {S_k}\} _{k \geqq 1}}
be a sequence of linear operators defined on
L
1
(
R
n
)
{L^1}({R^n})
such that for every
f
∈
L
1
(
R
n
)
,
S
k
f
=
f
∗
g
k
f \in {L^1}({R^n}),{S_k}f = f \ast {g_k}
for some
g
k
∈
L
1
(
R
n
)
,
k
=
1
,
2
,
⋯
{g_k} \in {L^1}({R^n}),k = 1,2, \cdots
, and
T
f
(
x
)
=
sup
k
≧
1
|
S
k
f
(
x
)
|
Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)|
. Then the inequality
m
{
x
∈
R
n
;
T
f
(
x
)
>
y
}
≦
C
y
−
1
∫
R
n
|
f
(
t
)
|
d
t
m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint _{{R^n}} {|f(t)|dt}
holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions
f
∈
L
1
(
R
n
)
f \in {L^1}({R^n})
(T is of weak type (1, 1)). In particular, if
S
k
f
{S_k}f
is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to
S
k
{S_k}
is not of restricted weak type (1, 1).