Hardy transitions on product lattices
Abstract
Boundary conditions can change the dimensional scale of a sharp functional inequality. We determine this change for inverse-square Hardy inequalities on the product lattices ℕk×ℤN−k. Their sharp constants satisfy CN,k∼ N2Ψ(k/N) uniformly for 1≤ k≤ N, where Ψ(α)∼2α at zero. The law gives linear growth for a fixed number of positive coordinates, every intermediate power for suitable sublinear choices, and quadratic growth for a positive proportion. We develop a method that connects exact uncertainty inequalities with Hardy inequalities through logarithmic localization, and combines this transfer with a block decomposition to cover all proportions. The transfer holds on weighted graphs with an explicit geometric error and controls the variational problem over arbitrary functions. A mixture of two Mathieu ground energies determines Ψ. At the fully positive endpoint, the variational problem defining κ=Ψ(1) has a unique nondegenerate minimum and a unique positive energy--moment minimizing profile. We also prove strict comparison with the continuum coefficient in every sector for N≥9, the optimal uniform dimension.
Related
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BibTeX
@misc{lizama2026hardy,
title = {{Hardy transitions on product lattices}},
author = {Carlos Lizama},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02206},
eprint = {2611.02206},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02206},
}
RIS
TY - UNPB
TI - Hardy transitions on product lattices
AU - Carlos Lizama
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02206
UR - https://arxiv.org/abs/2611.02206
AB - Boundary conditions can change the dimensional scale of a sharp functional inequality. We determine this change for inverse-square Hardy inequalities on the product lattices ℕ^k×ℤ^N−k. Their sharp constants satisfy C_N,k∼ N^2Ψ(k/N) uniformly for 1≤ k≤ N, where Ψ(α)∼2α at zero. The law gives linear growth for a fixed number of positive coordinates, every intermediate power for suitable sublinear choices, and quadratic growth for a positive proportion. We develop a method that connects exact uncertainty inequalities with Hardy inequalities through logarithmic localization, and combines this transfer with a block decomposition to cover all proportions. The transfer holds on weighted graphs with an explicit geometric error and controls the variational problem over arbitrary functions. A mixture of two Mathieu ground energies determines Ψ. At the fully positive endpoint, the variational problem defining κ=Ψ(1) has a unique nondegenerate minimum and a unique positive energy--moment minimizing profile. We also prove strict comparison with the continuum coefficient in every sector for N≥9, the optimal uniform dimension.
ER -