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arXiv · 2212.07728

On the Optimality and Decay of $p$-Hardy Weights on Graphs

Florian Fischer
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Abstract

We construct optimal Hardy weights to subcritical energy functionals $h$ associated with quasilinear Schr\"odinger operators on locally finite graphs. Here, optimality means that the weight $w$ is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional $h-w$ is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle and a Rellich-type inequality.

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Florian Fischer. 2022-12-15. On the Optimality and Decay of $p$-Hardy Weights on Graphs. https://arxiv.org/abs/2212.07728

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