Sharp discrete Hardy constants in low dimensions

For $d\ge 3$, let ${\lambda }_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on ${\mathbb{Z}}^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient $$C_d=\frac{(d-2{)}^2}{4}$$ is sharp in dimensions three and four, while it is not sharp for $d\ge 9$. We prove ${\lambda }_d=C_d$ for the remaining cases $5\le d\le 8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.