Ultracontractivity and functional inequalities on infinite graphs

TL;DR

This paper establishes the equivalence of ultracontractive bounds, heat kernel estimates, and functional inequalities on infinite graphs, linking geometric conditions with analytic properties of the heat semigroup.

Contribution

It proves the equivalence of various heat kernel bounds and functional inequalities on graphs under volume growth and curvature assumptions, extending classical results to discrete structures.

Findings

Ultracontractive bounds are equivalent to Nash inequalities on graphs.

Under volume growth and nonnegative curvature, multiple inequalities and heat kernel estimates hold.

The results unify geometric and analytic properties of infinite graphs.

Abstract

In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev inequality, Nash inequality, Faber-Krahn inequality, Log-Sobolev inequalities, discrete and continuous-time uniform upper estimate of heat kernel are all true on graph.