Sharp Inequalities for Maximal Operators on Finite Graphs

Abstract

Let $G=(V,E)$ be a finite graph (here V and E denote the set of vertices and edges of G respectively) and $M_G$ be the centered Hardy–Littlewood maximal operator defined there. We find the optimal value $\mathbf{{C}}_{G,p}$ such that the inequality

$$\begin{aligned} \mathrm{Var\,}_{p}M_{G}f\le \mathbf{C}_{G,p}\mathrm{Var\,}_{p}f \end{aligned}$$

holds for every $f:V\rightarrow {\mathbb {R}},$ where $\mathrm{Var\,}_p$ stands for the p-variation, when: (i) $G=K_n$ (complete graph) and $p\in [\frac{\log (4)}{\log (6)},\infty )$ or $G=K_4$ and $p\in (0,\infty )$; (ii) $G=S_n$(star graph) and $1\ge p\ge \frac{1}{2}$; $p\in (0,\frac{1}{2})$ and $n\ge C(p)$ or $G=S_3$ and $p\in (1,\infty ).$ We also find value of the norm $\Vert M_{G}\Vert _{2}$ when: (i) $G=K_n$ and $n\ge 3$; (ii) $G=S_n$ and $n\ge 3.$