Sharp constants for nonlinear discrete p-energies

Determine the exact sharp pure-power Hardy constants for nonlinear discrete p-energies.

Background

The paper studies a quadratic nearest-neighbor Dirichlet energy and establishes its sharp Euclidean inverse-square Hardy coefficient in dimension three. The authors explicitly identify determining the corresponding exact pure-power constants for nonlinear discrete p-energies as unresolved.

References

It remains to determine exact pure-power constants in fixed dimensions $d\geq4$ and for nonlinear discrete $p$-energies.

— The sharp discrete Hardy inequality on $\Z^3$  (2608.25262 - Alpay, 26 Aug 2026) in Remark ‘Further problems’

A natural conjecture is

C(N)=A_N\quad(3\le N\le8), \qquad C(N)<A_N\quad(N\ge9).

The substantive open part of this conjecture is the equality $C(N)=A_N$ for $5\le N\le8$.

— Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine  (2608.30831 - Lizama, 31 Aug 2026) in Section 6, “A spectral consequence and remaining dimensions”

Whether the constant associated with the corresponding weighted Birman inequality is optimal remains a open question.

— Improved Discrete Dual $p$-Hardy and Weighted Discrete $p$- Birman Inequalities  (2609.11890 - Das, 10 Sep 2026) in Section 1, Introduction, paragraph discussing Questions Q(a)–Q(c)