A note on the maximal operator on Banach function spaces

TL;DR

This paper confirms two conjectures regarding the maximal operator on rescaled Banach function spaces and introduces a new criterion for boundedness transfer to associate spaces.

Contribution

It provides a positive resolution to two conjectures and establishes a novel criterion for the boundedness of the maximal operator on associate spaces.

Findings

Confirmed two conjectures about the maximal operator on rescaled Banach spaces.

Established a new criterion for boundedness transfer to associate spaces.

Enhanced understanding of maximal operator behavior on Banach function spaces.

Abstract

In this note we answer positively to two conjectures proposed by Nieraeth (2023) about the maximal operator on rescaled Banach function spaces. We also obtain a new criterion saying when the maximal operator bounded on a Banach function space $X$ is also bounded on the associate space $X'$.