Abstract
We consider three kinds of discrete Sobolev inequalities corresponding to a graph Laplacian $\boldsymbol{A}$ on regular $M$-hedron for $M=4,6,8,12,20$. Discrete heat kernel $\boldsymbol{H}(t)=\exp(-t\boldsymbol{A})$, Green matrix $\boldsymbol{G}(a)=(\boldsymbol{A}+a\boldsymbol{I})^{-1}$ and pseudo Green matrix $\boldsymbol{G}_*$ are obtained and investigated in a detailed manner. The best constants of the inequalities are given by means of eigenvalues of $\boldsymbol{A}$.
Citation
Yoshinori KAMETAKA. Atsushi NAGAI. Kazuo TAKEMURA. Kohtaro WATANABE. Hiroyuki YAMAGISHI. "The Best Constant of Three Kinds of Discrete Sobolev Inequalities on Regular Polyhedron." Tokyo J. Math. 36 (1) 253 - 268, June 2013. https://doi.org/10.3836/tjm/1374497523
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