Inequalities on Infinite Networks

抄録・要旨

Inequalities on networks have played important roles in the theory of netwoks. We study several famous inequalities on networks such as Wirtinger's inequality, Hardy's inequality, Poincare-Sobolev's inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigenvalue of weighted discrete Laplacian. We discuss some relations between these inequalities and the potential-theorerteic magnitude of the ideal boundary of an infinite network.