Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs

TL;DR

This paper explores the relationship between Sobolev inequalities, Gaussian heat kernel bounds, and volume doubling on graphs, introducing new conditions and correction functions to understand their equivalence and behavior.

Contribution

It establishes the equivalence of Sobolev inequalities and Gaussian bounds on graphs under new local regularity conditions and introduces variable correction functions for dimensions.

Findings

Equivalence of Sobolev inequalities and Gaussian bounds under certain conditions

Introduction of new local regularity condition for arbitrary measures

Variable correction functions tend to one at infinity

Abstract

We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by imposing comparability of small balls and the vertex degree at their centers. If arbitrary measures are considered, we incorporate a new local regularity condition. Furthermore, new correction functions for the Gaussian, doubling, and Sobolev dimension are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Sobolev dimension can be related to the doubling dimension and the vertex degree growth.