Sharp Poincare-Wirtinger inequalities on complete graphs
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arXiv
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| Format: | Preprint |
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2024
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| author | González-Riquelme, Cristian Madrid, José |
| author_facet | González-Riquelme, Cristian Madrid, José |
| contents | Let $K_n=(V,E)$ be the complete graph with $n\geq 3$ vertices (here $V$ and $E$ denote the set of vertices and edges of $K_n$ respectively). We find the optimal value ${\bf{C}}_{n,p}$ such that the inequality $$\|f-m_f\|_p\le {\bf C}_{n,p}{\rm Var}_{p}f$$ holds for every $f:V\to \mathbb{R},$ where ${\rm Var}_p$ stands for the $p$-variation, and $m_f$ stands for the average value of $f$, for all $p\in[1,3+δ^1_n)\cup (3+δ^2_n,+\infty)$, for $δ^1_n=\frac{1}{2n^2\log(n)}+O(1/n^3)$ and $δ^2_n=\frac{2}{n}+O(1/n^2).$ Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals $(1,2)$, $(2,3+δ^{1}_n)$ and $(3+δ^{2}_n,\infty).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Poincare-Wirtinger inequalities on complete graphs González-Riquelme, Cristian Madrid, José Classical Analysis and ODEs Combinatorics 26A45, 39A12, 46E39, 05C12 Let $K_n=(V,E)$ be the complete graph with $n\geq 3$ vertices (here $V$ and $E$ denote the set of vertices and edges of $K_n$ respectively). We find the optimal value ${\bf{C}}_{n,p}$ such that the inequality $$\|f-m_f\|_p\le {\bf C}_{n,p}{\rm Var}_{p}f$$ holds for every $f:V\to \mathbb{R},$ where ${\rm Var}_p$ stands for the $p$-variation, and $m_f$ stands for the average value of $f$, for all $p\in[1,3+δ^1_n)\cup (3+δ^2_n,+\infty)$, for $δ^1_n=\frac{1}{2n^2\log(n)}+O(1/n^3)$ and $δ^2_n=\frac{2}{n}+O(1/n^2).$ Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals $(1,2)$, $(2,3+δ^{1}_n)$ and $(3+δ^{2}_n,\infty).$ |
| title | Sharp Poincare-Wirtinger inequalities on complete graphs |
| topic | Classical Analysis and ODEs Combinatorics 26A45, 39A12, 46E39, 05C12 |
| url | https://arxiv.org/abs/2411.12079 |