Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line
Abstract
For positive powers of the discrete Dirichlet Laplacian (−ΔD)σ, non-trivial Hardy weights exist if and only if 0<σ<3/2. For 0<σ≤ 1, an explicit optimal Hardy weight for (−ΔD)σ was obtained by Das and de la Fuente-Fernández. Their ground state argument does not apply when 1<σ<3/2, because the off-diagonal matrix entries of (−ΔD)σ are no longer all non-positive. Nevertheless, we show that their result is true for the entire subcriticality range 0<σ<3/2. For 1<σ<3/2, our argument combines the factorisation (−ΔD)σ=∇D*(−ΔN)σ−1∇D, where ∇D and −ΔN are the discrete Dirichlet gradient and the discrete Neumann Laplacian, respectively, with a suitable two-term ground state representation. In addition, we use these arguments to derive another explicit family of optimal Hardy weights for any σ∈(0,3/2).
Related
- On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
- Optimal design problems for the fractional g-Laplacian at prescribed energy level
- Optimal Lipschitz Thresholds for Weighted Dirichlet Problems, with an Explicit Bessel-Zero Bound for Power Weights
- Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
- The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations
BibTeX
@misc{stampach2026optimal,
title = {{Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line}},
author = {František Štampach and Jakub Waclawek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31171},
eprint = {2610.31171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31171},
}
RIS
TY - UNPB
TI - Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line
AU - František Štampach
AU - Jakub Waclawek
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31171
UR - https://arxiv.org/abs/2610.31171
AB - For positive powers of the discrete Dirichlet Laplacian (−Δ_D)^σ, non-trivial Hardy weights exist if and only if 0<σ<3/2. For 0<σ≤ 1, an explicit optimal Hardy weight for (−Δ_D)^σ was obtained by Das and de la Fuente-Fernández. Their ground state argument does not apply when 1<σ<3/2, because the off-diagonal matrix entries of (−Δ_D)^σ are no longer all non-positive. Nevertheless, we show that their result is true for the entire subcriticality range 0<σ<3/2. For 1<σ<3/2, our argument combines the factorisation (−Δ_D)^σ=∇_D^*(−Δ_N)^σ−1∇_D, where ∇_D and −Δ_N are the discrete Dirichlet gradient and the discrete Neumann Laplacian, respectively, with a suitable two-term ground state representation. In addition, we use these arguments to derive another explicit family of optimal Hardy weights for any σ∈(0,3/2).
ER -