From Hardy to Rellich inequalities on graphs

TL;DR

This paper demonstrates how Rellich inequalities can be derived from Hardy inequalities on infinite graphs, providing bounds involving Laplacians and weighted norms, with extensions to Schrödinger operators.

Contribution

It introduces a method to obtain Rellich inequalities from Hardy inequalities on graphs, including new weighted bounds and extensions to Schrödinger operators.

Findings

Rellich inequalities derived from Hardy inequalities on graphs

Weighted bounds involving Hardy weights and eikonal functions

Extension of results from Laplacians to Schrödinger operators

Abstract

We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schr\"odinger operators afterwards.