The Best Constant of Discrete Sobolev Inequality on Truncated Tetra-, Hexa- and Octa- Polyhedra(Theory)

Abstract

We have obtained the best constants of discrete Sobolev inequalities on the truncated regular M-hedron for M = 4, 6, 8. Giving appropriate indices on vertices of polyhedra and introducing discrete Laplacians A, we have obtained Green matrices G(a) = (A + aI)^<-1> (a > 0) and pseudo Green matrices G_*. (Pseudo) Green matrices are reproducing kernels by setting appropriate vector spaces and inner products. By applying Schwarz inequality to the reproducing relations, two types of discrete Sobolev inequalities are obtained. Diagonal values of (pseudo) Green matrices, which are identical, are best constants of these inequalities.