Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine

@misc{lizama2026sharp,
  title = {{Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine}},
  author = {Carlos Lizama},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30831},
  eprint = {2608.30831},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30831},
}

TY  - UNPB
TI  - Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine
AU  - Carlos Lizama
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30831
UR  - https://arxiv.org/abs/2608.30831
AB  - For N≥3, let C(N) be the optimal constant in the nearest-neighbour Hardy inequality on ℤ^N with u(0)=0, and set A_N=(N−2)^2/4. We prove that the continuum coefficient remains an upper bound, C(N)≤ A_N, in every dimension, and determine the exact values C(3)=A_3=1/4 and C(4)=A_4=1. In higher dimensions we show C(N)<A_N for N=9,10 and obtain the explicit bound C(N)≤ 3N−√(N^2+8N−8)<2N (N≥3), which lies below A_N from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh--Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.
ER  -