Let $Y$ be a unit cube in $\mathbb{R}^N$. Sobolev embedding theorem says that in the case $N=4$ one has the following embedding: \begin{equation} H^2(Y)\text{ is embedded continuously into }L^p(Y)\text{ provided }p<\infty. \end{equation} This is a peculiarity of the dimension $N=4$, for other dimensions there exists the largest value of $p$, for which the above embedding holds (namely, $p=2N/(N-4)$ as $N>4$, $p=\infty$ as $N=2,3$).
Question: is it possible to compute (or, at least, to estimate it from below) the norm of the above embedding, i.e. the smallest constant $C_p$ for which $$ \|u\|_{L^p(Y)} < C_p \|u\|_{H^2(Y)}, \forall f\in H^2(Y) $$ holds. Recall, that Y is a unit cube, so quite simple domain.