A hierarchical structure for the sharp constants of discrete Sobolev inequalities on a weighted complete graph
type: Item
instance of: scholarly article
title: A hierarchical structure for the sharp constants of discrete Sobolev inequalities on a weighted complete graph
zbMATH Open document ID: 1390.46040
author: Kazuo Takemura
author: Yoshinori Kametaka
author: Atsushi Nagai
published in: Symmetry
publication date: 2018-06-25T00:00:00Z
review text: Summary: This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian \(\boldsymbol{A} = \boldsymbol{I} - \boldsymbol{B}\) on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants \(C_0(N; a)\) and \(C_0(N)\) were calculated through the Green matrix \(\boldsymbol{G}(a) = (\boldsymbol{A} + a \boldsymbol{I})^{- 1}(0 < a < \infty)\) and the pseudo-Green matrix \(\boldsymbol{G}_\ast = \boldsymbol{A}^\dagger\). The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of \(\boldsymbol{A}\). Based on this new discovery, we provide the first proof that each set of the sharp constants \(\{C_0(n; a) \}_{n = 2}^N\) and \(\{C_0(n) \}_{n = 2}^N\) satisfies a certain hierarchical structure.
Mathematics Subject Classification ID: 46E39
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 6893757
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: sharp constant
zbMATH Keywords: complete graph
zbMATH Keywords: graph Laplacian
zbMATH Keywords: Green matrix
MaRDI profile type: Publication
full work available at URL: sym10010001
OpenAlex ID: W2777513486
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DOI: 10.3390/SYM10010001
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description: scientific article; zbMATH DE number 6893757