The Multilinear Spherical Maximal Function in one dimension

TL;DR

This paper establishes the boundedness of the multilinear spherical maximal function in one dimension across a broad range of Lebesgue space indices, and demonstrates the optimality of these results through counterexamples.

Contribution

It provides the first comprehensive boundedness results for the multilinear spherical maximal function in one dimension, including optimality via counterexamples.

Findings

Boundedness in the largest possible open set of indices

Counterexamples confirming optimality of results

Extension of multilinear spherical maximal function theory

Abstract

In dimension $n=1$ we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ 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.