论文标题
有限图上最大运算符的急剧不平等，ii
Sharp inequalities for maximal operators on finite graphs, II
论文作者
论文摘要
令$ m_ {g} $为有限图$ g $上的中心Hardy-Little Wood Maximal Operator。我们找到$ \ underSet {p \ to \ infty} {\ lim} \ | m_ {g} \ | _ {p}^{p}^{p} $ g $是$ g $是start graph（$ s_n $）（$ s_n $）和完整的图形（$ k_n $），并且我们完全描述了$ \ | m_} $ __n} \ | | | {s_n}} $ p \ in（1,2）$。我们证明$ \ unterSet {p \ to \ infty} {\ lim} \ | m_ {s_n} \ | _ {p}^{p} = \ frac {1+ \ sqrt {n}}}} {2} {2} {2} $ n \ n \ ge ge ge 25 $ 25 $。另外，我们计算最好的常数$ {\ bf c} _ {s_n，2} $，以便每$ f：v \ to \ to \ m athbb {r} $我们都有$ var_ {2} m_ {s_n} f \ le {我们证明$ {\ bf c} _ {s_n，2} = \ frac {（n^2-n-1）^{1/2}} {n} {n} {n} $ for All $ n \ geq 3 $ and the Extremizers。此外，当$ m $是$ \ mathbb {z} $上的强硬木材最大运算符时，我们计算了最佳常数$ {\ bf c} _ {p} $，以便$ var_ {p} mf \ le le {\ le {\ bf c} （\ frac {1} {2}，1）$，我们描述了极端化器。
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.
