We establish $L^{p_1}\times \cdots \times L^{p_k}\to L^r$ and ${\ell }^{p_1}\times \cdots \times {\ell }^{p_k}\to {\ell }^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex in both the continuous and discrete settings. These provide natural extensions of $L^p\to L^p$ and ${\ell }^p\to {\ell }^p$ bounds for Stein's spherical maximal operator and the discrete spherical maximal operator, with each of these results serving as a key ingredient of the respective proofs.

中文翻译：

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与单纯形相关的多重线性极大值算子

我们建立 ${升}^{{磷}_1}\times \cdots \times {升}^{{磷}_{克}}\to {升}^r$ 和 ${\ell }^{{磷}_1}\times \cdots \times {\ell }^{{磷}_{克}}\to {\ell }^r$ 与给定非退化等距副本的平均值相关联的多线性极大值算子的类型边界 $克$-simplex 在连续和离散设置中。这些提供了自然的延伸${升}^{磷}\to {升}^{磷}$ 和 ${\ell }^{磷}\to {\ell }^{磷}$ Stein 的球面极大算子和离散球面极大算子的界限，这些结果中的每一个都作为各自证明的关键成分。