AI 中文总结
该研究确定了$\Z^3$上带欧几里得反平方权重的最近邻Hardy不等式的精确常数，通过显式倒易边场、逐边配方法及顶点权重凹性论证，证明系数$1/4$的最优性且非零有限支撑函数无法取等。
AI 中文摘要
我们确定了带欧几里得反平方权重的$\Z^3$上最近邻Hardy不等式的精确常数。对于每个有限支撑函数$u:\Z^3\to\C$，我们证明：\\[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \\] 系数$1/4$是精确的，且非零有限支撑函数无法达到等号。证明过程用到了显式倒易边场、逐边配方法，以及相关顶点权重的凹性论证。
英文摘要
We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient $1/4$ is sharp, and equality is not attained by a nonzero finitely supported function. The proof uses an explicit reciprocal edge field and an edgewise completion of squares, together with a concavity argument for the associated vertex weight.
