Sharp Inequalities for maximal operators on finite graphs

Abstract

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value ${\mathbf{C}}_{\mathbf{G},\mathbf{p}}$ such that the inequality $${\text{Var}}_p(M_{G}f)\le {\text{C}}_{G,p}{\text{Var}}_p(f)$$ holds for every $f:V\to \mathbb{R},$ where ${\text{Var}}_p$ 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 the value of the norm $\Vert M_G{\Vert }_2$ when: (i) $G=K_n$ and $n\ge 3$; (ii) $G=S_n$ and $n\ge 3.$

27 pages. More details added. Improved presentation