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June 2014 The best constant of three kinds of the discrete Sobolev inequalities on the complete graph
Hiroyuki Yamagishi, Kohtaro Watanabe, Yoshinori Kametaka
Kodai Math. J. 37(2): 383-395 (June 2014). DOI: 10.2996/kmj/1404393893

Abstract

We introduce a discrete Laplacian A on the complete graph with N vertices, that is, KN. We obtain the best constants of three kinds of discrete Sobolev inequalities on KN. The background of the first inequality is the discrete heat operator (d/dt + A + a0I) ··· (d/dt + A + aM−1I) with positive distinct characteristic roots a0, ..., aM−1. The second one is the difference operator (A + a0I) ··· (A + aM−1I) and the third one is the discrete polyharmonic operator AM. Here A is an N × N real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is 1 = t(1, 1, ..., 1). A discrete heat kernel, a Green's matrix and a pseudo Green's matrix are obtained by means of A.

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Hiroyuki Yamagishi. Kohtaro Watanabe. Yoshinori Kametaka. "The best constant of three kinds of the discrete Sobolev inequalities on the complete graph." Kodai Math. J. 37 (2) 383 - 395, June 2014. https://doi.org/10.2996/kmj/1404393893

Information

Published: June 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1310.46037
MathSciNet: MR3229082
Digital Object Identifier: 10.2996/kmj/1404393893

Rights: Copyright © 2014 Tokyo Institute of Technology, Department of Mathematics

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Vol.37 • No. 2 • June 2014
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