Mathematical Journal of Okayama University volume64 issue1
2022-01 発行

The best constant of the discrete Sobolev inequalities on the complete bipartite graph

Yamagishi, Hiroyuki Tokyo Metropolitan College of Industrial Technology
Publication Date
2022-01
Abstract
We have the best constants of three kinds of discrete Sobolev inequalities on the complete bipartite graph with 2N vertices, that is, KN,N. We introduce a discrete Laplacian A on KN,N. A is a 2N ×2N real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is 1 = t(1, 1, … , 1)∈ C2N. A discrete heat kernel, a Green’s matrix and a pseudo Green’s matrix play important roles in giving the best constants.
Keywords
Discrete Sobolev inequality
Discrete Laplacian
Green’s matrix
Reproducing relation
Comments
Mathematics Subject Classification. Primary 46E39; Secondary 35K08.