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\(L^p\) estimates for the iterated Hardy-Littlewood maximal operator on \(\mathbb{R}^n\) and \(K^n,K\) a local field. (English) Zbl 0921.42013

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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
43A70 Analysis on specific locally compact and other abelian groups
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