Multilinear Spherical Maximal Function

TL;DR

This paper establishes optimal boundedness results for the multilinear spherical maximal function across various Lebesgue spaces in higher dimensions, including endpoint estimates and counterexamples.

Contribution

It provides the first comprehensive boundedness characterization for the multilinear spherical maximal function in high dimensions, including endpoint and Lorentz space estimates.

Findings

Proved boundedness in the largest possible index range.

Established weak type and Lorentz space estimates.

Provided counterexamples demonstrating optimality.

Abstract

In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases.