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.
Citation
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
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