Best constants for the Hardy-Littlewood maximal operator on finite graphs

2014/10/22 by Javier Soria, Soria, Javier, Pedro Tradacete +1
Mathematics · #05C12 #05C75 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #math.CA #math.CO #msc:05C12 #msc:05C75

paper · pdf · doi:10.48550/arxiv.1410.6204

arxiv created 2014/10/22 · arxiv updated 2014/10/24

Abstract

We study the behavior of averages for functions defined on finite graphs G, in terms of the Hardy-Littlewood maximal operator M_G. We explore the relationship between the geometry of a graph and its maximal operator and prove that M_G 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.

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