Remarks on the Hardy–Littlewood maximal function

Abstract

We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).