Kodai Mathematical Journal

The best constant of three kinds of the discrete Sobolev inequalities on the complete graph

Hiroyuki Yamagishi, Kohtaro Watanabe, and Yoshinori Kametaka

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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.

Article information

Source
Kodai Math. J., Volume 37, Number 2 (2014), 383-395.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1404393893

Digital Object Identifier
doi:10.2996/kmj/1404393893

Mathematical Reviews number (MathSciNet)
MR3229082

Zentralblatt MATH identifier
1310.46037

Citation

Yamagishi, Hiroyuki; Watanabe, Kohtaro; Kametaka, Yoshinori. The best constant of three kinds of the discrete Sobolev inequalities on the complete graph. Kodai Math. J. 37 (2014), no. 2, 383--395. doi:10.2996/kmj/1404393893. https://projecteuclid.org/euclid.kmj/1404393893


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