Let $\sigma =({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_n)\in {\mathbb{S}}^{n-1}$ and $d\sigma$ denote the normalized Lebesgue measure on ${\mathbb{S}}^{n-1},n⩾2$. For functions $f_1,f_2,\cdots ,f_n$ defined on $\mathbb{R}$, consider the multilinear operator given by

$$T(f_1,f_2,\cdots ,f_n)(x)={\int }_{{\mathbb{S}}^{n-1}}\prod _{j=1}^n{f}_j(x-{\sigma }_j)d\sigma ,x\in \mathbb{R}.$$

In this paper, we obtain necessary and sufficient conditions on exponents $p_1,p_2,\cdots ,p_n$ 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 $1⩽p_j,r⩽\infty ,j=1,2,\cdots ,n$. This generalizes the results obtained in (Bak and Shim, J. Funct. Anal. 157 (1998) 534–553; Oberlin, Trans. Amer. Math. Soc. 310 (1988) 821–835).

中文翻译：

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用球形测度定义的多线性卷积算子的 Lp 估计

让 $\sigma =({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_n)\in {\mathbb{秒}}^{n-1}$ 和 $d\sigma$ 表示标准化的 Lebesgue 测度 ${\mathbb{秒}}^{n-1},n⩾2$. 对于函数$f_1,f_2,\cdots ,f_n$ 定义于 $\mathbb{电阻}$，考虑由下式给出的多重线性算子

$$吨(f_1,f_2,\cdots ,f_n)(\times )={\int }_{{\mathbb{秒}}^{n-1}}\prod _{j=1}^n{f}_j(\times -{\sigma }_j)d\sigma ,\times \in \mathbb{电阻}.$$

在本文中，我们获得了指数的充分必要条件 ${磷}_1,{磷}_2,\cdots ,{磷}_n$ 和 $r$ 运营商 $吨$ 有界于 ${\prod }_{j=1}^n{升}^{{磷}_j}(\mathbb{电阻})\to {升}^r(\mathbb{电阻})$，哪里 $1⩽{磷}_j,r⩽\infty ,j=1,2,\cdots ,n$. 这概括了 (Bak and Shim, J. Funct. Anal . 157 (1998) 534-553; Oberlin, Trans. Amer. Math. Soc . 310 (1988) 821-835) 中获得的结果。