Dual boundedness of the maximal operator on concave Banach function spaces

Abstract

We prove a conjecture of Nieraeth asserting that if the Hardy--Littlewood maximal operator $M$ is bounded on an $s$-concave Banach function space $X$ for some $s\in (1,\infty)$, then $M$ is also bounded on the associate space $X'$. The proof is based on a new argument establishing the Fefferman--Stein inequality on $X$.