Sharp inequalities for maximal operators on finite graphs, II

Abstract

Let $M_G$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty }{\lim }\Vert M_G{\Vert }_p^p$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe $\Vert M_{S_n}{\Vert }_p$ and the corresponding extremizers for $p\in (1,2)$. We prove that $\underset{p\to \infty }{\lim }\Vert M_{S_n}{\Vert }_p^p=\frac{1+\sqrt{n}}{2}$ when $n\ge 25$. Also, we compute the best constant ${\mathbf{C}}_{S_n,2}$ such that for every $f:V\to \mathbb{R}$ we have $Va{r}_2{M}_{S_n}f\le {\mathbf{C}}_{S_n,2}Va{r}_{2}f$. We prove that ${\mathbf{C}}_{S_n,2}=\frac{(n^2-n-1{)}^{1/2}}{n}$ for all $n\ge 3$ and characterize the extremizers. Moreover, when $M$ is the Hardy-Littlewood maximal operator on $\mathbb{Z}$, we compute the best constant ${\mathbf{C}}_p$ such that $Va{r}_{p}Mf\le {\mathbf{C}}_p\Vert f{\Vert }_p$ for $p\in (\frac{1}{2},1)$ and we describe the extremizers.

23 pages