Can somebody help me with this doubt?
Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality?
$\vert v(a) \vert \leq C \| v \|_p \quad \forall a \in A, \; \forall v \in H^p(\Omega).$
($\| \cdot \|_p$ denotes the usual Sobolev norm.)
Regards
The inequality holds if $H^p(\Omega) \subset C^\gamma(\Omega)$ for some $\gamma > 0$ (the $C^\gamma$-norm bounds all $v(a)$ since $A$ is bounded). That is the case for $p$ large enough (depending on the space dimension).
See http://en.wikipedia.org/wiki/Sobolev_embedding (Morreys inequality) or https://mathoverflow.net/questions/25470/when-is-sobolev-space-a-subset-of-the-continuous-functions.
The rules given in the mathoverflow post are sharp, I believe, but I don't know any counterexamples.