Remarks about Hardy inequalities on metric trees

TL;DR

This paper establishes precise growth conditions for metric trees to satisfy Hardy inequalities for the Neumann Laplacian, and demonstrates magnetic field effects on loop graphs.

Contribution

It provides sharp criteria for Hardy inequalities on metric trees and explores magnetic field influences on loop graphs.

Findings

Sharp growth conditions for metric trees to satisfy Hardy inequalities

Magnetic fields induce Hardy inequalities on loop graphs

Results apply to homogeneous metric trees and Aharonov-Bohm magnetic fields

Abstract

We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.