Restricted testing conditions for the multilinear maximal operator

TL;DR

This paper explores restricted testing conditions for the multilinear maximal operator, building on prior work that established testing conditions for the linear case, and extends these ideas to the multilinear setting.

Contribution

It introduces a multilinear version of the restricted testing condition, expanding the theoretical framework for maximal operators.

Findings

Established a multilinear restricted testing condition

Extended linear testing results to multilinear operators

Provided new theoretical insights into maximal operator bounds

Abstract

Restricted testing conditions were considered recently. For the maximal operator, Li and Sawyer first obtained parental testing condition. Later, they showed that it suffices to restrict testing to doubling cubes. Chen and Lacey gave a similar restricted testing condition. In our paper, we discuss a version of the latter in the multilinear setting.