Asymptotically sharp Hardy-Rellich inequalities on lattices

Technical Summary: Asymptotically Sharp Hardy-Rellich Inequalities on Lattices

Problem Statement
The paper addresses the determination of the sharp asymptotic behavior of the optimal constants, denoted $C_{m,d}$, in discrete Hardy-Rellich inequalities on the integer lattice $\mathbb{Z}^d$ as the dimension $d$ tends to infinity. For a fixed integer order $m \ge 1$, the inequality is defined as:
$$\sum_{n \in \mathbb{Z}^d} |D_m u(n)|^2 \ge C_{m,d} \sum_{n \in \mathbb{Z}^d \setminus \{0\}} \frac{|u(n)|^2}{|n|^{2m}}, \quad \forall u \in E(\mathbb{Z}^d),$$
where $E(\mathbb{Z}^d)$ is the space of compactly supported functions vanishing at the origin, and $D_m$ represents the discrete differential operator of order $m$.

While sharp constants for these inequalities are well-understood in the continuous Euclidean setting $\mathbb{R}^d$ (where they scale as $d^{2m}$), the discrete case presents significant challenges due to the lack of scaling arguments and integration by parts. Previous work by Gupta established that $C_{m,d}$ scales as $d^m$ for large $d$, but the exact asymptotic limit remained an open question. Specifically, it was known that $0 < \liminf_{d\to\infty} C_{m,d}/d^m \le \limsup_{d\to\infty} C_{m,d}/d^m < \infty$. This paper aims to resolve the exact value of this limit.

Methodology
The authors employ a multi-step approach combining Fourier analysis, functional identities, and probabilistic concentration estimates:

1. Fourier Reduction: Following Gupta's approach, the discrete problem on $\mathbb{Z}^d$ is transformed into an equivalent weighted integral inequality on the flat torus $\mathbb{T}^d = [-\pi, \pi]^d$. The discrete sums are converted into integrals involving a specific anisotropic weight function:
   $$\omega(x) = \sum_{j=1}^d \left(\sin \frac{x_j}{2}\right)^2.$$
   The problem reduces to finding the best constant for weighted inequalities of the form:
   $$\int_{\mathbb{T}^d} |D_k \phi|^2 \omega^\gamma dx \ge C_{d,k,\ell,\gamma} \int_{\mathbb{T}^d} |D_\ell \phi|^2 \omega^{\gamma-k+\ell} dx.$$

2. Generalized Hardy-Rellich Identities: To handle the error terms arising from the anisotropic weight $\omega^\gamma$, the authors utilize general first and second-order Hardy-Rellich identities. A key novelty is the introduction of a second-order Rellich-type identity (Lemma 2.1, Eq. 2.2) that couples the Hessian matrix with a vector field via tensor products. This allows for the precise management of the coupling between different spatial directions, which standard Bessel-pair approaches fail to address in this context.

3. Probabilistic Concentration Estimates: The weight $\omega$ is interpreted as an empirical mean of independent random variables. The authors leverage this probabilistic structure to control the degenerate regions of the weight (where $\omega \to 0$) as $d \to \infty$.

   • Hoeffding's Inequality: Used to estimate the measure of the set where the weight deviates significantly from its mean, ensuring that the "bad" regions become negligible in high dimensions.
   • Entropy Methods: The authors employ the logarithmic Sobolev inequality on the torus, noting that the best constant is dimension-independent. This allows for uniform entropy estimates across dimensions, which are crucial for bounding the weighted integrals.

Key Contributions and Results
The primary result of the paper is the determination of the exact asymptotic limit of the optimal constants:

Theorem 1.1: For every fixed integer $m \ge 1$, the sharp constant $C_{m,d}$ satisfies:
$$\lim_{d \to \infty} \frac{C_{m,d}}{d^m} = 2^m.$$

To achieve this, the authors first establish Theorem 1.3, which provides the asymptotic behavior for the general weighted inequalities on the torus. They prove that for non-negative integers $\ell, k$ with $0 < k - \ell \le \gamma$:
$$\lim_{d \to \infty} \frac{C_{d,k,\ell,\gamma}}{d^{k-\ell}} = 2^{\ell-k}.$$

The proof relies on Proposition 1.4, which establishes sharp estimates for the weighted norms involving the gradient and Laplacian. Specifically, they show that for large $d$:

• $\int |\nabla \phi|^2 \omega^\gamma \approx \int \phi^2 \omega^\gamma$ (up to lower order terms).
• $\int |\nabla \phi|^2 \omega^\gamma \approx \frac{d}{2} \int \phi^2 \omega^{\gamma-1}$.
• $\int |\Delta \phi|^2 \omega^\gamma \approx \int |\nabla \phi|^2 \omega^\gamma$.
• $\int |\Delta \phi|^2 \omega^\gamma \approx \frac{d}{2} \int |\nabla \phi|^2 \omega^{\gamma-1}$.

These estimates are derived by combining the new Rellich identities with the probabilistic concentration bounds (Hoeffding and entropy) to handle the anisotropy of $\omega$.

Significance
The paper claims to provide a complete answer to a question left open by Gupta regarding the precise asymptotic behavior of discrete Hardy-Rellich constants. The result highlights a fundamental difference between discrete and continuous settings: while optimal constants in $\mathbb{R}^d$ grow as $d^{2m}$, those on the lattice $\mathbb{Z}^d$ grow as $d^m$. The authors note that their approach, particularly the use of probabilistic concentration estimates to handle anisotropic weights in a dimension-uniform manner, offers a new toolkit for analyzing high-dimensional discrete inequalities.

The paper also acknowledges that after its completion, Gupta independently obtained the same result using a different approach. The authors' contribution lies in the specific combination of Fourier reduction, new second-order identities, and probabilistic methods to derive these sharp estimates.