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Superharmonicity of the fractional ground state
Abstract
We prove that the positive ground state u of the fractional Laplacian (−Δ)s, s∈ (0,1), on an arbitrary open set Ω satisfies −Δu>λs(Ω)1/su in Ω. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for log u, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of ψ(−Δ) for complete Bernstein functions ψ.
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BibTeX
@misc{nitti2026superharmonicity,
title = {{Superharmonicity of the fractional ground state}},
author = {Nicola De Nitti and Xavier Fernández-Real and Florian Grube and Harsh Prasad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11686},
eprint = {2608.11686},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11686},
}
RIS
TY - UNPB
TI - Superharmonicity of the fractional ground state
AU - Nicola De Nitti
AU - Xavier Fernández-Real
AU - Florian Grube
AU - Harsh Prasad
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.11686
UR - https://arxiv.org/abs/2608.11686
AB - We prove that the positive ground state u of the fractional Laplacian (−Δ)^s, s∈ (0,1), on an arbitrary open set Ω satisfies −Δu>λ_s(Ω)^1/su in Ω. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for log u, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of ψ(−Δ) for complete Bernstein functions ψ.
ER -