It is shown that multilinear operators of the form
T
(
f
1
,
.
.
.
,
f
k
)
(
x
)
T(f_1,...,f_k)(x)
=
∫
R
n
K
(
x
,
y
1
,
.
.
.
,
y
k
)
f
1
(
y
1
)
.
.
.
f
k
(
y
k
)
d
y
1
.
.
.
d
y
k
=\!\int _{\mathbb {R}^n}\!K(x,y_1,...,y_k)f_1(y_1)... f_k(y_k)dy_1...dy_k
of restricted weak type
(
1
,
.
.
.
,
1
,
q
)
(1,...,1,q)
are always of weak type
(
1
,
.
.
.
,
1
,
q
)
(1,...,1,q)
whenever the map
x
→
K
x
x\to K_x
is a locally integrable
L
1
(
R
n
)
L^1(\mathbb {R}^n)
-valued function.