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arXiv · 1410.6204

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

Javier Soria·Pedro Tradacete
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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 $0<p\le1$ 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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Javier Soria, Pedro Tradacete. 2014-10-22. Best constants for the Hardy-Littlewood maximal operator on finite graphs. https://arxiv.org/abs/1410.6204

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