Let \({\mathcal Q}\) be the family of all finite cubes Q in \(R^ n\) with sides parallel to the axes and let \(M_{{\mathcal Q}}\) denote the Hardy-Littlewood maximal operator. According to a fundamental result of Muckenhoupt, \(M_{{\mathcal Q}}\) is a bounded operator on the Lebesgue-space \(L^ p(d\mu)\), \(1