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Strictly positive optimal Hardy weights on manifolds and graphs via Green potential
Abstract
Consider a subcritical functional Q associated with a p-Schrödinger operator defined on a domain Ω in noncompact Riemannian manifold. Employing the supersolution construction, we obtain a family of strictly positive critical Hardy weights for Q. The construction is based on the Green potential, with a strictly positive charge, which is assumed to vanish at infinity. An interesting new feature of this family, depending on a parameter 0≤ ε≤ 1, is the transition from positive-criticality to null-criticality at ε=0. Moreover, we show that the best Hardy constant for this family is given by c=[(p−1)/p]p−1, and that for ε=0 the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.
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BibTeX
@misc{das2026strictly,
title = {{Strictly positive optimal Hardy weights on manifolds and graphs via Green potential}},
author = {Ujjal Das and Matthias Keller and Yehuda Pinchover},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02365},
eprint = {2610.02365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02365},
}
RIS
TY - UNPB
TI - Strictly positive optimal Hardy weights on manifolds and graphs via Green potential
AU - Ujjal Das
AU - Matthias Keller
AU - Yehuda Pinchover
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02365
UR - https://arxiv.org/abs/2610.02365
AB - Consider a subcritical functional Q associated with a p-Schrödinger operator defined on a domain Ω in noncompact Riemannian manifold. Employing the supersolution construction, we obtain a family of strictly positive critical Hardy weights for Q. The construction is based on the Green potential, with a strictly positive charge, which is assumed to vanish at infinity. An interesting new feature of this family, depending on a parameter 0≤ ε≤ 1, is the transition from positive-criticality to null-criticality at ε=0. Moreover, we show that the best Hardy constant for this family is given by c=[(p−1)/p]^p−1, and that for ε=0 the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.
ER -