Abstract
We study the behavior of averages for functions defined on finite graphs G, in terms of the Hardy–Littlewood maximal operator MG. We explore the relationship between the geometry of a graph and its maximal operator and prove that MG completely determines G (even though embedding properties for the graphs do not imply pointwise inequalities for the maximal operators). Optimal bounds for the p-(quasi)norm of a general graph G in the range 0<p≤1 are given, and it is shown that the complete graph Kn and the star graph Sn are the extremal graphs attaining, respectively, the lower and upper estimates. Finally, we study weak-type estimates and some connections with the dilation and overlapping indices of a graph.
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