The discrete Hardy inequality on the integer lattice 
bounds a weighted sum of squared function values by a sum of squared nearest-neighbor
differences. For 
, let 
be the largest constant such that
 |
for every finitely supported complex function 
with 
,
where 
are the standard basis vectors and 
is the Euclidean norm.
Alpay (2026) proved the sharp value 
for dimensions 5, 6, 7, and 8. Together with
the previously known cases 3 and 4, this establishes equality with the continuous
Hardy constant in dimensions 3 through 8. For 
the sharp discrete constant is strictly smaller (Alpay
2026).
