Multilinear maximal operators associated to simplices

Abstract

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.

Expanded to include new results in the continuous setting. Figures added. New example added