Sharp inequalities for maximal operators on finite graphs, II

TL;DR

This paper investigates sharp inequalities for maximal operators on finite graphs and the integers, providing explicit limits, constants, and extremizers for various cases and parameters.

Contribution

It derives exact limits, constants, and extremizers for maximal operators on specific finite graphs and on integers, extending the understanding of their behavior.

Findings

Limit of the operator norm on star and complete graphs as p→∞

Exact value of the best constant for variance inequalities on star graphs

Explicit extremizers for maximal operators on graphs and integers

Abstract

Let $M_{G}$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty}{\lim}\|M_{G}\|_{p}^{p }$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe $\|M_{S_n}\|_{p}$ and the corresponding extremizers for $p\in (1,2)$. We prove that $\underset{p\to \infty}{\lim}\|M_{S_n}\|_{p}^{p }=\frac{1+\sqrt{n}}{2}$ when $n\ge 25$. Also, we compute the best constant ${\bf C}_{S_n,2}$ such that for every $f:V\to \mathbb{R}$ we have $Var_{2}M_{S_n}f\le {\bf C}_{S_n,2} Var_{2}f$. We prove that ${\bf C}_{S_n,2}=\frac{(n^2-n-1)^{1/2}}{n}$ for all $n\geq 3$ and characterize the extremizers. Moreover, when $M$ is the Hardy-Littlewood maximal operator on $\mathbb{Z}$, we compute the best constant ${\bf C}_{p}$ such that $Var_{p}Mf\le {\bf C}_{p}\|f\|_{p}$ for $p\in (\frac{1}{2},1)$ and we describe the extremizers.