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arXiv · 2407.02116

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

Ujjal Das·Matthias Keller·Yehuda Pinchover
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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.

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Ujjal Das, Matthias Keller, Yehuda Pinchover. 2024-07-02. The space of Hardy-weights for quasilinear operators on discrete graphs. https://arxiv.org/abs/2407.02116

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