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

TL;DR

This paper investigates the Hardy-Littlewood maximal operator on finite graphs, revealing how the graph's geometry influences the operator and identifying extremal graphs for certain bounds.

Contribution

It establishes that the maximal operator uniquely determines the graph's structure and provides optimal bounds for the operator's norms on various graphs.

Findings

Complete graph and star graph are extremal for bounds.

Optimal bounds for the p-(quasi)norm are derived.

Connections between maximal operator, dilation, and overlapping indices are explored.

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.