The space of Hardy-weights for quasilinear operators on discrete graphs

TL;DR

This paper characterizes Hardy weights for p-Schrödinger operators on weighted graphs using a generalized capacity, providing new insights into their structure and conditions for minimizer existence.

Contribution

It introduces a Maz'ya-type characterization of Hardy weights via a generalized capacity and demonstrates compatibility of the energy functional with normal contractions.

Findings

Characterization of Hardy weights via generalized capacity

Necessary integrability criterion for Hardy weights

Existence conditions for minimizers in Hardy inequalities

Abstract

We study Hardy inequalities for $p$-Schr\"odinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for $ p $-Schr\"odinger operators via a generalized capacity. The novel ingredient in the proof is the demonstration that the simplified energy of the $ p $-Schr\"odinger energy functional is compatible with certain normal contractions. As a consequence, we obtain a necessary integrability criterion for Hardy weights. Finally, using some tools of criticality theory, we investigate the existence of minimizers in the Hardy inequalities and discuss relations to Cheeger type estimates.