A Note on the Maximal Operator on Banach Function Spaces

Abstract 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 $$X'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> .