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

Abstract

Let Md be the centered Hardy-Littlewood maximal function associated to cubes in Rd with Lebesgue measure, and let cd denote the lowest constant appearing in the weak type (1,1) inequality satisfied by Md. We show that cd → ∞ as d → ∞, thus answering, for the case of cubes, a longstanding open question of E. M. Stein and J. O. Str̈omberg.