Let $\Omega$ be a domain of $R^n, \; n\ge 1.$
I am wondering about the existence of an open subset $\omega$ of $\Omega$ such that $ \|y\|_{L^\infty(\omega)} \le C \|y\|_{H^2(\Omega)},\; \forall y\in H^2(\Omega) $
for some constant $C>0?$
In the case $n=1,$ the response is yes with $\omega=\Omega.$ For $n\ge 2$, the inequality is not true for $\omega=\Omega.$
Thanks!