Multilinear maximal operators associated to simplices

TL;DR

This paper proves new bounds for multilinear maximal operators related to averaging over simplices, extending classical spherical maximal operator results to multilinear and discrete settings.

Contribution

It establishes novel $L^{p_1} imes \

Findings

Derived $L^{p_1} imes o L^r$ bounds for multilinear operators.

Extended classical spherical maximal bounds to multilinear and discrete contexts.

Provided key tools for proofs of these bounds.

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.