Why is sobolev space $H^m$ embedded in the $C^k$ space?
Solution 1:
The key is that $u\in H^m$ for ALL $m$. For any $k$, let $m$ be large so that
$$m-[\frac{n}{2}]-1\ge k$$
Then since $u\in H^m$, $u\in C^k$ by the Sobolev embedding.
The key is that $u\in H^m$ for ALL $m$. For any $k$, let $m$ be large so that
$$m-[\frac{n}{2}]-1\ge k$$
Then since $u\in H^m$, $u\in C^k$ by the Sobolev embedding.