On the variation of the Hardy-Littlewood maximal functions on finite graphs
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.
scientific article; zbMATH DE number 7378259
scientific article