L p estimates for the iterated Hardy-Littlewood maximal operator on\mathbb{R}^n and K n , k a local fieldand K n , k a local field

In light of two measure estimate inequalities from 4] for the iterated Hardy-Littlewood maximal operatorM ^k f, certain equivalence betweenM ^k f and the Zygmund classLLog ^a L are established on $\mathbb{R}^n $ , so that we generalize Stein'sLLogL theorem. In Section 3, a simple induction enables us to prove such extensions onK ⁿ, the n-dimensional linear space over a local fieldK, without recoursing to Leckband's result.