Sobolev spaces on graphs

Abstract

The present paper is devoted to discrete analogues of Sobolev spaces of smooth functions. The discrete analogues that we consider are spaces of functions on vertex sets of graphs. Such spaces have applications in Graph Theory, Metric Geometry and Convex Geometry. We present known and prove some new results on Banach-space-theoretical properties of such spaces.
Keywords: Sobolev space on a graph; isoperimetric constant of a graph; Cheeger constant; Banach-Mazur distance

Quaestiones Mathematicae 28(2005), 501–523.