Abstract
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.
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Acknowledgements
The authors would like to express their sincerely thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable. The authors also want to thank Professor Javier Soria for helpful comments during the preparation of this manuscript. The first author was supported partly by NSFC (Grant No. 11701333) and SP-OYSTTT-CMSS (Grant No. Sxy2016K01). The second author was supported partly by NSFC (Grant Nos. 11471041, 11671039, 11871101) and NSFC-DFG (Grant No. 11761131002).
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Liu, F., Xue, Q. On the variation of the Hardy–Littlewood maximal functions on finite graphs. Collect. Math. 72, 333–349 (2021). https://doi.org/10.1007/s13348-020-00290-6
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DOI: https://doi.org/10.1007/s13348-020-00290-6