- April 2016
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- 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 K-n and the star graph S-n 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. (C) 2015 Elsevier Inc. All rights reserved.
- finite graph; maximal operator; l(p)-estimate; weak-type (1;1); metric spaces