PreprintCC BY 4.0arXiv
Sharp discrete Hardy constants in low dimensions
Abstract
For d≥3, let λd denote the sharp constant in the nearest-neighbor Hardy inequality on ℤd with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient Cd=((d−2)2)/4 is sharp in dimensions three and four, while it is not sharp for d≥9. We prove λd=Cd for the remaining cases 5≤ d≤8. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.
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BibTeX
@misc{alpay2026sharp,
title = {{Sharp discrete Hardy constants in low dimensions}},
author = {Natanael Alpay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05421},
eprint = {2610.05421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05421},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp discrete Hardy constants in low dimensions
AU - Natanael Alpay
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05421
UR - https://arxiv.org/abs/2610.05421
AB - For d≥3, let λ_d denote the sharp constant in the nearest-neighbor Hardy inequality on ℤ^d with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient C_d=((d−2)^2)/4 is sharp in dimensions three and four, while it is not sharp for d≥9. We prove λ_d=C_d for the remaining cases 5≤ d≤8. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.
ER -