On the Discretization Technique for the Hardy-Littlewood Maximal Operators

Abstract

We extend the discretization method of de Guzmán to the setting of general metric measure spaces with mild assumptions on their structures. This method allows one to relate the best constants in the weak type $(1,1)$ inequalities for the relevant centered and uncentered Hardy-Littlewood maximal operators with the analogous constants received by applying the maximal operators to sums of Dirac deltas rather than to $L^1$ functions.