Multilinear convolutions defined by measures on spheres

Abstract:

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