Asymptotically sharp Hardy-Rellich inequalities on lattices

Abstract

We determine the sharp asymptotic behavior of the optimal constants in the discrete Hardy-Rellich inequalities on the lattice ${\mathbb{Z}}^d$. For every fixed integer $m\ge 1$, let ${\mathcal{C}}_{m,d}$ be the best constant in the $m$-th order Hardy-Rellich inequality. We prove that $$\underset{d\to \infty }{\lim }\frac{{\mathcal{C}}_{m,d}}{d^m}=2^m.$$ 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 $\ddot{I}{‰}^3$ $({\hat{I}}^3\ge 1)$ where $$\ddot{I}‰(x)=\sum _{j=1}^d(\sin \frac{x_j}{2}{)}^2$$ in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.