Is it true that: $u \in C^0(\bar U) \cap C^{\infty} (U) \Rightarrow u \in W^{k-1/p,p}(\partial U)$?

Question

Let $U$ be an open, bounded subset of $\mathbb R^3$ with a $C^2-$boundary. For $k\ge 2$ and $1\le p<\infty$ is it true that:

If $u \in C^0(\bar U) \cap C^{\infty} (U)$ then $u \in W^{k-1/p,p}(\partial U)$?

My Thought: If $u \in C^0(\bar U) \cap C^{\infty} (U)$ then $u \in W^{k,p}(U)$. Also by the trace theorem we have that $W^{k,p}(U)$ is embedded into $W^{k-1/p,p}(\partial U)$ and thus we finished.

Maybe my question is quite trivial or elementary but I have always doubts when it comes to sobolev embeddings. I would appreciate if somebody could confirm this idea or fix it (if possible).

Thanks in advance!

Answer 1

You certainly cannot get $u \in W^{k,p}(U)$ since you do not have any control on the derivatives near the boundary, e.g., $$u(t) = t \, \sin( 1 / t^{42} )$$ belongs to $C(\bar U) \cap C^\infty(U)$ for $U = (0,1)$, but not to $W^{k,p}(U)$ (at least for some large $k$, $p$).

I doubt that you can get the regularity for the trace but I do not have a counterexample.