$L^p$ estimates for multilinear convolution operators defined with spherical measure

Abstract

Let $\sigma =({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n)\in {\mathbb{S}}^{n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on ${\mathbb{S}}^{n-1},n\ge 2$. For functions $f_1,f_2,\dots ,f_n$ defined on $\mathbb{R}$ consider the multilinear operator given by $$T(f_1,f_2,\dots ,f_n)(x)={\int }_{{\mathbb{S}}^{n-1}}\prod _{j=1}^n{f}_j(x-{\sigma }_j)d\sigma ,x\in \mathbb{R}.$$ 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 ${\prod }_{j=1}^n{L}^{p_j}(\mathbb{R})\to L^r(\mathbb{R}),$ where $1\le p_j,r\le \infty ,j=1,2,\dots ,n.$ This generalizes the results obtained in~\cite{jbak,oberlin}.