论文标题
有限图上最大运算符的尖锐不平等
Sharp Inequalities for maximal operators on finite graphs
论文作者
论文摘要
令$ g =（v，e）$为有限的图表，$ m_g $为中心的Hardy-Little Wood Wood Maximal Operator在此处定义。我们找到最佳值$ \ bf {c} _ {g，p} $，使得不平等$ \ text {var} _ {p}（m_ {g} f）\ leq {\ leq {\ textbf {c}} _ \ Mathbb {r}，$ whene $ \ text {var} _p $代表$ p $ - 变量，何时：（i）$ g = k_n $（完整的图形）和$ p \ in [\ frac {\ log log（4）} （ii）$ g = s_n $（star Graph）和$ 1 \ ge p \ ge \ frac {1} {2} $; $ p \ in（0，\ frac {1} {2}）$和$ n \ ge c（p）$或$ g = s_3 $和$ p \ in（1，\ infty）。$我们还找到了norm $ $ $ $ $ \ | m_ {g} g}的价值， （ii）$ g = s_n $和$ n \ ge 3. $
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 $\bf{C}_{G,p}$ such that the inequality $$\text{Var}_{p}(M_{G}f)\leq {\textbf{C}}_{G,p}\text{Var}_{p}(f)$$ 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 $\|M_{G}\|_{2}$ when: (i) $G=K_n$ and $n\ge 3$; (ii) $G=S_n$ and $n\ge 3.$
