格点上渐近尖锐的哈代 - 雷利奇不等式
Asymptotically sharp Hardy-Rellich inequalities on lattices
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中文总结 AI 辅助
研究格点\(\mathbb{Z}^d\)上离散哈代 - 雷利奇不等式最优常数的渐近行为，通过傅里叶约简、哈代 - 雷利奇恒等式及概率集中估计等方法，证明\(\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m\)。
中文摘要 AI 辅助
我们确定了格点\(\mathbb{Z}^d\)上离散哈代 - 雷利奇不等式中最优常数的尖锐渐近行为。对于每个固定整数\(m\geq1\)，设\({\mathcal C}_{m,d}\)为\(m\)阶哈代 - 雷利奇不等式中的最佳常数。我们证明\(\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m\)。我们的方法将傅里叶约简为带权不等式与平坦环面以及一阶和二阶一般带权哈代 - 雷利奇恒等式相结合。一个关键的新颖之处是使用概率集中估计，特别是霍夫丁不等式和熵方法，以维度一致的方式处理涉及各向异性权重\(\omega^\gamma\) \((\gamma\geq1)\)的估计，其中\(\omega(x)=\sum_{j=1}^{d} \Big(\sin\frac{x_j}{2}\Big)^2\)。这些工具在环面上产生渐近尖锐的带权估计，通过迭代可得到格点不等式。
英文摘要
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 $$\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m.$$ 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 $ω^γ$ $(γ\geq 1)$ where $$ω(x)=\sum_{j=1}^{d} \Big(\sin\frac{x_j}{2}\Big)^2$$ in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.
