2026/07/16 by Shubham Gupta · 1 citation
#math.FA #math.AP
We study the optimal constants in higher-order Hardy inequalities on the lattice ℤd. For each fixed ℓ ∈ ℕ, we prove that the optimal constant Copt^ℓ(d) in ∑n ∈ ℤd |Δℓ/2u(n)|2 ≥ Copt^ℓ(d)∑n ∈ ℤd \frac|u(n)|2|n|2ℓ. satisfies limd→∞(Copt^ℓ(d))/(d^ℓ) =2^ℓ. The proof is based on a Fourier reduction to a family of singular Hardy inequalities on the flat torus, involving the weight ω(x)-2ℓ, ω(x)2=∑j=1dsin2((xj)/(2)), and zero average condition on admissible functions. We establish these torus inequalities by combining a ground state representation formula with a weighted integrated Bochner identity in an iterative scheme. The method yields explicit constants, defined recursively in the order ℓ, and requires only the classical unweighted Poincaré inequality on the torus. The appearance of the limiting constant 2^ℓ is particularly striking, as it suggests that, in the high dimensional regime, the optimizers are localized near the unit sphere \n∈ℤd:|n|=1\ in ℤd.