Collectanea Mathematica ( IF 0.7 ) Pub Date : 2020-05-25 , DOI: 10.1007/s13348-020-00290-6 Feng Liu , Qingying Xue
Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let \(M_{G}\) be the Hardy–Littlewood maximal function defined on graph G and \(M_{\upalpha ,G}\)\((0\le \upalpha <1)\) be its fractional version. In this paper, the regularity problems related to \(M_{G}\) and \(M_{\upalpha ,G}\) will be studied. We show that \(M_{G}:\mathrm{BV}_p (G)\rightarrow \mathrm{BV}_p(G)\) is bounded and \(M_{\upalpha ,G}: \ell ^p(V)\rightarrow \mathrm{BV}_q(G)\) is bounded and continuous for all \(0<p,\,q\le \infty \). Here \(\mathrm{BV}_p(G)\) is the set of all functions of bounded p-variation on V. The operator norms of \(M_{G}\) and \(M_{\upalpha ,G}\) have also been investigated.
中文翻译:
关于有限图上的Hardy–Littlewood最大函数的变化
令G为具有一组顶点V和一组边缘E的连通有限图。令\(M_ {G} \)为在图G上定义的Hardy–Littlewood最大值函数,\(M _ {\ upalpha,G} \)\((0 \ le \ upalpha <1)\)为其分数形式。在本文中,将研究与\(M_ {G} \)和\(M _ {\ upalpha,G} \)相关的正则性问题。我们证明\(M_ {G}:\ mathrm {BV} _p(G)\ rightarrow \ mathrm {BV} _p(G)\)是有界的,而\(M _ {\ upalpha,G}:\ ell ^ p( V)\ rightarrow \ mathrm {BV} _q(G)\)对于所有\(0 <p,\,q \ le \ infty \)是有界的和连续的。这里\(\ mathrm {BV} _p(G)\)是V上有界p变量的所有函数的集合。\(M_ {G} \)和\(M _ {\ upalpha,G} \)的算子范数也已研究。