Heat kernel estimates and related inequalities on metric graphs

TL;DR

This paper investigates heat kernel estimates and fundamental inequalities on metric graphs with Kirchhoff boundary conditions, employing Dirichlet form techniques to establish key properties like the Harnack inequality.

Contribution

It introduces new heat kernel estimates and inequalities for metric graphs, expanding the understanding of their analytical and geometric properties.

Findings

Established volume doubling and Poincaré inequalities

Proved a parabolic Harnack inequality

Derived Sobolev and Nash inequalities

Abstract

We consider metric graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincar\'e inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from the theory of strongly local Dirichlet forms. Along our way we show Sobolev and Nash type inequalities and related heat kernel estimates.