Let $\sigma=(\sigma_{ 1},\sigma_{ 2},\dots,\sigma_{ n})\in \mathbb{ S}^{ n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on $\mathbb{ S}^{ n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear operator given by $$T(f_{ 1},f_{ 2},\dots,f_{ n})(x)=\int_{ \mathbb{ S}^{ n-1}}\prod^{ n}_{ j=1}f_{ j}(x-\sigma_j)d\sigma, ~x\in \R.$$ In this paper we obtain necessary and sufficient conditions on exponents $p_1,p_2,\dots,p_n$ and $r$ for which the operator $T$ is bounded from $\prod_{ j=1}^n L^{ p_j}(\R)\rightarrow L^r(\R),$ where $1\leq p_j,r\leq \infty, j=1,2,\dots,n.$ This generalizes the results obtained in~\cite{ jbak,oberlin}.
Categories: math.CA