L~p ESTIMATES FOR THE ITERATED HARDY-LITTLEWOOD MAXIMAL OPERATOR ON R~n AND K~n,KALOCAL FIELD

In light of two measure estimate inequalities from 4] for the iterated Hardy-Littlewood maximal operator M~kf,certain equivalence between M~kf andthe Zygmund class LLog~aL are established on R~n,so that we generalizeStein's LLogL theorem.In Section 3,a simple induction enables us toprove such extensions on K~n,the n-dimensional linear space over a localfield K,without recoursing to Leckband's result.