Let
σ
\sigma
be Lebesgue measure on
Σ
n
−
1
{\Sigma _{n - 1}}
and write
σ
=
(
σ
1
,
…
,
σ
n
)
\sigma = ({\sigma _1}, \ldots ,{\sigma _n})
for an element of
Σ
n
−
1
{\Sigma _{n - 1}}
. For functions
f
1
,
…
,
f
n
{f_1}, \ldots ,{f_n}
on
R
{\mathbf {R}}
, define
\[
T
(
f
1
,
…
,
f
n
)
(
x
)
=
∫
Σ
n
−
1
f
1
(
x
−
σ
1
)
⋯
f
n
(
x
−
σ
n
)
d
σ
,
x
∈
R
.
T({f_1}, \ldots ,{f_n})(x) = \int _{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf {R}}.}
\]
This paper partially answers the question: for which values of
p
p
and
q
q
is there an inequality
\[
|
|
T
(
f
1
,
…
,
f
n
)
|
|
q
⩽
C
|
|
f
1
|
|
p
⋯
|
|
f
n
|
|
p
?
||T({f_1}, \ldots ,{f_n})|{|_q} \leqslant C||{f_1}|{|_p} \cdots ||{f_n}|{|_p}?
\]