On the variation of the Hardy-Littlewood maximal functions on finite graphs
type: Item
instance of: scholarly article
title: On the variation of the Hardy-Littlewood maximal functions on finite graphs
zbMATH Open document ID: 1475.42035
author: Feng Liu
author: Qingying Xue
published in: Collectanea Mathematica
publication date: 2021-08-04T00:00:00Z
review text: Let \(G=(V,E)\) be an undirected combinatorial graph with the set of vertices \(V\) and the set of edges \(E\). Let \(B(v,r)\) be a ball centered at \(v\), with radius \(r\) on the graph, equipped with the metric \(d_G\) induced by the edges in \(E\). For \(0 \leq \alpha < 1\), the fractional Hardy-Littlewood maximal function defined on the graph \(G\) is given by \[ M_{\alpha, G}f(v)=\sup_{r\geq 0}\frac{1}{|B(v,r)|^{1-\alpha}}\sum_{w\in B(v,r)}|f(w)|. \] Denote by \(M_{G}\) the Hardy-Littlewood maximal function when \(\alpha=0\) in \(M_{\alpha, G}\). In the paper under review, the authors obtain the regularity properties for \(M_G\) and \(M_{\alpha,G}\) on the spaces of bounded \(p\)-variation functions \({BV}_p (G)\). The corresponding operator norm of \(M_{\alpha, G}\) on \({BV}_p (G)\) is also worked out.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 7378259
zbMATH Keywords: finite graphs
zbMATH Keywords: maximal functions
zbMATH Keywords: bounded variation
zbMATH Keywords: continuity
reviewed by: Fayou Zhao
MaRDI profile type: Publication
full work available at URL: s13348-020-00290-6
OpenAlex ID: W3031360695
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DOI: 10.1007/S13348-020-00290-6
description: scientific article; zbMATH DE number 7378259