A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph
A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph
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DOI：
10.3390/sym10010001
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
Kazuo Takemura;Y. Kametaka;A. Nagai
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文献类型：
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作者：
Kazuo Takemura;Y. Kametaka;A. Nagai
本文阐明了加权完全图上离散Sobolev不等式的锐常数的层次结构。为此，我们在图上引入了广义图Laplacian A = I − B，并研究了两类离散Sobolev不等式.锐常数C 0（N ; a）和C 0（N）通过绿色矩阵G（a）=（A + ai）− 1（0 < a < ∞）和伪格林矩阵G = A †计算。尖锐常数用A的特征多项式的展开系数表示。在此基础上，首次证明了每一组锐常数{ C 0（n ; a）} n = 2N和{ C 0（n）} n = 2N满足一定的层次结构.
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 A = I − 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 G ( a ) = ( A + a I ) − 1 ( 0 < a < ∞ ) and the pseudo-Green matrix G ∗ = A † . The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of 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.