An extension of a critical Hardy--Rellich inequality: explicit constants and the sharp weight range

We revisit the critical Hardy–Rellich inequalities recently established by Castro in “A critical Hardy–Rellich inequality (preprint, arXiv:2511.16537, 2025)”. Using the classical one-dimensional weighted Hardy inequality in Emden–Fowler variables, we prove that the weighted inequality holds exactly for $a\mathrm{\ne }N$, with $a=N$ as the unique critical value. We derive explicit constants for $N\ge 2$, obtain the sharp constant in dimension one, and extend the $\mathrm{\Delta }u$ formulation to the Muckenhoupt range $-N<a<N(N-1)$, $a\mathrm{\ne }N$. This provides a partial solution to an open problem raised in Castro’s work.