The sharp discrete Hardy inequality on $\Z^3$

Abstract: We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z<sup>3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z<sup>3\to\C$, 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.