A New Proof of the Hardy‐Littlewood Maximal Theorem

if A > 0. The standard proof of (1) is based on a covering lemma of Vitali type. For details see [2, Chapter 1]. Here we will give a different proof of (1) based on a result of de Guzman N [1, Theorem 4.1.1]. Let 0 = £ bk5ak, where dak is the Dirac delta function at ak. Set N fc=l ll^lli — Z l̂ kl d> for A > 0, Ex = {x: M$(x) > X}. In view of de Guzman's result, to prove (1) it is enough to consider linear combinations of delta functions. Thus the maximal theorem follows from the following