Surjectivity of the trace operator in Sobolev spaces

At least in the best circumstances, it is easy to prove that the trace/restriction map loses ${\ell\over p}+\epsilon$ in "Sobolev units" (for arbitrarily small $\epsilon>0$) where codimension is $\ell$. And an easy extension of Sobolev imbedding theorems shows that (for example, for $L^2$ Sobolev spaces so I don't mess up the indexing shift...) $H^{k+{n\over 2}+\epsilon}\subset C^{k,\epsilon}$, the latter being $C^k$ functions with an $\epsilon$ Lipschitz condition.

Thus, with $p=2$ for example, $H^1(\Omega)$ maps to $C^{0,{1\over 2}-\epsilon}(\partial \Omega)$ for every $0<\epsilon<{1\over 2}$.