The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

Abstract

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a long standing open question of E. M. Stein and J. O. Str\"{o}mberg.