Zheng, Shijun; Su, Weiyi \(L^p\) estimates for the iterated Hardy-Littlewood maximal operator on \(\mathbb{R}^n\) and \(K^n,K\) a local field. (English) Zbl 0921.42013 Approximation Theory Appl. 14, No. 3, 36-54 (1998). By means of two inequalities of Leckband on the iterated Hardy-Littlewood maximal operator \(M^kf\) with any positive integer \(k\), the authors show that if \(f\in L^1({\mathbf{R}}^n)\), \(\text{ supp} f\subset S\), \(S\) is a ball of finite measure, then \(M^kf\in L^1(S)\) if and only if \(| f| (\log| f|)^k\in L^1(S)\); and for \(0<p<1\), \((M^kf)^p\in L^1(S)\) if and only if \(| f| (\log| f|)^{k-1} \in L^1(S)\). The authors also establish a similar result to this on the \(n-\)dimensional linear space \({\mathbf{K}}^n\) over a local field \(\mathbf K\) without relating to Leckband’s result. Moreover, the authors find a largest subspace in some sense of the Hardy space \(H^p({\mathbf{K}}^n)\) for \(0<p<1\) via the Herz space. Reviewer: Yang Dachun (Beijing) MSC: 42B25 Maximal functions, Littlewood-Paley theory 43A70 Analysis on specific locally compact and other abelian groups Keywords:Hardy-Littlewood maximal operator; Euclidean space; local field; Herz space; Hardy space PDFBibTeX XMLCite \textit{S. Zheng} and \textit{W. Su}, Approximation Theory Appl. 14, No. 3, 36--54 (1998; Zbl 0921.42013)