Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs

TL;DR

This paper establishes sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on finite graphs, providing new bounds, computational results, and conjectures for specific graph classes.

Contribution

It proves sharp $p$-variational inequalities on complete graphs, computes sharp constants for small graphs, and constructs graphs with arbitrarily large variational constants.

Findings

Confirmed sharp inequalities on complete graphs.

Computed sharp constants for all connected graphs with up to five vertices.

Constructed graphs with arbitrarily large $p$-variational constants.

Abstract

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in $1$-variational inequalities for all connected graphs on at most five vertices and pose a conjecture on the corresponding sharp constants for path graphs. Finally, we construct finite graphs with arbitrarily large $p$-variational constants.