Asymptotically sharp Hardy-Rellich inequalities on lattices

2026/07/25 by Xia Huang, Dong Ye
#math.AP

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Abstract

We determine the sharp asymptotic behavior of the optimal constants in the discrete Hardy-Rellich inequalities on the lattice ℤ^d. For every fixed integer m≥ 1, let \mathcal C_m,d be the best constant in the m-th order Hardy-Rellich inequality. We prove that lim_d→∞\frac\mathcal C_m,dd^m=2^m. Our approach combines a Fourier reduction to weighted inequalities with the flat torus and general weighted Hardy-Rellich identities of first and second order. A key novelty is the use of probabilistic concentration estimates, specifically Hoeffding's inequality and entropy methods, to handle estimates involving the anisotropic weight ω^γ (γ≥ 1) where ω(x)=∑_j=1^d (sin(x_j)/(2))² in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.

Citations

• Sharp asymptotics for higher-order Hardy constants on lattices
• Higher order Hardy-Rellich identities
• Optimal discrete Hardy-Rellich-Birman inequalities
• First order Hardy inequalities revisited
• An improved discrete Hardy inequality
• Optimal Hardy inequalities for Schrödinger operators on graphs
• Weighted inequalities and Stein-Weiss potentials
• Best Constants in the Hardy-Rellich Inequalities and Related Improvements
• The Hardy-Rellich Inequality for Polyharmonic Operators

Related

• Sharp asymptotics for higher-order Hardy constants on lattices
• Optimal Hardy weights on the Euclidean lattice
• Discrete functional inequalities on lattice graphs
• Hardy inequalities, Rellich inequalities and local Dirichlet forms
• A critical Hardy-Rellich inequality