In this paper we study the convolution operator given on the Fourier transform side by multiplication by
\[
m
α
(
x
,
z
)
=
ϕ
(
z
)
(
1
−
|
x
|
/
z
)
+
α
,
(
x
,
z
)
∈
R
2
×
R
,
α
>
0
,
{m_\alpha }(x,z) = \phi (z)(1 - |x|/z)_ + ^\alpha ,\qquad (x,z) \in {{\mathbf {R}}^2} \times {\mathbf {R}},\;\alpha > 0,
\]
where
ϕ
∈
C
0
∞
(
1
,
2
)
\phi \in C_0^\infty (1,2)
. We will prove that
m
α
{m_\alpha }
defines a bounded operator on
L
4
(
R
3
)
{L^4}({{\mathbf {R}}^3})
if
α
>
1
8
\alpha > \tfrac {1} {8}
. Furthermore, as a generalization of a result of C. Fefferman (Acta Math. 124 (1970), 9-36), we will show that an
(
L
2
,
L
p
)
({L^2},{L^p})
restriction theorem for compact
C
∞
{C^\infty }
submanifolds
M
⊂
R
n
M \subset {{\mathbf {R}}^n}
of arbitrary codimension imply results for multipliers having a singularity of the form
dist
(
x
,
M
)
α
\operatorname {dist} {(x,M)^\alpha }
near
M
M
.