A Note on the Maximal Operator on Banach Function Spaces

In this note we answer positively to two conjectures proposed by Nieraeth (J Funct Anal 285:109, 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 $$X'$$ X ′ .